Transcript powerpoint file

Quantum One: Lecture 27
1
2
The Harmonic Oscillator
3
In the last lecture, we completed our formal development of the postulates of
quantum mechanics as they apply to arbitrary quantum mechanical systems, by
considering the evolution of the state vector for conservative systems.
We saw how, for a conservative system, equations of motion for the time
dependent expansion coefficients representing the state vector in the energy
representation, i.e., in the basis of eigenstates of the Hamiltonian, decouple.
Working in this representation, we obtained explicit expressions for the evolution
operator, as a ket-bra expansion, and as a simple exponential operator valued
function of the Hamiltonian operator itself.
Finally, we derived an integral equation for the evolution operator, which we
formally solved by iteration, obtaining an expansion in powers of the Hamiltonian.
4
The Harmonic Oscillator - We now consider an extended example which allows
us to apply the theoretical apparatus constructed up to this point.
The example we choose, that of a particle subjected to a linear restoring force the so-called harmonic oscillator - is important for several reasons.
First, it is one of the relatively small number of quantum mechanical problems
that can be solved exactly and completely.
In addition, the problem provides a basis for our understanding of many
important physical problems, including molecular vibrations, the vibrational
excitations of solids (i.e., phonons), and the quantization of the electromagnetic
field (photons).
In a real sense, the one-dimensional harmonic oscillator is the main building block
of a great deal of quantum field theory
5
The Harmonic Oscillator - We now consider an extended example which allows
us to apply the theoretical apparatus constructed up to this point.
The example we choose, that of a particle subjected to a linear restoring force the so-called harmonic oscillator - is important for several reasons.
First, it is one of the relatively small number of quantum mechanical problems
that can be solved exactly and completely.
In addition, the problem provides a basis for our understanding of many
important physical problems, including molecular vibrations, the vibrational
excitations of solids (i.e., phonons), and the quantization of the electromagnetic
field (photons).
In a real sense, the one-dimensional harmonic oscillator is the main building block
of a great deal of quantum field theory
6
The Harmonic Oscillator - We now consider an extended example which allows
us to apply the theoretical apparatus constructed up to this point.
The example we choose, that of a particle subjected to a linear restoring force the so-called harmonic oscillator - is important for several reasons.
First, it is one of the relatively small number of quantum mechanical problems
that can be solved exactly and completely.
In addition, the problem provides a basis for our understanding of many
important physical problems, including molecular vibrations, the vibrational
excitations of solids (i.e., phonons), and the quantization of the electromagnetic
field (photons).
In a real sense, the one-dimensional harmonic oscillator is the main building block
of a great deal of quantum field theory
7
The Harmonic Oscillator - We now consider an extended example which allows
us to apply the theoretical apparatus constructed up to this point.
The example we choose, that of a particle subjected to a linear restoring force the so-called harmonic oscillator - is important for several reasons.
First, it is one of the relatively small number of quantum mechanical problems
that can be solved exactly and completely.
In addition, the problem provides a basis for our understanding of many
important physical problems, including molecular vibrations, the vibrational
excitations of solids (i.e., phonons), and the quantization of the electromagnetic
field (photons).
In a real sense, the one-dimensional harmonic oscillator is the main building block
of a great deal of quantum field theory
8
The Harmonic Oscillator - We now consider an extended example which allows
us to apply the theoretical apparatus constructed up to this point.
The example we choose, that of a particle subjected to a linear restoring force the so-called harmonic oscillator - is important for several reasons.
First, it is one of the relatively small number of quantum mechanical problems
that can be solved exactly and completely.
In addition, the problem provides a basis for our understanding of many
important physical problems, including molecular vibrations, the vibrational
excitations of solids (i.e., phonons), and the quantization of the electromagnetic
field (photons).
In a real sense, the one-dimensional harmonic oscillator is the main building block
of a great deal of quantum field theory
9
The Harmonic Oscillator – Statement of the Problem
We consider a particle of mass 𝑚 subject to a linear restoring force 𝐹 = −𝑘𝑥,
corresponding to the quadratic potential
In the Hamiltonian description of classical mechanics, the system is described by
the dynamical variables {𝑥, 𝑝}, and the evolution is governed by the Hamiltonian
Hamilton's equations of motion for the system are
10
The Harmonic Oscillator – Statement of the Problem
We consider a particle of mass 𝑚 subject to a linear restoring force 𝐹 = −𝑘𝑥,
corresponding to the quadratic potential
In the Hamiltonian description of classical mechanics, the system is described by
the dynamical variables {𝑥, 𝑝}, and the evolution is governed by the Hamiltonian
Hamilton's equations of motion for the system are
11
The Harmonic Oscillator – Statement of the Problem
We consider a particle of mass 𝑚 subject to a linear restoring force 𝐹 = −𝑘𝑥,
corresponding to the quadratic potential
In the Hamiltonian description of classical mechanics, the system is described by
the dynamical variables {𝑥, 𝑝}, and the evolution is governed by the Hamiltonian
Hamilton's equations of motion for the system are
12
The Harmonic Oscillator – Statement of the Problem
Upon taking a second derivative, these are equivalent to the familiar Newtonian
equations
𝑥 + 𝜔²𝑥 = 0
𝑝 + 𝜔²𝑝 = 0
whose solutions lead to the familiar oscillatory behavior
𝑥(𝑡) = 𝐴𝑠𝑖𝑛(𝜔𝑡 + 𝛿)
𝑝(𝑡) = 𝐴𝑚𝜔cos(𝜔𝑡 + 𝛿).
In passing from a classical treatment to a quantum mechanical one, the
dynamical variables are replaced by operators
𝑥→𝑋
𝑝 → 𝑃 = ℏ𝐾
which obey the canonical commutation relations
[𝑋, 𝑃] = 𝑖ℏ.
13
The Harmonic Oscillator – Statement of the Problem
Upon taking a second derivative, these are equivalent to the familiar Newtonian
equations
𝑥 + 𝜔²𝑥 = 0
𝑝 + 𝜔²𝑝 = 0
whose solutions lead to the familiar oscillatory behavior
𝑥 𝑡 = 𝐴 sin(𝜔𝑡 + 𝛿)
𝑝 𝑡 = 𝐴𝑚𝜔 cos(𝜔𝑡 + 𝛿).
In passing from a classical treatment to a quantum mechanical one, the
dynamical variables are replaced by operators
𝑥→𝑋
𝑝 → 𝑃 = ℏ𝐾
which obey the canonical commutation relations
[𝑋, 𝑃] = 𝑖ℏ.
14
The Harmonic Oscillator – Statement of the Problem
Upon taking a second derivative, these are equivalent to the familiar Newtonian
equations
𝑥 + 𝜔²𝑥 = 0
𝑝 + 𝜔²𝑝 = 0
whose solutions lead to the familiar oscillatory behavior
𝑥 𝑡 = 𝐴 sin(𝜔𝑡 + 𝛿)
𝑝 𝑡 = 𝐴𝑚𝜔 cos(𝜔𝑡 + 𝛿).
In passing from a classical treatment to a quantum mechanical one, the
dynamical variables are replaced by operators
𝑥→𝑋
𝑝 → 𝑃 = ℏ𝐾
which obey the canonical commutation relations
[𝑋, 𝑃] = 𝑖ℏ.
15
The Harmonic Oscillator – Statement of the Problem
Upon taking a second derivative, these are equivalent to the familiar Newtonian
equations
𝑥 + 𝜔²𝑥 = 0
𝑝 + 𝜔²𝑝 = 0
whose solutions lead to the familiar oscillatory behavior
𝑥 𝑡 = 𝐴 sin(𝜔𝑡 + 𝛿)
𝑝 𝑡 = 𝐴𝑚𝜔 cos(𝜔𝑡 + 𝛿).
In passing from a classical treatment to a quantum mechanical one, the
dynamical variables are replaced by operators
𝑥→𝑋
𝑝 → 𝑃 = ℏ𝐾
which obey the canonical commutation relations
[𝑋, 𝑃] = 𝑖ℏ.
16
The Harmonic Oscillator – Evolution of the quantum mechanical system is
governed by the associated Hamiltonian operator
Since the system is conservative (𝜕𝐻/𝜕𝑡 = 0), this evolution is best considered in
the basis of the eigenstates
of the Hamiltonian, which are assumed to span
the space of a single particle moving in one-dimension, and which obey the
energy eigenvalue equation
As with any eigenvalue problem, we need an initial representation in which to
work.
17
The Harmonic Oscillator – Evolution of the quantum mechanical system is
governed by the associated Hamiltonian operator
Since the system is conservative (𝜕𝐻/𝜕𝑡 = 0), this evolution is best considered in
the basis of the eigenstates
of the Hamiltonian, which are assumed to span
the space of a single particle moving in one-dimension, and which obey the
energy eigenvalue equation
As with any eigenvalue problem, we need an initial representation in which to
work.
18
The Harmonic Oscillator – Evolution of the quantum mechanical system is
governed by the associated Hamiltonian operator
Since the system is conservative (𝜕𝐻/𝜕𝑡 = 0), this evolution is best considered in
the basis of the eigenstates
of the Hamiltonian, which are assumed to span
the space of a single particle moving in one-dimension, and which obey the
energy eigenvalue equation
As with any eigenvalue problem, we need an initial representation in which to
work.
19
The Harmonic Oscillator - In the |𝑥〉 representation, associated with the
eigenstates of the position operator 𝑋, this becomes a differential equation
for the eigenfunctions
.
The notation that we have introduced suggests a discrete spectrum, and, indeed
it can be anticipated that all of the eigenstates of the harmonic potential must be
bound states.
This follows from the observation that the potential energy of the oscillator
becomes infinite as |𝑥| → ∞.
Thus, the wave function must go to zero at large distances from the origin in
order for the energy of the system to remain finite. Thus,
as |𝑥| →
∞, and acceptable solutions correspond to bound, square-normalizable states.
20
The Harmonic Oscillator - In the |𝑥〉 representation, associated with the
eigenstates of the position operator 𝑋, this becomes a differential equation
for the eigenfunctions
.
The notation that we have introduced suggests a discrete spectrum, and, indeed
it can be anticipated that all of the eigenstates of the harmonic potential must be
bound states.
This follows from the observation that the potential energy of the oscillator
becomes infinite as |𝑥| → ∞.
Thus, the wave function must go to zero at large distances from the origin in
order for the energy of the system to remain finite. Thus,
as |𝑥| →
∞, and acceptable solutions correspond to bound, square-normalizable states.
21
The Harmonic Oscillator - In the |𝑥〉 representation, associated with the
eigenstates of the position operator 𝑋, this becomes a differential equation
for the eigenfunctions
.
The notation that we have introduced suggests a discrete spectrum, and, indeed
it can be anticipated that all of the eigenstates of the harmonic potential must be
bound states.
This follows from the observation that the potential energy of the oscillator
becomes infinite as |𝑥| → ∞.
Thus, the wave function must go to zero at large distances from the origin in
order for the energy of the system to remain finite. Thus,
as |𝑥| →
∞, and acceptable solutions correspond to bound, square-normalizable states.
22
The Harmonic Oscillator - In the |𝑥〉 representation, associated with the
eigenstates of the position operator 𝑋, this becomes a differential equation
for the eigenfunctions
.
The notation that we have introduced suggests a discrete spectrum, and, indeed
it can be anticipated that all of the eigenstates of the harmonic potential must be
bound states.
This follows from the observation that the potential energy of the oscillator
becomes infinite as |𝑥| → ∞.
Thus, the wave function must go to zero at large distances from the origin in
order for the energy of the system to remain finite. Thus,
as |𝑥| →
∞, and acceptable solutions correspond to bound, square-normalizable states.
23
The Harmonic Oscillator - In the |𝑥〉 representation, associated with the
eigenstates of the position operator 𝑋, this becomes a differential equation
for the eigenfunctions
.
The notation that we have introduced suggests a discrete spectrum, and, indeed
it can be anticipated that all of the eigenstates of the harmonic potential must be
bound states.
This follows from the observation that the potential energy of the oscillator
becomes infinite as |𝑥| → ∞.
Thus, the wave function must go to zero at large distances from the origin in
order for the energy of the system to remain finite. Thus,
as |𝑥| →
∞, and acceptable solutions correspond to bound, square-normalizable states.
24
The Harmonic Oscillator - Of course, it is also possible to solve the eigenvalue
equation in the wave vector or momentum representation.
Indeed, in the |𝑘〉 representation, the eigenvalue equation for the harmonic
oscillator is also a 2nd order differential equation:
due to the fact that 𝑥 → 𝑖𝑑/𝑑𝑘 acts as a differential operator in that
representation.
Again, this eigenvalue equation is to be solved under the requirement that the
solution vanish as |𝑘| → ∞, so that the energy of the system (in this case the
kinetic energy) be finite.
25
The Harmonic Oscillator - Of course, it is also possible to solve the eigenvalue
equation in the wave vector or momentum representation.
Indeed, in the |𝑘〉 representation, the eigenvalue equation for the harmonic
oscillator is also a 2nd order differential equation:
due to the fact that 𝑥 → 𝑖𝑑/𝑑𝑘 acts as a differential operator in that
representation.
Again, this eigenvalue equation is to be solved under the requirement that the
solution vanish as |𝑘| → ∞, so that the energy of the system (in this case the
kinetic energy) be finite.
26
The Harmonic Oscillator - Of course, it is also possible to solve the eigenvalue
equation in the wave vector or momentum representation.
Indeed, in the |𝑘〉 representation, the eigenvalue equation for the harmonic
oscillator is also a 2nd order differential equation:
due to the fact that 𝑥 → 𝑖𝑑/𝑑𝑘 acts as a differential operator in that
representation.
Again, this eigenvalue equation is to be solved under the requirement that the
solution vanish as |𝑘| → ∞, so that the energy of the system (in this case the
kinetic energy) be finite.
27
The Harmonic Oscillator - A traditional approach commonly taken to solve
either of these equations is the so-called power series method, the basic steps of
which we enumerate for the spatial eigenfunctions below:
1. Determine for large x that the solution has the asymptotic form
where 𝛼 = 𝑚𝜔/2ℏ, and 𝐴(𝑥) is slowly varying in 𝑥.
2. Assume a power series solution of the form
to describe the slowly-varying function 𝐴(𝑥),
obtaining a recursion relation for the coefficients
,
.
3. Show that the series produces a solution that diverges as
for large 𝑥,
unless it terminates. Deduce that physically acceptable solutions must terminate,
so 𝐴(𝑥) is a polynomial in 𝑥.
4. Deduce the values of energy for which the series terminates, thereby solving
the eigenvalue problem.
28
The Harmonic Oscillator - A traditional approach commonly taken to solve
either of these equations is the so-called power series method, the basic steps of
which we enumerate for the spatial eigenfunctions below:
1. Determine for large 𝑥 that the solution has the asymptotic form
where 𝛼 = 𝑚𝜔/2ℏ, and 𝐴(𝑥) is slowly varying in 𝑥.
2. Assume a power series solution of the form
to describe the slowly-varying function 𝐴(𝑥),
obtaining a recursion relation for the coefficients
,
.
3. Show that the series produces a solution that diverges as
for large 𝑥,
unless it terminates. Deduce that physically acceptable solutions must terminate,
so 𝐴(𝑥) is a polynomial in 𝑥.
4. Deduce the values of energy for which the series terminates, thereby solving
the eigenvalue problem.
29
The Harmonic Oscillator - A traditional approach commonly taken to solve
either of these equations is the so-called power series method, the basic steps of
which we enumerate for the spatial eigenfunctions below:
1. Determine for large 𝑥 that the solution has the asymptotic form
where 𝛼 = 𝑚𝜔/2ℏ, and 𝐴(𝑥) is slowly varying in 𝑥.
2. Assume a power series solution of the form
to describe the slowly-varying function 𝐴(𝑥),
obtaining a recursion relation for the coefficients
,
.
3. Show that the series produces a solution that diverges as
for large 𝑥,
unless it terminates. Deduce that physically acceptable solutions must terminate,
so that 𝐴(𝑥) is a polynomial in 𝑥.
4. Deduce the values of energy for which the series terminates, thereby solving
the eigenvalue problem.
30
The Harmonic Oscillator - A traditional approach commonly taken to solve
either of these equations is the so-called power series method, the basic steps of
which we enumerate for the spatial eigenfunctions below:
1. Determine for large 𝑥 that the solution has the asymptotic form
where 𝛼 = 𝑚𝜔/2ℏ, and 𝐴(𝑥) is slowly varying in 𝑥.
2. Assume a power series solution of the form
to describe the slowly-varying function 𝐴(𝑥),
obtaining a recursion relation for the coefficients
,
.
3. Show that the series produces a solution that diverges as
for large 𝑥,
unless it terminates. Deduce that physically acceptable solutions must terminate,
so that 𝐴(𝑥) is a polynomial in 𝑥.
4. Deduce the values of energy for which the series terminates, thereby solving
the eigenvalue problem.
31
Algebraic Approach to the Quantum Harmonic Oscillator
In what follows we take a different approach, due to Dirac, that allows us
ultimately to obtain all the eigenfunctions from the solution to a simple firstorder differential equation.
This algebraic method uses the fundamental commutation relations to directly
deduce the spectrum and degeneracy of the harmonic oscillator Hamiltonian.
To facilitate our study we begin by introducing some simplifying notation.
We observe first that the classical harmonic oscillator possesses a natural
frequency 𝜔.
Quantum mechanically this implies the existence of a natural energy scale 𝜀₀ =
ℏ𝜔.
Thus, the Hamiltonian, which itself has units of energy, can be written in terms of
this energy in the form . . .
32
Algebraic Approach to the Quantum Harmonic Oscillator
In what follows we take a different approach, due to Dirac, that allows us
ultimately to obtain all the eigenfunctions from the solution to a simple firstorder differential equation.
This algebraic method uses the fundamental commutation relations to directly
deduce the spectrum and degeneracy of the harmonic oscillator Hamiltonian.
To facilitate our study we begin by introducing some simplifying notation.
We observe first that the classical harmonic oscillator possesses a natural
frequency 𝜔.
Quantum mechanically this implies the existence of a natural energy scale 𝜀₀ =
ℏ𝜔.
Thus, the Hamiltonian, which itself has units of energy, can be written in terms of
this energy in the form . . .
33
Algebraic Approach to the Quantum Harmonic Oscillator
In what follows we take a different approach, due to Dirac, that allows us
ultimately to obtain all the eigenfunctions from the solution to a simple firstorder differential equation.
This algebraic method uses the fundamental commutation relations to directly
deduce the spectrum and degeneracy of the harmonic oscillator Hamiltonian.
To facilitate our study we begin by introducing some simplifying notation.
We observe first that the classical harmonic oscillator possesses a natural
frequency 𝜔.
Quantum mechanically this implies the existence of a natural energy scale 𝜀₀ =
ℏ𝜔.
Thus, the Hamiltonian, which itself has units of energy, can be written in terms of
this energy in the form . . .
34
Algebraic Approach to the Quantum Harmonic Oscillator
In what follows we take a different approach, due to Dirac, that allows us
ultimately to obtain all the eigenfunctions from the solution to a simple firstorder differential equation.
This algebraic method uses the fundamental commutation relations to directly
deduce the spectrum and degeneracy of the harmonic oscillator Hamiltonian.
To facilitate our study we begin by introducing some simplifying notation.
We observe first that the classical harmonic oscillator possesses a natural
frequency 𝜔.
Quantum mechanically this implies the existence of a natural energy scale 𝜀₀ =
ℏ𝜔.
Thus, the Hamiltonian, which itself has units of energy, can be written in terms of
this energy in the form . . .
35
Algebraic Approach to the Quantum Harmonic Oscillator
In what follows we take a different approach, due to Dirac, that allows us
ultimately to obtain all the eigenfunctions from the solution to a simple firstorder differential equation.
This algebraic method uses the fundamental commutation relations to directly
deduce the spectrum and degeneracy of the harmonic oscillator Hamiltonian.
To facilitate our study we begin by introducing some simplifying notation.
We observe first that the classical harmonic oscillator possesses a natural
frequency 𝜔.
Quantum mechanically this implies the existence of a natural energy scale 𝜀₀ =
ℏ𝜔.
Thus, the Hamiltonian, which itself has units of energy, can be written in terms of
this energy in the form . . .
36
Algebraic Approach to the Quantum Harmonic Oscillator
In what follows we take a different approach, due to Dirac, that allows us
ultimately to obtain all the eigenfunctions from the solution to a simple firstorder differential equation.
This algebraic method uses the fundamental commutation relations to directly
deduce the spectrum and degeneracy of the harmonic oscillator Hamiltonian.
To facilitate our study we begin by introducing some simplifying notation.
We observe first that the classical harmonic oscillator possesses a natural
frequency 𝜔.
Quantum mechanically this implies the existence of a natural energy scale 𝜀₀ =
ℏ𝜔.
Thus, the Hamiltonian, which itself has units of energy, can be written in terms of
this energy in the form . . .
37
Algebraic Approach to the Quantum Harmonic Oscillator
or more simply
in which
represent dimensionless momentum and position operators, respectively.
38
Algebraic Approach to the Quantum Harmonic Oscillator
or more simply
in which
represent dimensionless momentum and position operators, respectively.
39
Algebraic Approach to the Quantum Harmonic Oscillator
or more simply
in which
represent dimensionless momentum and position operators, respectively.
40
Algebraic Approach to the Quantum Harmonic Oscillator
or more simply
in which
represent dimensionless momentum and position operators, respectively.
(We are relaxing our convention of representing operators by capital letters.)
41
Algebraic Approach to the Quantum Harmonic Oscillator
or more simply
in which
represent dimensionless momentum and position operators, respectively.
(We are relaxing our convention of representing operators by capital letters.)
42
Algebraic Approach to the Quantum Harmonic Oscillator
It is readily verified that these new operators obey a dimensionless form
of the canonical commutation relations, and apart from a slightly different
normalization, the eigenstates {|𝑞〉} and {|𝑝〉} of these operators are essentially
those of their dimensionally correct counterparts {|𝑥〉} and {|𝑘〉}.
There is a representation associated with each set of states, so that
in terms of which we can expand an arbitrary state of the system.
43
Algebraic Approach to the Quantum Harmonic Oscillator
It is readily verified that these new operators obey a dimensionless form
of the canonical commutation relations, and apart from a slightly different
normalization, the eigenstates {|𝑞〉} and {|𝑝〉} of these operators are essentially
those of their dimensionally correct counterparts {|𝑥〉} and {|𝑘〉}.
There is a representation associated with each set of states, so that
in terms of which we can expand an arbitrary state of the system.
44
Algebraic Approach to the Quantum Harmonic Oscillator
It is readily verified that these new operators obey a dimensionless form
of the canonical commutation relations, and apart from a slightly different
normalization, the eigenstates {|𝑞〉} and {|𝑝〉} of these operators are essentially
those of their dimensionally correct counterparts {|𝑥〉} and {|𝑘〉}.
There is a representation associated with each set of states, so that
in terms of which we can expand an arbitrary state of the system.
45
Algebraic Approach to the Quantum Harmonic Oscillator
so, we can write
which define dimensionless position and momentum wave functions, 𝜓(𝑞) and
𝜓(𝑝), respectively.
We can also expand each basis ket in terms of the basis vectors of the other
representation:
46
Algebraic Approach to the Quantum Harmonic Oscillator
so, we can write
which define dimensionless position and momentum wave functions, 𝜓(𝑞) and
𝜓(𝑝), respectively.
We can also expand each basis ket in terms of the basis vectors of the other
representation:
47
Algebraic Approach to the Quantum Harmonic Oscillator
so, we can write
which define dimensionless position and momentum wave functions, 𝜓(𝑞) and
𝜓(𝑝), respectively.
We can also expand each basis ket in terms of the basis vectors of the other
representation:
48
Algebraic Approach to the Quantum Harmonic Oscillator
so, we can write
which define dimensionless position and momentum wave functions, 𝜓(𝑞) and
𝜓(𝑝), respectively.
We can also expand each basis ket in terms of the basis vectors of the other
representation:
49
Algebraic Approach to the Quantum Harmonic Oscillator
Finally, it is straightforward to show that in the |𝑞〉 representation
and in the |𝑝〉 representation
50
Algebraic Approach to the Quantum Harmonic Oscillator
Finally, it is straightforward to show that in the |𝑞〉 representation
and in the |𝑝〉 representation
51
Algebraic Approach to the Quantum Harmonic Oscillator
Finally, it is straightforward to show that in the |𝑞〉 representation
and in the |𝑝〉 representation
52
Algebraic Approach to the Quantum Harmonic Oscillator
Finally, it is straightforward to show that in the |𝑞〉 representation
and in the |𝑝〉 representation
53
Algebraic Approach to the Quantum Harmonic Oscillator
To proceed, it is useful to note that the Hamiltonian for the corresponding
classical problem is factorable, i.e., if 𝑞 and 𝑝 were classical variables we could
write
The fact that 𝑞 and 𝑝 do not commute renders this factorization invalid, but it
does lead us to consider the non-Hermitian operators
in terms of which our dimensionless operators 𝑞 and 𝑝 can be written
54
Algebraic Approach to the Quantum Harmonic Oscillator
To proceed, it is useful to note that the Hamiltonian for the corresponding
classical problem is factorable, i.e., if 𝑞 and 𝑝 were classical variables we could
write
The fact that 𝑞 and 𝑝 do not commute renders this factorization invalid, but it
does lead us to consider the non-Hermitian operators
in terms of which our dimensionless operators 𝑞 and 𝑝 can be written
55
Algebraic Approach to the Quantum Harmonic Oscillator
To proceed, it is useful to note that the Hamiltonian for the corresponding
classical problem is factorable, i.e., if 𝑞 and 𝑝 were classical variables we could
write
The fact that 𝑞 and 𝑝 do not commute renders this factorization invalid, but it
does lead us to consider the non-Hermitian operators
in terms of which our dimensionless operators 𝑞 and 𝑝 can be written
56
Algebraic Approach to the Quantum Harmonic Oscillator
To proceed, it is useful to note that the Hamiltonian for the corresponding
classical problem is factorable, i.e., if 𝑞 and 𝑝 were classical variables we could
write
The fact that 𝑞 and 𝑝 do not commute renders this factorization invalid, but it
does lead us to consider the non-Hermitian operators
in terms of which our dimensionless operators 𝑞 and 𝑝 can be written
57
Algebraic Approach to the Quantum Harmonic Oscillator
To proceed, it is useful to note that the Hamiltonian for the corresponding
classical problem is factorable, i.e., if 𝑞 and 𝑝 were classical variables we could
write
The fact that 𝑞 and 𝑝 do not commute renders this factorization invalid, but it
does lead us to consider the non-Hermitian operators
in terms of which our dimensionless operators 𝑞 and 𝑝 can be written
58
Algebraic Approach to the Quantum Harmonic Oscillator
The product of 𝑎⁺ and 𝑎 is easily evaluated:
Recognizing the commutator [𝑞, 𝑝] = 𝑖 in this last expression we find that
which allows us to express the harmonic oscillator Hamiltonian in the form
59
Algebraic Approach to the Quantum Harmonic Oscillator
The product of 𝑎⁺ and 𝑎 is easily evaluated:
Recognizing the commutator [𝑞, 𝑝] = 𝑖 in this last expression we find that
which allows us to express the harmonic oscillator Hamiltonian in the form
60
Algebraic Approach to the Quantum Harmonic Oscillator
The product of 𝑎⁺ and 𝑎 is easily evaluated:
Recognizing the commutator [𝑞, 𝑝] = 𝑖 in this last expression we find that
which allows us to express the harmonic oscillator Hamiltonian in the form
61
Algebraic Approach to the Quantum Harmonic Oscillator
The product of 𝑎⁺ and 𝑎 is easily evaluated:
Recognizing the commutator [𝑞, 𝑝] = 𝑖 in this last expression we find that
which allows us to express the harmonic oscillator Hamiltonian in the form
62
Algebraic Approach to the Quantum Harmonic Oscillator
The product of 𝑎⁺ and 𝑎 is easily evaluated:
Recognizing the commutator [𝑞, 𝑝] = 𝑖 in this last expression we find that
which allows us to express the harmonic oscillator Hamiltonian in the form
63
Algebraic Approach to the Quantum Harmonic Oscillator
Introducing one further bit of simplifying notation, we denote by
the operator product of 𝑎⁺ and 𝑎 .
Thus, the Hamiltonian 𝐻 can be written in the following simple form
It is obvious that the eigenstates of the (manifestly Hermitian) operator 𝑁 = 𝑎⁺𝑎
are also eigenstates of 𝐻. Indeed, if we can find a complete set of eigenstates
{|𝑛〉} such that
then
where
.
64
Algebraic Approach to the Quantum Harmonic Oscillator
Introducing one further bit of simplifying notation, we denote by
the operator product of 𝑎⁺ and 𝑎 .
Thus, the Hamiltonian 𝐻 can be written in the following simple form
It is obvious that the eigenstates of the (manifestly Hermitian) operator 𝑁 = 𝑎⁺𝑎
are also eigenstates of 𝐻. Indeed, if we can find a complete set of eigenstates
{|𝑛〉} such that
then
where
.
65
Algebraic Approach to the Quantum Harmonic Oscillator
Introducing one further bit of simplifying notation, we denote by
the operator product of 𝑎⁺ and 𝑎 .
Thus, the Hamiltonian 𝐻 can be written in the following simple form
It is obvious that the eigenstates of the (manifestly Hermitian) operator 𝑁 = 𝑎⁺𝑎
are also eigenstates of 𝐻. Indeed, if we can find a complete set of eigenstates
{|𝑛〉} such that
then
where
.
66
Algebraic Approach to the Quantum Harmonic Oscillator
Introducing one further bit of simplifying notation, we denote by
the operator product of 𝑎⁺ and 𝑎 .
Thus, the Hamiltonian 𝐻 can be written in the following simple form
It is obvious that the eigenstates of the (manifestly Hermitian) operator 𝑁 = 𝑎⁺𝑎
are also eigenstates of 𝐻. Indeed, if we can find a complete set of eigenstates
{|𝑛〉} such that
then
where
.
67
Algebraic Approach to the Quantum Harmonic Oscillator
Introducing one further bit of simplifying notation, we denote by
the operator product of 𝑎⁺ and 𝑎 .
Thus, the Hamiltonian 𝐻 can be written in the following simple form
It is obvious that the eigenstates of the (manifestly Hermitian) operator 𝑁 = 𝑎⁺𝑎
are also eigenstates of 𝐻. Indeed, if we can find a complete set of eigenstates
{|𝑛〉} such that
then
where
.
68
Algebraic Approach to the Quantum Harmonic Oscillator
Introducing one further bit of simplifying notation, we denote by
the operator product of 𝑎⁺ and 𝑎 .
Thus, the Hamiltonian 𝐻 can be written in the following simple form
It is obvious that the eigenstates of the (manifestly Hermitian) operator 𝑁 = 𝑎⁺𝑎
are also eigenstates of 𝐻. Indeed, if we can find a complete set of eigenstates
{|𝑛〉} such that
then
where
.
69
Algebraic Approach to the Quantum Harmonic Oscillator
Thus, we simplify our original notation for the energy eigenstates, setting
It is important to stress that, at this point, we haven't really done anything, since
we still don't know eigenvalues of the operator 𝑁 = 𝑎⁺𝑎.
But we have traded in our original operators {𝑋, 𝑃, 𝐻} with all their dimensions
and constants, for a dimensionless set of operators {𝑎, 𝑎⁺, 𝑁} and now have a
mathematically cleaner eigenvalue problem to deal with.
We thus transfer the eigenvalue problem that we have to solve to that of the
operator 𝑁, rather than the operator 𝐻.
We will refer to 𝑁 = 𝑎⁺𝑎 as the number operator because, as we will see, it
counts the number of energy quanta in the system, in units of ℏ𝜔.
70
Algebraic Approach to the Quantum Harmonic Oscillator
Thus, we simplify our original notation for the energy eigenstates, setting
It is important to stress that, at this point, we haven't really done anything, since
we still don't know eigenvalues of the operator 𝑁 = 𝑎⁺𝑎.
But we have traded in our original operators {𝑋, 𝑃, 𝐻} with all their dimensions
and constants, for a dimensionless set of operators {𝑎, 𝑎⁺, 𝑁} and now have a
mathematically cleaner eigenvalue problem to deal with.
We thus transfer the eigenvalue problem that we have to solve to that of the
operator 𝑁, rather than the operator 𝐻.
We will refer to 𝑁 = 𝑎⁺𝑎 as the number operator because, as we will see, it
counts the number of energy quanta in the system, in units of ℏ𝜔.
71
Algebraic Approach to the Quantum Harmonic Oscillator
Thus, we simplify our original notation for the energy eigenstates, setting
It is important to stress that, at this point, we haven't really done anything, since
we still don't know eigenvalues of the operator 𝑁 = 𝑎⁺𝑎.
But we have traded in our original operators {𝑋, 𝑃, 𝐻} with all their dimensions
and constants, for a dimensionless set of operators {𝑎, 𝑎⁺, 𝑁} and now have a
mathematically cleaner eigenvalue problem to deal with.
We thus transfer the eigenvalue problem that we have to solve to that of the
operator 𝑁, rather than the operator 𝐻.
We will refer to 𝑁 = 𝑎⁺𝑎 as the number operator because, as we will see, it
counts the number of energy quanta in the system, in units of ℏ𝜔.
72
Algebraic Approach to the Quantum Harmonic Oscillator
Thus, we simplify our original notation for the energy eigenstates, setting
It is important to stress that, at this point, we haven't really done anything, since
we still don't know eigenvalues of the operator 𝑁 = 𝑎⁺𝑎.
But we have traded in our original operators {𝑋, 𝑃, 𝐻} with all their dimensions
and constants, for a dimensionless set of operators {𝑎, 𝑎⁺, 𝑁} and now have a
mathematically cleaner eigenvalue problem to deal with.
We thus transfer the eigenvalue problem that we have to solve to that of the
operator 𝑁, rather than the operator 𝐻.
We will refer to 𝑁 = 𝑎⁺𝑎 as the number operator because, as we will see, it
counts the number of energy quanta in the system, in units of ℏ𝜔.
73
Algebraic Approach to the Quantum Harmonic Oscillator
Thus, we simplify our original notation for the energy eigenstates, setting
It is important to stress that, at this point, we haven't really done anything, since
we still don't know eigenvalues of the operator 𝑁 = 𝑎⁺𝑎.
But we have traded in our original operators {𝑋, 𝑃, 𝐻} with all their dimensions
and constants, for a dimensionless set of operators {𝑎, 𝑎⁺, 𝑁} and now have a
mathematically cleaner eigenvalue problem to deal with.
We thus transfer the eigenvalue problem that we have to solve to that of the
operator 𝑁, rather than the operator 𝐻.
We will refer to 𝑁 = 𝑎⁺𝑎 as the number operator because, as we will see, it
counts the number of energy quanta in the system, in units of ℏ𝜔.
74
Algebraic Approach to the Quantum Harmonic Oscillator
Our goal in what follows is to use the commutation relations obeyed by the new
operators 𝑎, 𝑎⁺, and 𝑁 = 𝑎⁺𝑎, to deduce the structure of the energy eigenstates
of this system.
The commutation relations that we will need are easily obtained.
We note, first, that
which the canonical commutation relations reduce to
A direct consequence of this relation is that
find that
, from which we
75
Algebraic Approach to the Quantum Harmonic Oscillator
Our goal in what follows is to use the commutation relations obeyed by the new
operators 𝑎, 𝑎⁺, and 𝑁 = 𝑎⁺𝑎, to deduce the structure of the energy eigenstates
of this system.
The commutation relations that we will need are easily obtained.
We note, first, that
which the canonical commutation relations reduce to
A direct consequence of this relation is that
find that
, from which we
76
Algebraic Approach to the Quantum Harmonic Oscillator
Our goal in what follows is to use the commutation relations obeyed by the new
operators 𝑎, 𝑎⁺, and 𝑁 = 𝑎⁺𝑎, to deduce the structure of the energy eigenstates
of this system.
The commutation relations that we will need are easily obtained.
We note, first, that
which the canonical commutation relations reduce to
A direct consequence of this relation is that
find that
, from which we
77
Algebraic Approach to the Quantum Harmonic Oscillator
Our goal in what follows is to use the commutation relations obeyed by the new
operators 𝑎, 𝑎⁺, and 𝑁 = 𝑎⁺𝑎, to deduce the structure of the energy eigenstates
of this system.
The commutation relations that we will need are easily obtained.
We note, first, that
which the canonical commutation relations reduce to
A direct consequence of this relation is that
find that
, from which we
78
Algebraic Approach to the Quantum Harmonic Oscillator
Our goal in what follows is to use the commutation relations obeyed by the new
operators 𝑎, 𝑎⁺, and 𝑁 = 𝑎⁺𝑎, to deduce the structure of the energy eigenstates
of this system.
The commutation relations that we will need are easily obtained.
We note, first, that
which the canonical commutation relations reduce to
A direct consequence of this relation is that
find that
, from which we
79
Algebraic Approach to the Quantum Harmonic Oscillator
Our goal in what follows is to use the commutation relations obeyed by the new
operators 𝑎, 𝑎⁺, and 𝑁 = 𝑎⁺𝑎, to deduce the structure of the energy eigenstates
of this system.
The commutation relations that we will need are easily obtained.
We note, first, that
which the canonical commutation relations reduce to
A direct consequence of this relation is that
find that
, from which we
80
Algebraic Approach to the Quantum Harmonic Oscillator
Our goal in what follows is to use the commutation relations obeyed by the new
operators 𝑎, 𝑎⁺, and 𝑁 = 𝑎⁺𝑎, to deduce the structure of the energy eigenstates
of this system.
The commutation relations that we will need are easily obtained.
We note, first, that
which the canonical commutation relations reduce to
A direct consequence of this relation is that
find that
, from which we
81
Algebraic Approach to the Quantum Harmonic Oscillator
Our goal in what follows is to use the commutation relations obeyed by the new
operators 𝑎, 𝑎⁺, and 𝑁 = 𝑎⁺𝑎, to deduce the structure of the energy eigenstates
of this system.
The commutation relations that we will need are easily obtained.
We note, first, that
which the canonical commutation relations reduce to
A direct consequence of this relation is that
find that
, from which we
82
Algebraic Approach to the Quantum Harmonic Oscillator
Next, we evaluate the commutator
[𝑁, 𝑎] = [𝑎⁺𝑎, 𝑎] = 𝑎⁺[𝑎, 𝑎] + [𝑎⁺, 𝑎]𝑎
which our previous result reduces to
[𝑁, 𝑎] = −𝑎.
We note that this commutation relation, 𝑁𝑎 − 𝑎𝑁 = −𝑎, can be written in the
form
𝑁𝑎 = 𝑎𝑁 − 𝑎
= 𝑎(𝑁 − 1)
which shows that we can "push" 𝑁 to the right through 𝑎, where it pops out as
𝑁 − 1. I like to think of this as a kind of quantum mechanics magic trick…
83
Algebraic Approach to the Quantum Harmonic Oscillator
Next, we evaluate the commutator
[𝑁, 𝑎] = [𝑎⁺𝑎, 𝑎] = 𝑎⁺[𝑎, 𝑎] + [𝑎⁺, 𝑎]𝑎
which our previous result reduces to
[𝑁, 𝑎] = −𝑎.
We note that this commutation relation, 𝑁𝑎 − 𝑎𝑁 = −𝑎, can be written in the
form
𝑁𝑎 = 𝑎𝑁 − 𝑎
= 𝑎(𝑁 − 1)
which shows that we can "push" 𝑁 to the right through 𝑎, where it pops out as
𝑁 − 1. I like to think of this as a kind of quantum mechanics magic trick…
84
Algebraic Approach to the Quantum Harmonic Oscillator
Next, we evaluate the commutator
[𝑁, 𝑎] = [𝑎⁺𝑎, 𝑎] = 𝑎⁺[𝑎, 𝑎] + [𝑎⁺, 𝑎]𝑎
which our previous result reduces to
[𝑁, 𝑎] = −𝑎.
We note that this commutation relation, 𝑁𝑎 − 𝑎𝑁 = −𝑎, can be written in the
form
𝑁𝑎 = 𝑎𝑁 − 𝑎
= 𝑎(𝑁 − 1)
which shows that we can "push" 𝑁 to the right through 𝑎, where it pops out as
𝑁 − 1. I like to think of this as a kind of quantum mechanics magic trick…
85
Algebraic Approach to the Quantum Harmonic Oscillator
Next, we evaluate the commutator
[𝑁, 𝑎] = [𝑎⁺𝑎, 𝑎] = 𝑎⁺[𝑎, 𝑎] + [𝑎⁺, 𝑎]𝑎
which our previous result reduces to
[𝑁, 𝑎] = −𝑎.
We note that this commutation relation, 𝑁𝑎 − 𝑎𝑁 = −𝑎, can be written in the
form
𝑁𝑎 = 𝑎𝑁 − 𝑎
= 𝑎(𝑁 − 1)
which shows that we can "push" 𝑁 to the right through 𝑎, where it pops out as
𝑁 − 1. I like to think of this as a kind of quantum mechanics magic trick…
86
Algebraic Approach to the Quantum Harmonic Oscillator
Next, we evaluate the commutator
[𝑁, 𝑎] = [𝑎⁺𝑎, 𝑎] = 𝑎⁺[𝑎, 𝑎] + [𝑎⁺, 𝑎]𝑎
which our previous result reduces to
[𝑁, 𝑎] = −𝑎.
We note that this commutation relation, 𝑁𝑎 − 𝑎𝑁 = −𝑎, can be written in the
form
𝑁𝑎 = 𝑎𝑁 − 𝑎
= 𝑎(𝑁 − 1)
which shows that we can "push" 𝑁 to the right through 𝑎, where it pops out as
𝑁 − 1. I like to think of this as a kind of quantum mechanics magic trick…
87
Algebraic Approach to the Quantum Harmonic Oscillator
Next, we evaluate the commutator
[𝑁, 𝑎] = [𝑎⁺𝑎, 𝑎] = 𝑎⁺[𝑎, 𝑎] + [𝑎⁺, 𝑎]𝑎
which our previous result reduces to
[𝑁, 𝑎] = −𝑎.
We note that this commutation relation, 𝑁𝑎 − 𝑎𝑁 = −𝑎, can be written in the
form
𝑁𝑎 = 𝑎𝑁 − 𝑎
= 𝑎(𝑁 − 1)
which shows that we can "push" 𝑁 to the right through 𝑎, where it pops out as
𝑁 − 1. I like to think of this as a kind of quantum mechanics magic trick…
88
Algebraic Approach to the Quantum Harmonic Oscillator
Next, we evaluate the commutator
[𝑁, 𝑎] = [𝑎⁺𝑎, 𝑎] = 𝑎⁺[𝑎, 𝑎] + [𝑎⁺, 𝑎]𝑎
which our previous result reduces to
[𝑁, 𝑎] = −𝑎.
We note that this commutation relation, 𝑁𝑎 − 𝑎𝑁 = −𝑎, can be written in the
form
𝑁𝑎 = 𝑎𝑁 − 𝑎
= 𝑎(𝑁 − 1)
which shows that we can "push" 𝑁 to the right through 𝑎, where it pops out as
𝑁 − 1. I like to think of this as a kind of quantum mechanics magic trick…
89
Algebraic Approach to the Quantum Harmonic Oscillator
Next, we evaluate the commutator
[𝑁, 𝑎⁺] = [𝑎⁺𝑎, 𝑎⁺] = 𝑎⁺[𝑎, 𝑎⁺] + [𝑎⁺, 𝑎⁺]𝑎
which our previous result reduces to
[𝑁, 𝑎⁺] = 𝑎⁺.
This also can be expressed in a convenient form for calculations, namely
𝑁𝑎+ = 𝑎+ 𝑁 + 𝑎⁺
= 𝑎⁺(𝑁 + 1)
which shows that we can also, "push" 𝑁 to the right through 𝑎⁺, but now it pops
out the other side as 𝑁 + 1.
90
Algebraic Approach to the Quantum Harmonic Oscillator
Next, we evaluate the commutator
[𝑁, 𝑎⁺] = [𝑎⁺𝑎, 𝑎⁺] = 𝑎⁺[𝑎, 𝑎⁺] + [𝑎⁺, 𝑎⁺]𝑎
which our previous result reduces to
[𝑁, 𝑎⁺] = 𝑎⁺.
This also can be expressed in a convenient form for calculations, namely
𝑁𝑎+ = 𝑎+ 𝑁 + 𝑎⁺
= 𝑎⁺(𝑁 + 1)
which shows that we can also, "push" 𝑁 to the right through 𝑎⁺, but now it pops
out the other side as 𝑁 + 1.
91
Algebraic Approach to the Quantum Harmonic Oscillator
Next, we evaluate the commutator
[𝑁, 𝑎⁺] = [𝑎⁺𝑎, 𝑎⁺] = 𝑎⁺[𝑎, 𝑎⁺] + [𝑎⁺, 𝑎⁺]𝑎
which our previous result reduces to
[𝑁, 𝑎⁺] = 𝑎⁺.
This also can be expressed in a convenient form for calculations, namely
𝑁𝑎+ = 𝑎+ 𝑁 + 𝑎⁺
= 𝑎⁺(𝑁 + 1)
which shows that we can also, "push" 𝑁 to the right through 𝑎⁺, but now it pops
out the other side as 𝑁 + 1.
92
Algebraic Approach to the Quantum Harmonic Oscillator
Next, we evaluate the commutator
[𝑁, 𝑎⁺] = [𝑎⁺𝑎, 𝑎⁺] = 𝑎⁺[𝑎, 𝑎⁺] + [𝑎⁺, 𝑎⁺]𝑎
which our previous result reduces to
[𝑁, 𝑎⁺] = 𝑎⁺.
This also can be expressed in a convenient form for calculations, namely
𝑁𝑎+ = 𝑎+ 𝑁 + 𝑎⁺
= 𝑎⁺(𝑁 + 1)
which shows that we can also, "push" 𝑁 to the right through 𝑎⁺, but now it pops
out the other side as 𝑁 + 1.
93
Algebraic Approach to the Quantum Harmonic Oscillator
Next, we evaluate the commutator
[𝑁, 𝑎⁺] = [𝑎⁺𝑎, 𝑎⁺] = 𝑎⁺[𝑎, 𝑎⁺] + [𝑎⁺, 𝑎⁺]𝑎
which our previous result reduces to
[𝑁, 𝑎⁺] = 𝑎⁺.
This also can be expressed in a convenient form for calculations, namely
𝑁𝑎+ = 𝑎+ 𝑁 + 𝑎⁺
= 𝑎⁺(𝑁 + 1)
which shows that we can also, "push" 𝑁 to the right through 𝑎⁺, but now it pops
out the other side as 𝑁 + 1.
94
Algebraic Approach to the Quantum Harmonic Oscillator
Next, we evaluate the commutator
[𝑁, 𝑎⁺] = [𝑎⁺𝑎, 𝑎⁺] = 𝑎⁺[𝑎, 𝑎⁺] + [𝑎⁺, 𝑎⁺]𝑎
which our previous result reduces to
[𝑁, 𝑎⁺] = 𝑎⁺.
This also can be expressed in a convenient form for calculations, namely
𝑁𝑎+ = 𝑎+ 𝑁 + 𝑎⁺
= 𝑎⁺(𝑁 + 1)
which shows that we can also, "push" 𝑁 to the right through 𝑎⁺, but now it pops
out the other side as 𝑁 + 1.
95
Algebraic Approach to the Quantum Harmonic Oscillator
Combining these relations, we have the following closed algebra of commutation
relations,
[𝑎, 𝑎⁺] = 1
𝑁, 𝑎 = −𝑎
[𝑁, 𝑎⁺] = 𝑎⁺.
and the expressions to which they are equivalent,
𝑁 + 1 = 𝑎𝑎⁺
𝑁𝑎 = 𝑎(𝑁 − 1)
𝑁𝑎⁺ = 𝑎⁺(𝑁 + 1).
which we can use to determine the eigenstates and eigenvalues of 𝑁 = 𝑎⁺𝑎.
96
Algebraic Approach to the Quantum Harmonic Oscillator
Combining these relations, we have the following closed algebra of commutation
relations,
[𝑎, 𝑎⁺] = 1
𝑁, 𝑎 = −𝑎
[𝑁, 𝑎⁺] = 𝑎⁺.
and the expressions to which they are equivalent,
𝑁 + 1 = 𝑎𝑎⁺
𝑁𝑎 = 𝑎(𝑁 − 1)
𝑁𝑎⁺ = 𝑎⁺(𝑁 + 1).
which we can use to determine the eigenstates and eigenvalues of 𝑁 = 𝑎⁺𝑎.
97
Algebraic Approach to the Quantum Harmonic Oscillator
Combining these relations, we have the following closed algebra of commutation
relations,
[𝑎, 𝑎⁺] = 1
𝑁, 𝑎 = −𝑎
[𝑁, 𝑎⁺] = 𝑎⁺.
and the expressions to which they are equivalent,
𝑁 + 1 = 𝑎𝑎⁺
𝑁𝑎 = 𝑎(𝑁 − 1)
𝑁𝑎⁺ = 𝑎⁺(𝑁 + 1).
which we can use to determine the eigenstates and eigenvalues of 𝑁 = 𝑎⁺𝑎.
98
Spectrum of the Number Operator Using the commutation relations obtained above, we now deduce a number of
basic properties associated with the eigenstates of the number operator 𝑁 and,
hence, of the eigenstates of the harmonic oscillator Hamiltonian 𝐻.
In what follows, we begin by simply assuming the existence of at least one
nonzero eigenvector |𝑛〉 of the observable 𝑁.
This is a trivial assumption, since 𝑁 is a simple function of the observable 𝐻, and
is therefore an observable of the system.
This assumption then allows us to prove the following
1. Positivity of eigenvalues: If |𝑛〉 is an eigenvector of 𝑁 with eigenvalue 𝑛,
then 𝑛 ≥ 0. This is obvious because, as you have shown, any operator of the
form 𝑁 = 𝑎⁺𝑎 is a positive operator, and positive operators have positive
eigenvalues. We then have the following….
99
Spectrum of the Number Operator Using the commutation relations obtained above, we now deduce a number of
basic properties associated with the eigenstates of the number operator 𝑁 and,
hence, of the eigenstates of the harmonic oscillator Hamiltonian 𝐻.
In what follows, we begin by simply assuming the existence of at least one
nonzero eigenvector |𝑛〉 of the number operator 𝑁.
This is a trivial assumption, since 𝑁 is a simple function of the observable 𝐻, and
is therefore an observable of the system.
This assumption then allows us to prove the following
1. Positivity of eigenvalues: If |𝑛〉 is an eigenvector of 𝑁 with eigenvalue 𝑛,
then 𝑛 ≥ 0. This is obvious because, as you have shown, any operator of the
form 𝑁 = 𝑎⁺𝑎 is a positive operator, and positive operators have positive
eigenvalues. We then have the following….
100
Spectrum of the Number Operator Using the commutation relations obtained above, we now deduce a number of
basic properties associated with the eigenstates of the number operator 𝑁 and,
hence, of the eigenstates of the harmonic oscillator Hamiltonian 𝐻.
In what follows, we begin by simply assuming the existence of at least one
nonzero eigenvector |𝑛〉 of the number operator 𝑁.
This is non-controversial, since 𝑁 is a simple function of the observable 𝐻, and is
therefore an observable of the system.
This assumption then allows us to prove the following
1. Positivity of eigenvalues: If |𝑛〉 is an eigenvector of 𝑁 with eigenvalue 𝑛,
then 𝑛 ≥ 0. This is obvious because, as you have shown, any operator of the
form 𝑁 = 𝑎⁺𝑎 is a positive operator, and positive operators have positive
eigenvalues. We then have the following….
101
Spectrum of the Number Operator Using the commutation relations obtained above, we now deduce a number of
basic properties associated with the eigenstates of the number operator 𝑁 and,
hence, of the eigenstates of the harmonic oscillator Hamiltonian 𝐻.
In what follows, we begin by simply assuming the existence of at least one
nonzero eigenvector |𝑛〉 of the number operator 𝑁.
This is non-controversial, since 𝑁 is a simple function of the observable 𝐻, and is
therefore an observable of the system.
This assumption then allows us to prove the following statements:
1. Positivity of eigenvalues: If |𝑛〉 is an eigenvector of 𝑁 with eigenvalue 𝑛,
then 𝑛 ≥ 0. This is obvious because, as you have shown, any operator of the
form 𝑁 = 𝑎⁺𝑎 is a positive operator, and positive operators have positive
eigenvalues. We then have the following….
102
Spectrum of the Number Operator Using the commutation relations obtained above, we now deduce a number of
basic properties associated with the eigenstates of the number operator 𝑁 and,
hence, of the eigenstates of the harmonic oscillator Hamiltonian 𝐻.
In what follows, we begin by simply assuming the existence of at least one
nonzero eigenvector |𝑛〉 of the number operator 𝑁.
This is non-controversial, since 𝑁 is a simple function of the observable 𝐻, and is
therefore an observable of the system.
This assumption then allows us to prove the following statements:
1. Positivity of eigenvalues: If |𝑛〉 is an eigenvector of 𝑁 with eigenvalue 𝑛,
then 𝑛 ≥ 0. This is obvious because, as you have shown, any operator of the
form 𝑁 = 𝑎⁺𝑎 is a positive operator, and positive operators have positive
eigenvalues. We then have the following….
103
Spectrum of the Number Operator Using the commutation relations obtained above, we now deduce a number of
basic properties associated with the eigenstates of the number operator 𝑁 and,
hence, of the eigenstates of the harmonic oscillator Hamiltonian 𝐻.
In what follows, we begin by simply assuming the existence of at least one
nonzero eigenvector |𝑛〉 of the number operator 𝑁.
This is non-controversial, since 𝑁 is a simple function of the observable 𝐻, and is
therefore an observable of the system.
This assumption then allows us to prove the following statements:
1. Positivity of eigenvalues: If |𝑛〉 is an eigenvector of 𝑁 with eigenvalue 𝑛,
then 𝑛 ≥ 0. This is obvious because, as you have shown, any operator of the
form 𝑁 = 𝑎⁺𝑎 is a positive operator, and positive operators have positive
eigenvalues. We then have the following….
104
Spectrum of the Number Operator 2. Action of 𝑎⁺ on 𝑛 :
If 𝑛 is an eigenvector of the operator 𝑁 with eigenvalue 𝑛, then the vector
𝑎⁺ 𝑛 never vanishes; moreover 𝑎⁺ 𝑛 is an eigenvector of with eigenvalue
𝑛 + 1.
Proof: We consider
𝑁 𝑎+ 𝑛 = 𝑁𝑎+ 𝑛 = 𝑎+ 𝑁 + 1 𝑛
= 𝑎+ 𝑛 + 1 𝑛 = 𝑛 + 1 [𝑎+ 𝑛 ]
where we have used quantum magic trick associated with the commutation
relation [𝑁, 𝑎⁺] = 𝑎+ to push 𝑁 through 𝑎+ where it became 𝑁 + 1.
So 𝑎+ 𝑛 satisfies the eigenvalue equation, and its length never vanishes
since
〈𝑛|𝑎𝑎⁺|𝑛〉 = 〈𝑛|𝑁 + 1|𝑛〉 = (𝑛 + 1)〈𝑛|𝑛〉 > 0
105
Spectrum of the Number Operator 2. Action of 𝑎⁺ on 𝑛 :
If 𝑛 is an eigenvector of the operator 𝑁 with eigenvalue 𝑛, then the vector
𝑎⁺ 𝑛 never vanishes; moreover 𝑎⁺ 𝑛 is an eigenvector of 𝑁 with eigenvalue
𝑛 + 1.
Proof: We consider
𝑁 𝑎+ 𝑛 = 𝑁𝑎+ 𝑛 = 𝑎+ 𝑁 + 1 𝑛
= 𝑎+ 𝑛 + 1 𝑛 = 𝑛 + 1 [𝑎+ 𝑛 ]
where we have used quantum magic trick associated with the commutation
relation [𝑁, 𝑎⁺] = 𝑎+ to push 𝑁 through 𝑎+ where it became 𝑁 + 1.
So 𝑎+ 𝑛 satisfies the eigenvalue equation, and its length never vanishes
since
〈𝑛|𝑎𝑎⁺|𝑛〉 = 〈𝑛|𝑁 + 1|𝑛〉 = (𝑛 + 1)〈𝑛|𝑛〉 > 0
106
Spectrum of the Number Operator 2. Action of 𝑎⁺ on 𝑛 :
If 𝑛 is an eigenvector of the operator 𝑁 with eigenvalue 𝑛, then the vector
𝑎⁺ 𝑛 never vanishes; moreover 𝑎⁺ 𝑛 is an eigenvector of 𝑁 with eigenvalue
𝑛 + 1.
Proof: We consider
𝑁 𝑎+ 𝑛 = 𝑁𝑎+ 𝑛 = 𝑎+ 𝑁 + 1 𝑛
= 𝑎+ 𝑛 + 1 𝑛 = 𝑛 + 1 [𝑎+ 𝑛 ]
where we have used quantum magic trick associated with the commutation
relation [𝑁, 𝑎⁺] = 𝑎+ to push 𝑁 through 𝑎+ where it became 𝑁 + 1.
So 𝑎+ 𝑛 satisfies the eigenvalue equation, and its length never vanishes
since
〈𝑛|𝑎𝑎⁺|𝑛〉 = 〈𝑛|𝑁 + 1|𝑛〉 = (𝑛 + 1)〈𝑛|𝑛〉 > 0
107
Spectrum of the Number Operator 2. Action of 𝑎⁺ on 𝑛 :
If 𝑛 is an eigenvector of the operator 𝑁 with eigenvalue 𝑛, then the vector
𝑎⁺ 𝑛 never vanishes; moreover 𝑎⁺ 𝑛 is an eigenvector of 𝑁 with eigenvalue
𝑛 + 1.
Proof: We consider
𝑁 𝑎+ 𝑛 = 𝑁𝑎+ 𝑛 = 𝑎+ 𝑁 + 1 𝑛
= 𝑎+ 𝑛 + 1 𝑛 = 𝑛 + 1 [𝑎+ 𝑛 ]
where we have used quantum magic trick associated with the commutation
relation [𝑁, 𝑎⁺] = 𝑎+ to push 𝑁 through 𝑎+ where it became 𝑁 + 1.
So 𝑎+ 𝑛 satisfies the eigenvalue equation, and its length never vanishes
since
〈𝑛|𝑎𝑎⁺|𝑛〉 = 〈𝑛|𝑁 + 1|𝑛〉 = (𝑛 + 1)〈𝑛|𝑛〉 > 0
108
Spectrum of the Number Operator 2. Action of 𝑎⁺ on 𝑛 :
If 𝑛 is an eigenvector of the operator 𝑁 with eigenvalue 𝑛, then the vector
𝑎⁺ 𝑛 never vanishes; moreover 𝑎⁺ 𝑛 is an eigenvector of 𝑁 with eigenvalue
𝑛 + 1.
Proof: We consider
𝑁 𝑎+ 𝑛 = 𝑁𝑎+ 𝑛 = 𝑎+ 𝑁 + 1 𝑛
= 𝑎+ 𝑛 + 1 𝑛 = 𝑛 + 1 [𝑎+ 𝑛 ]
where we have used quantum magic trick associated with the commutation
relation [𝑁, 𝑎⁺] = 𝑎+ to push 𝑁 through 𝑎+ where it became 𝑁 + 1.
So 𝑎+ 𝑛 satisfies the eigenvalue equation, and its length never vanishes
since
〈𝑛|𝑎𝑎⁺|𝑛〉 = 〈𝑛|𝑁 + 1|𝑛〉 = (𝑛 + 1)〈𝑛|𝑛〉 > 0
109
Spectrum of the Number Operator 2. Action of 𝑎⁺ on 𝑛 :
If 𝑛 is an eigenvector of the operator 𝑁 with eigenvalue 𝑛, then the vector
𝑎⁺ 𝑛 never vanishes; moreover 𝑎⁺ 𝑛 is an eigenvector of 𝑁 with eigenvalue
𝑛 + 1.
Proof: We consider
𝑁 𝑎+ 𝑛 = 𝑁𝑎+ 𝑛 = 𝑎+ 𝑁 + 1 𝑛
= 𝑎+ 𝑛 + 1 𝑛 = 𝑛 + 1 [𝑎+ 𝑛 ]
where we have used quantum magic trick associated with the commutation
relation [𝑁, 𝑎⁺] = 𝑎+ to push 𝑁 through 𝑎+ where it became 𝑁 + 1.
So 𝑎+ 𝑛 satisfies the eigenvalue equation, and its length never vanishes
since
〈𝑛|𝑎𝑎⁺|𝑛〉 = 〈𝑛|𝑁 + 1|𝑛〉 = (𝑛 + 1)〈𝑛|𝑛〉 > 0
110
Spectrum of the Number Operator 2. Action of 𝑎⁺ on 𝑛 :
If 𝑛 is an eigenvector of the operator 𝑁 with eigenvalue 𝑛, then the vector
𝑎⁺ 𝑛 never vanishes; moreover 𝑎⁺ 𝑛 is an eigenvector of 𝑁 with eigenvalue
𝑛 + 1.
Proof: We consider
𝑁 𝑎+ 𝑛 = 𝑁𝑎+ 𝑛 = 𝑎+ 𝑁 + 1 𝑛
= 𝑎+ 𝑛 + 1 𝑛 = 𝑛 + 1 [𝑎+ 𝑛 ]
where we have used quantum magic trick associated with the commutation
relation [𝑁, 𝑎⁺] = 𝑎+ to push 𝑁 through 𝑎+ where it became 𝑁 + 1.
So 𝑎+ 𝑛 satisfies the eigenvalue equation, and its length never vanishes
since
〈𝑛|𝑎𝑎⁺|𝑛〉 = 〈𝑛|𝑁 + 1|𝑛〉 = (𝑛 + 1)〈𝑛|𝑛〉 > 0
111
Spectrum of the Number Operator 2. Action of 𝑎⁺ on 𝑛 :
If 𝑛 is an eigenvector of the operator 𝑁 with eigenvalue 𝑛, then the vector
𝑎⁺ 𝑛 never vanishes; moreover 𝑎⁺ 𝑛 is an eigenvector of 𝑁 with eigenvalue
𝑛 + 1.
Proof: We consider
𝑁 𝑎+ 𝑛 = 𝑁𝑎+ 𝑛 = 𝑎+ 𝑁 + 1 𝑛
= 𝑎+ 𝑛 + 1 𝑛 = 𝑛 + 1 [𝑎+ 𝑛 ]
where we have used quantum magic trick associated with the commutation
relation [𝑁, 𝑎⁺] = 𝑎+ to push 𝑁 through 𝑎+ where it became 𝑁 + 1.
So 𝑎+ 𝑛 satisfies the eigenvalue equation, and its length never vanishes
since
〈𝑛|𝑎𝑎⁺|𝑛〉 = 〈𝑛|𝑁 + 1|𝑛〉 = (𝑛 + 1)〈𝑛|𝑛〉 > 0
112
Spectrum of the Number Operator 2. Action of 𝑎⁺ on 𝑛 :
If 𝑛 is an eigenvector of the operator 𝑁 with eigenvalue 𝑛, then the vector
𝑎⁺ 𝑛 never vanishes; moreover 𝑎⁺ 𝑛 is an eigenvector of 𝑁 with eigenvalue
𝑛 + 1.
Proof: We consider
𝑁 𝑎+ 𝑛 = 𝑁𝑎+ 𝑛 = 𝑎+ 𝑁 + 1 𝑛
= 𝑎+ 𝑛 + 1 𝑛 = 𝑛 + 1 [𝑎+ 𝑛 ]
where we have used quantum magic trick associated with the commutation
relation [𝑁, 𝑎⁺] = 𝑎+ to push 𝑁 through 𝑎+ where it became 𝑁 + 1.
So 𝑎+ 𝑛 satisfies the eigenvalue equation, and its length never vanishes
since
〈𝑛|𝑎𝑎⁺|𝑛〉 = 〈𝑛|𝑁 + 1|𝑛〉 = (𝑛 + 1)〈𝑛|𝑛〉 > 0
113
Spectrum of the Number Operator 2. Action of 𝑎⁺ on 𝑛 :
If 𝑛 is an eigenvector of the operator 𝑁 with eigenvalue 𝑛, then the vector
𝑎⁺ 𝑛 never vanishes; moreover 𝑎⁺ 𝑛 is an eigenvector of 𝑁 with eigenvalue
𝑛 + 1.
Proof: We consider
𝑁 𝑎+ 𝑛 = 𝑁𝑎+ 𝑛 = 𝑎+ 𝑁 + 1 𝑛
= 𝑎+ 𝑛 + 1 𝑛 = 𝑛 + 1 [𝑎+ 𝑛 ]
where we have used quantum magic trick associated with the commutation
relation [𝑁, 𝑎⁺] = 𝑎+ to push 𝑁 through 𝑎+ where it became 𝑁 + 1.
So 𝑎+ 𝑛 satisfies the eigenvalue equation, and its length never vanishes
since
〈𝑛|𝑎𝑎⁺|𝑛〉 = 〈𝑛|𝑁 + 1|𝑛〉 = (𝑛 + 1)〈𝑛|𝑛〉 > 0
114
Spectrum of the Number Operator 2. Action of 𝑎⁺ on 𝑛 :
If 𝑛 is an eigenvector of the operator 𝑁 with eigenvalue 𝑛, then the vector
𝑎⁺ 𝑛 never vanishes; moreover 𝑎⁺ 𝑛 is an eigenvector of 𝑁 with eigenvalue
𝑛 + 1.
Proof: We consider
𝑁 𝑎+ 𝑛 = 𝑁𝑎+ 𝑛 = 𝑎+ 𝑁 + 1 𝑛
= 𝑎+ 𝑛 + 1 𝑛 = 𝑛 + 1 [𝑎+ 𝑛 ]
where we have used quantum magic trick associated with the commutation
relation [𝑁, 𝑎⁺] = 𝑎+ to push 𝑁 through 𝑎+ where it became 𝑁 + 1.
So 𝑎+ 𝑛 satisfies the eigenvalue equation, and its length never vanishes
since
〈𝑛|𝑎𝑎⁺|𝑛〉 = 〈𝑛|𝑁 + 1|𝑛〉 = (𝑛 + 1)〈𝑛|𝑛〉 > 0
115
Spectrum of the Number Operator 3. Action of 𝑎 on 𝑛 :
If 𝑛 is an eigenvector of 𝑁 with eigenvalue 𝑛, then the vector 𝑎 𝑛 vanishes
if and only if 𝑛 = 0; moreover if 𝑛 ≠ 0 then 𝑎 𝑛 is an eigenvector of 𝑁 with
eigenvalue 𝑛 − 1.
Proof: We consider
𝑁 𝑎 𝑛 = 𝑁𝑎 𝑛 = 𝑎+ 𝑁 − 1 𝑛
= 𝑎 𝑛 − 1 𝑛 = 𝑛 − 1 [𝑎 𝑛 ]
where we have used quantum magic trick associated with the commutation
relation 𝑁, 𝑎 = −𝑎 to push 𝑁 through 𝑎 where it became 𝑁 − 1.
So 𝑎 𝑛 satisfies the eigenvalue equation with eigenvalue 𝑛 − 1, and its
length, as shown below, vanishes if and only if 𝑛 = 0,
〈𝑛|𝑎⁺𝑎|𝑛〉 = 〈𝑛|𝑁|𝑛〉 = 𝑛〈𝑛|𝑛〉 > 0
116
Spectrum of the Number Operator 3. Action of 𝑎 on 𝑛 :
If 𝑛 is an eigenvector of 𝑁 with eigenvalue 𝑛, then the vector 𝑎 𝑛 vanishes
if and only if 𝑛 = 0; moreover if 𝑛 ≠ 0 then 𝑎 𝑛 is an eigenvector of 𝑁 with
eigenvalue 𝑛 − 1.
Proof: We consider
𝑁 𝑎 𝑛 = 𝑁𝑎 𝑛 = 𝑎+ 𝑁 − 1 𝑛
= 𝑎 𝑛 − 1 𝑛 = 𝑛 − 1 [𝑎 𝑛 ]
where we have used quantum magic trick associated with the commutation
relation 𝑁, 𝑎 = −𝑎 to push 𝑁 through 𝑎 where it became 𝑁 − 1.
So 𝑎 𝑛 satisfies the eigenvalue equation with eigenvalue 𝑛 − 1, and its
length, as shown below, vanishes if and only if 𝑛 = 0,
〈𝑛|𝑎⁺𝑎|𝑛〉 = 〈𝑛|𝑁|𝑛〉 = 𝑛〈𝑛|𝑛〉 > 0
117
Spectrum of the Number Operator 3. Action of 𝑎 on 𝑛 :
If 𝑛 is an eigenvector of 𝑁 with eigenvalue 𝑛, then the vector 𝑎 𝑛 vanishes
if and only if 𝑛 = 0; moreover if 𝑛 ≠ 0 then 𝑎 𝑛 is an eigenvector of 𝑁 with
eigenvalue 𝑛 − 1.
Proof: We consider
𝑁 𝑎 𝑛 = 𝑁𝑎 𝑛 = 𝑎+ 𝑁 − 1 𝑛
= 𝑎 𝑛 − 1 𝑛 = 𝑛 − 1 [𝑎 𝑛 ]
where we have used quantum magic trick associated with the commutation
relation 𝑁, 𝑎 = −𝑎 to push 𝑁 through 𝑎 where it became 𝑁 − 1.
So 𝑎 𝑛 satisfies the eigenvalue equation with eigenvalue 𝑛 − 1, and its
length, as shown below, vanishes if and only if 𝑛 = 0,
〈𝑛|𝑎⁺𝑎|𝑛〉 = 〈𝑛|𝑁|𝑛〉 = 𝑛〈𝑛|𝑛〉 > 0
118
Spectrum of the Number Operator 3. Action of 𝑎 on 𝑛 :
If 𝑛 is an eigenvector of 𝑁 with eigenvalue 𝑛, then the vector 𝑎 𝑛 vanishes
if and only if 𝑛 = 0; moreover if 𝑛 ≠ 0 then 𝑎 𝑛 is an eigenvector of 𝑁 with
eigenvalue 𝑛 − 1.
Proof: We consider
𝑁 𝑎 𝑛 = 𝑁𝑎 𝑛 = 𝑎+ 𝑁 − 1 𝑛
= 𝑎 𝑛 − 1 𝑛 = 𝑛 − 1 [𝑎 𝑛 ]
where we have used quantum magic trick associated with the commutation
relation 𝑁, 𝑎 = −𝑎 to push 𝑁 through 𝑎 where it became 𝑁 − 1.
So 𝑎 𝑛 satisfies the eigenvalue equation with eigenvalue 𝑛 − 1, and its
length, as shown below, vanishes if and only if 𝑛 = 0,
〈𝑛|𝑎⁺𝑎|𝑛〉 = 〈𝑛|𝑁|𝑛〉 = 𝑛〈𝑛|𝑛〉 > 0
119
Spectrum of the Number Operator 3. Action of 𝑎 on 𝑛 :
If 𝑛 is an eigenvector of 𝑁 with eigenvalue 𝑛, then the vector 𝑎 𝑛 vanishes
if and only if 𝑛 = 0; moreover if 𝑛 ≠ 0 then 𝑎 𝑛 is an eigenvector of 𝑁 with
eigenvalue 𝑛 − 1.
Proof: We consider
𝑁 𝑎 𝑛 = 𝑁𝑎 𝑛 = 𝑎+ 𝑁 − 1 𝑛
= 𝑎 𝑛 − 1 𝑛 = 𝑛 − 1 [𝑎 𝑛 ]
where we have used quantum magic trick associated with the commutation
relation 𝑁, 𝑎 = −𝑎 to push 𝑁 through 𝑎 where it became 𝑁 − 1.
So 𝑎 𝑛 satisfies the eigenvalue equation with eigenvalue 𝑛 − 1, and its
length, as shown below, vanishes if and only if 𝑛 = 0,
〈𝑛|𝑎⁺𝑎|𝑛〉 = 〈𝑛|𝑁|𝑛〉 = 𝑛〈𝑛|𝑛〉 > 0
120
Spectrum of the Number Operator 3. Action of 𝑎 on 𝑛 :
If 𝑛 is an eigenvector of 𝑁 with eigenvalue 𝑛, then the vector 𝑎 𝑛 vanishes
if and only if 𝑛 = 0; moreover if 𝑛 ≠ 0 then 𝑎 𝑛 is an eigenvector of 𝑁 with
eigenvalue 𝑛 − 1.
Proof: We consider
𝑁 𝑎 𝑛 = 𝑁𝑎 𝑛 = 𝑎+ 𝑁 − 1 𝑛
= 𝑎 𝑛 − 1 𝑛 = 𝑛 − 1 [𝑎 𝑛 ]
where we have used quantum magic trick associated with the commutation
relation 𝑁, 𝑎 = −𝑎 to push 𝑁 through 𝑎 where it became 𝑁 − 1.
So 𝑎 𝑛 satisfies the eigenvalue equation with eigenvalue 𝑛 − 1, and its
length, as shown below, vanishes if and only if 𝑛 = 0,
〈𝑛|𝑎⁺𝑎|𝑛〉 = 〈𝑛|𝑁|𝑛〉 = 𝑛〈𝑛|𝑛〉 > 0
121
Spectrum of the Number Operator 3. Action of 𝑎 on 𝑛 :
If 𝑛 is an eigenvector of 𝑁 with eigenvalue 𝑛, then the vector 𝑎 𝑛 vanishes
if and only if 𝑛 = 0; moreover if 𝑛 ≠ 0 then 𝑎 𝑛 is an eigenvector of 𝑁 with
eigenvalue 𝑛 − 1.
Proof: We consider
𝑁 𝑎 𝑛 = 𝑁𝑎 𝑛 = 𝑎+ 𝑁 − 1 𝑛
= 𝑎 𝑛 − 1 𝑛 = 𝑛 − 1 [𝑎 𝑛 ]
where we have used quantum magic trick associated with the commutation
relation 𝑁, 𝑎 = −𝑎 to push 𝑁 through 𝑎 where it became 𝑁 − 1.
So 𝑎 𝑛 satisfies the eigenvalue equation with eigenvalue 𝑛 − 1, and its
length, as shown below, vanishes if and only if 𝑛 = 0,
〈𝑛|𝑎⁺𝑎|𝑛〉 = 〈𝑛|𝑁|𝑛〉 = 𝑛〈𝑛|𝑛〉 > 0
122
Spectrum of the Number Operator 3. Action of 𝑎 on 𝑛 :
If 𝑛 is an eigenvector of 𝑁 with eigenvalue 𝑛, then the vector 𝑎 𝑛 vanishes
if and only if 𝑛 = 0; moreover if 𝑛 ≠ 0 then 𝑎 𝑛 is an eigenvector of 𝑁 with
eigenvalue 𝑛 − 1.
Proof: We consider
𝑁 𝑎 𝑛 = 𝑁𝑎 𝑛 = 𝑎+ 𝑁 − 1 𝑛
= 𝑎 𝑛 − 1 𝑛 = 𝑛 − 1 [𝑎 𝑛 ]
where we have used quantum magic trick associated with the commutation
relation 𝑁, 𝑎 = −𝑎 to push 𝑁 through 𝑎 where it became 𝑁 − 1.
So 𝑎 𝑛 satisfies the eigenvalue equation with eigenvalue 𝑛 − 1, and its
squared length, shown below, vanishes if and only if 𝑛 = 0,
〈𝑛|𝑎⁺𝑎|𝑛〉 = 〈𝑛|𝑁|𝑛〉 = 𝑛〈𝑛|𝑛〉 > 0
123
Spectrum of the Number Operator 3. Action of 𝑎 on 𝑛 :
If 𝑛 is an eigenvector of 𝑁 with eigenvalue 𝑛, then the vector 𝑎 𝑛 vanishes
if and only if 𝑛 = 0; moreover if 𝑛 ≠ 0 then 𝑎 𝑛 is an eigenvector of 𝑁 with
eigenvalue 𝑛 − 1.
Proof: We consider
𝑁 𝑎 𝑛 = 𝑁𝑎 𝑛 = 𝑎+ 𝑁 − 1 𝑛
= 𝑎 𝑛 − 1 𝑛 = 𝑛 − 1 [𝑎 𝑛 ]
where we have used quantum magic trick associated with the commutation
relation 𝑁, 𝑎 = −𝑎 to push 𝑁 through 𝑎 where it became 𝑁 − 1.
So 𝑎 𝑛 satisfies the eigenvalue equation with eigenvalue 𝑛 − 1, and its
squared length, shown below, vanishes if and only if 𝑛 = 0,
〈𝑛|𝑎⁺𝑎|𝑛〉 = 〈𝑛|𝑁|𝑛〉 = 𝑛〈𝑛|𝑛〉 > 0
124
Spectrum of the Number Operator 3. Action of 𝑎 on 𝑛 :
If 𝑛 is an eigenvector of 𝑁 with eigenvalue 𝑛, then the vector 𝑎 𝑛 vanishes
if and only if 𝑛 = 0; moreover if 𝑛 ≠ 0 then 𝑎 𝑛 is an eigenvector of 𝑁 with
eigenvalue 𝑛 − 1.
Proof: We consider
𝑁 𝑎 𝑛 = 𝑁𝑎 𝑛 = 𝑎+ 𝑁 − 1 𝑛
= 𝑎 𝑛 − 1 𝑛 = 𝑛 − 1 [𝑎 𝑛 ]
where we have used quantum magic trick associated with the commutation
relation 𝑁, 𝑎 = −𝑎 to push 𝑁 through 𝑎 where it became 𝑁 − 1.
So 𝑎 𝑛 satisfies the eigenvalue equation with eigenvalue 𝑛 − 1, and its
squared length, shown below, vanishes if and only if 𝑛 = 0,
〈𝑛|𝑎⁺𝑎|𝑛〉 = 〈𝑛|𝑁|𝑛〉 = 𝑛〈𝑛|𝑛〉 > 0
125
Spectrum of the Number Operator 3. Action of 𝑎 on 𝑛 :
If 𝑛 is an eigenvector of 𝑁 with eigenvalue 𝑛, then the vector 𝑎 𝑛 vanishes
if and only if 𝑛 = 0; moreover if 𝑛 ≠ 0 then 𝑎 𝑛 is an eigenvector of 𝑁 with
eigenvalue 𝑛 − 1.
Proof: We consider
𝑁 𝑎 𝑛 = 𝑁𝑎 𝑛 = 𝑎+ 𝑁 − 1 𝑛
= 𝑎 𝑛 − 1 𝑛 = 𝑛 − 1 [𝑎 𝑛 ]
where we have used quantum magic trick associated with the commutation
relation 𝑁, 𝑎 = −𝑎 to push 𝑁 through 𝑎 where it became 𝑁 − 1.
So 𝑎 𝑛 satisfies the eigenvalue equation with eigenvalue 𝑛 − 1, and its
squared length, shown below, vanishes if and only if 𝑛 = 0,
〈𝑛|𝑎⁺𝑎|𝑛〉 = 〈𝑛|𝑁|𝑛〉 = 𝑛〈𝑛|𝑛〉 > 0
126
Spectrum of the Number Operator 4. The eigenvalues of 𝑁 are the non-negative integers - We now assert that the
spectrum of the number operator 𝑁 consists of the numbers 𝑛 = 0,1,2, ⋯.
To prove this, assume that there exists a (necessarily positive) eigenvalue
𝑛 ≥ 0 of 𝑁 that is not in this set.
Given any nonzero eigenvector |𝑛〉 with this eigenvalue, we could then
produce a sequence of non-vanishing eigenstates of 𝑁
with eigenvalues
|𝑛〉 ,
𝑎|𝑛〉 , 𝑎2 |𝑛〉 , ⋯
𝑎𝑚 |𝑛〉 ⋯
𝑛,
𝑛 − 1,
𝑛 − 2, ⋯
𝑛−𝑚 ⋯
that decrease without bound, becoming negative when 𝑚 > 𝑛 .
This would contradict our proof that any eigenvalue must be non-negative.
Hence any eigenvalue of 𝑛 of 𝑁 must lie in the set 𝑛 = 0,1,2, ⋯.
127
Spectrum of the Number Operator 4. The eigenvalues of 𝑁 are the non-negative integers - We now assert that the
spectrum of the number operator 𝑁 consists of the numbers 𝑛 = 0,1,2, ⋯.
To prove this, assume that there exists a (necessarily positive) eigenvalue
𝑛 ≥ 0 of 𝑁 that is not in this set.
Given any nonzero eigenvector |𝑛〉 with this eigenvalue, we could then
produce a sequence of non-vanishing eigenstates of 𝑁
with eigenvalues
|𝑛〉 ,
𝑎|𝑛〉 , 𝑎2 |𝑛〉 , ⋯
𝑎𝑚 |𝑛〉 ⋯
𝑛,
𝑛 − 1,
𝑛 − 2, ⋯
𝑛−𝑚 ⋯
that decrease without bound, becoming negative when 𝑚 > 𝑛 .
This would contradict our proof that any eigenvalue must be non-negative.
Hence any eigenvalue of 𝑛 of 𝑁 must lie in the set 𝑛 = 0,1,2, ⋯.
128
Spectrum of the Number Operator 4. The eigenvalues of 𝑁 are the non-negative integers - We now assert that the
spectrum of the number operator 𝑁 consists of the numbers 𝑛 = 0,1,2, ⋯.
To prove this, assume that there exists a (necessarily positive) eigenvalue
𝑛 ≥ 0 of 𝑁 that is not in this set.
Given any nonzero eigenvector |𝑛〉 with this eigenvalue, we could then
produce a sequence of non-vanishing eigenstates of 𝑁
with eigenvalues
|𝑛〉 ,
𝑎|𝑛〉 , 𝑎2 |𝑛〉 , ⋯
𝑎𝑚 |𝑛〉 ⋯
𝑛,
𝑛 − 1,
𝑛 − 2, ⋯
𝑛−𝑚 ⋯
that decrease without bound, becoming negative when 𝑚 > 𝑛 .
This would contradict our proof that any eigenvalue must be non-negative.
Hence any eigenvalue of 𝑛 of 𝑁 must lie in the set 𝑛 = 0,1,2, ⋯.
129
Spectrum of the Number Operator 4. The eigenvalues of 𝑁 are the non-negative integers - We now assert that the
spectrum of the number operator 𝑁 consists of the numbers 𝑛 = 0,1,2, ⋯.
To prove this, assume that there exists a (necessarily positive) eigenvalue
𝑛 ≥ 0 of 𝑁 that is not in this set.
Given any nonzero eigenvector |𝑛〉 with this eigenvalue, we could then
produce a sequence of non-vanishing eigenstates of 𝑁
with eigenvalues
|𝑛〉 ,
𝑎|𝑛〉 , 𝑎2 |𝑛〉 , ⋯
𝑎𝑚 |𝑛〉 ⋯
𝑛,
𝑛 − 1,
𝑛 − 2, ⋯
𝑛−𝑚 ⋯
that decrease without bound, becoming negative when 𝑚 > 𝑛 .
This would contradict our proof that any eigenvalue must be non-negative.
Hence any eigenvalue of 𝑛 of 𝑁 must lie in the set 𝑛 = 0,1,2, ⋯.
130
Spectrum of the Number Operator 4. The eigenvalues of 𝑁 are the non-negative integers - We now assert that the
spectrum of the number operator 𝑁 consists of the numbers 𝑛 = 0,1,2, ⋯.
To prove this, assume that there exists a (necessarily positive) eigenvalue
𝑛 ≥ 0 of 𝑁 that is not in this set.
Given any nonzero eigenvector |𝑛〉 with this eigenvalue, we could then
produce a sequence of non-vanishing eigenstates of 𝑁
with eigenvalues
|𝑛〉 ,
𝑎|𝑛〉 , 𝑎2 |𝑛〉 , ⋯
𝑎𝑚 |𝑛〉 ⋯
𝑛,
𝑛 − 1,
𝑛 − 2, ⋯
𝑛−𝑚 ⋯
that decrease without bound, becoming negative when 𝑚 > 𝑛 .
This would contradict our proof that any eigenvalue must be non-negative.
Hence any eigenvalue of 𝑛 of 𝑁 must lie in the set 𝑛 = 0,1,2, ⋯.
131
Spectrum of the Number Operator 4. The eigenvalues of 𝑁 are the non-negative integers - We now assert that the
spectrum of the number operator 𝑁 consists of the numbers 𝑛 = 0,1,2, ⋯.
To prove this, assume that there exists a (necessarily positive) eigenvalue
𝑛 ≥ 0 of 𝑁 that is not in this set.
Given any nonzero eigenvector |𝑛〉 with this eigenvalue, we could then
produce a sequence of non-vanishing eigenstates of 𝑁
with eigenvalues
|𝑛〉 ,
𝑎|𝑛〉 , 𝑎2 |𝑛〉 , ⋯
𝑎𝑚 |𝑛〉 ⋯
𝑛,
𝑛 − 1,
𝑛 − 2, ⋯
𝑛−𝑚 ⋯
that decrease without bound, becoming negative when 𝑚 > 𝑛 .
This would contradict our proof that any eigenvalue must be non-negative.
Hence any eigenvalue of 𝑛 of 𝑁 must lie in the set 𝑛 = 0,1,2, ⋯.
132
Spectrum of the Number Operator 4. The eigenvalues of 𝑁 are the non-negative integers - We now assert that the
spectrum of the number operator 𝑁 consists of the numbers 𝑛 = 0,1,2, ⋯.
To prove this, assume that there exists a (necessarily positive) eigenvalue
𝑛 ≥ 0 of 𝑁 that is not in this set.
Given any nonzero eigenvector |𝑛〉 with this eigenvalue, we could then
produce a sequence of non-vanishing eigenstates of 𝑁
with eigenvalues
|𝑛〉 ,
𝑎|𝑛〉 , 𝑎2 |𝑛〉 , ⋯
𝑎𝑚 |𝑛〉 ⋯
𝑛,
𝑛 − 1,
𝑛 − 2, ⋯
𝑛−𝑚 ⋯
that decrease without bound, becoming negative when 𝑚 > 𝑛 .
This would contradict our proof that any eigenvalue must be non-negative.
Hence any eigenvalue of 𝑛 of 𝑁 must lie in the set 𝑛 = 0,1,2, ⋯.
133
Spectrum of the Number Operator 4. The eigenvalues of 𝑁 are the non-negative integers - We now assert that the
spectrum of the number operator 𝑁 consists of the numbers 𝑛 = 0,1,2, ⋯.
To prove this, assume that there exists a (necessarily positive) eigenvalue
𝑛 ≥ 0 of 𝑁 that is not in this set.
Given any nonzero eigenvector |𝑛〉 with this eigenvalue, we could then
produce a sequence of non-vanishing eigenstates of 𝑁
with eigenvalues
|𝑛〉 ,
𝑎|𝑛〉 , 𝑎2 |𝑛〉 , ⋯
𝑎𝑚 |𝑛〉 ⋯
𝑛,
𝑛 − 1,
𝑛 − 2, ⋯
𝑛−𝑚 ⋯
that decrease without bound, becoming negative when 𝑚 > 𝑛 .
This would contradict our proof that any eigenvalue must be non-negative.
Hence any eigenvalue of 𝑛 of 𝑁 must lie in the set 𝑛 = 0,1,2, ⋯.
134
Spectrum of the Number Operator 4. The eigenvalues of 𝑁 are the non-negative integers - We now assert that the
spectrum of the number operator 𝑁 consists of the numbers 𝑛 = 0,1,2, ⋯.
To prove this, assume that there exists a (necessarily positive) eigenvalue
𝑛 ≥ 0 of 𝑁 that is not in this set.
Given any nonzero eigenvector |𝑛〉 with this eigenvalue, we could then
produce a sequence of non-vanishing eigenstates of 𝑁
with eigenvalues
|𝑛〉 ,
𝑎|𝑛〉 , 𝑎2 |𝑛〉 , ⋯
𝑎𝑚 |𝑛〉 ⋯
𝑛,
𝑛 − 1,
𝑛 − 2, ⋯
𝑛−𝑚 ⋯
that decrease without bound, becoming negative when 𝑚 > 𝑛 .
This would contradict our proof that any eigenvalue must be non-negative.
Hence any eigenvalue of 𝑛 of 𝑁 must lie in the set 𝑛 = 0,1,2, ⋯.
135
Spectrum of the Number Operator 4. The eigenvalues of 𝑁 are the non-negative integers - We now assert that the
spectrum of the number operator 𝑁 consists of the numbers 𝑛 = 0,1,2, ⋯.
To prove this, assume that there exists a (necessarily positive) eigenvalue
𝑛 ≥ 0 of 𝑁 that is not in this set.
Given any nonzero eigenvector |𝑛〉 with this eigenvalue, we could then
produce a sequence of non-vanishing eigenstates of 𝑁
with eigenvalues
|𝑛〉 ,
𝑎|𝑛〉 , 𝑎2 |𝑛〉 , ⋯
𝑎𝑚 |𝑛〉 ⋯
𝑛,
𝑛 − 1,
𝑛 − 2, ⋯
𝑛−𝑚 ⋯
that decrease without bound, becoming negative when 𝑚 > 𝑛 .
This would contradict our proof that any eigenvalue must be non-negative.
Hence any eigenvalue of 𝑛 of 𝑁 must lie in the set 𝑛 = 0,1,2, ⋯.
136
Spectrum of the Number Operator 4. The eigenvalues of 𝑁 are the non-negative integers - We now assert that the
spectrum of the number operator 𝑁 consists of the numbers 𝑛 = 0,1,2, ⋯.
To prove this, assume that there exists a (necessarily positive) eigenvalue
𝑛 ≥ 0 of 𝑁 that is not in this set.
Given any nonzero eigenvector |𝑛〉 with this eigenvalue, we could then
produce a sequence of non-vanishing eigenstates of 𝑁
with eigenvalues
|𝑛〉 ,
𝑎|𝑛〉 , 𝑎2 |𝑛〉 , ⋯
𝑎𝑚 |𝑛〉 ⋯
𝑛,
𝑛 − 1,
𝑛 − 2, ⋯
𝑛−𝑚 ⋯
that decrease without bound, becoming negative when 𝑚 > 𝑛 .
This would contradict our proof that any eigenvalue must be non-negative.
Hence any eigenvalue of 𝑛 of 𝑁 must lie in the set 𝑛 = 0,1,2, ⋯.
137
Spectrum of the Number Operator 4. The eigenvalues of 𝑁 are the non-negative integers - We now assert that the
spectrum of the number operator 𝑁 consists of the numbers 𝑛 = 0,1,2, ⋯.
To prove this, assume that there exists a (necessarily positive) eigenvalue
𝑛 ≥ 0 of 𝑁 that is not in this set.
Given any nonzero eigenvector |𝑛〉 with this eigenvalue, we could then
produce a sequence of non-vanishing eigenstates of 𝑁
with eigenvalues
|𝑛〉 ,
𝑎|𝑛〉 , 𝑎2 |𝑛〉 , ⋯
𝑎𝑚 |𝑛〉 ⋯
𝑛,
𝑛 − 1,
𝑛 − 2, ⋯
𝑛−𝑚 ⋯
that decrease without bound, becoming negative when 𝑚 > 𝑛 .
This would contradict our proof that any eigenvalue must be non-negative.
Hence any eigenvalue of 𝑛 of 𝑁 must lie in the set 𝑛 = 0,1,2, ⋯.
138
Spectrum of the Number Operator 4. The eigenvalues of 𝑁 are the non-negative integers - We now assert that the
spectrum of the number operator 𝑁 consists of the numbers 𝑛 = 0,1,2, ⋯.
To prove this, assume that there exists a (necessarily positive) eigenvalue
𝑛 ≥ 0 of 𝑁 that is not in this set.
Given any nonzero eigenvector |𝑛〉 with this eigenvalue, we could then
produce a sequence of non-vanishing eigenstates of 𝑁
with eigenvalues
|𝑛〉 ,
𝑎|𝑛〉 , 𝑎2 |𝑛〉 , ⋯
𝑎𝑚 |𝑛〉 ⋯
𝑛,
𝑛 − 1,
𝑛 − 2, ⋯
𝑛−𝑚 ⋯
that decrease without bound, becoming negative when 𝑚 > 𝑛 .
This would contradict our proof that any eigenvalue must be non-negative.
Hence any eigenvalue of 𝑛 of 𝑁 must lie in the set 𝑛 = 0,1,2, ⋯.
139
Spectrum of the Number Operator 4. The eigenvalues of 𝑁 are the non-negative integers - We now assert that the
spectrum of the number operator 𝑁 consists of the numbers 𝑛 = 0,1,2, ⋯.
To prove this, assume that there exists a (necessarily positive) eigenvalue
𝑛 ≥ 0 of 𝑁 that is not in this set.
Given any nonzero eigenvector |𝑛〉 with this eigenvalue, we could then
produce a sequence of non-vanishing eigenstates of 𝑁
with eigenvalues
|𝑛〉 ,
𝑎|𝑛〉 , 𝑎2 |𝑛〉 , ⋯
𝑎𝑚 |𝑛〉 ⋯
𝑛,
𝑛 − 1,
𝑛 − 2, ⋯
𝑛−𝑚 ⋯
that decrease without bound, becoming negative when 𝑚 > 𝑛 .
This would contradict our proof that any eigenvalue must be non-negative.
Hence any eigenvalue 𝑛 of 𝑁 must lie in the set 𝑛 = 0,1,2, ⋯.
140
Spectrum of the Number Operator 4. The eigenvalues of 𝑁 are the non-negative integers – For any eigenvalue 𝑛
that does lie in the set of non-negative integers, the process naturally
terminates since, at the 𝑛th step, the vector 𝑎𝑛 |𝑛〉 is an eigenstate of 𝑁 with
eigenvalue 𝑛 − 𝑛 = 0. Application of 𝑎 to this state takes it onto the null
vector, which terminates the sequence.
In the process, states are produced with eigenvalues corresponding to all the
integers 𝑛 ≥ 𝑚 ≥ 0. Repeated application of the operator 𝑎⁺ to any one of
these will generate a sequence of eigenstates, with all integers greater than 𝑛.
Thus, all integers 𝑛 = 0,1,2, ⋯ must lie in the spectrum of the number
operator 𝑁. As a consequence of their effect on the eigenstates of 𝑁, the
operators 𝑎⁺ and 𝑎 are referred to, respectively, as raising and lowering
operators.
141
Spectrum of the Number Operator 4. The eigenvalues of 𝑁 are the non-negative integers – For any eigenvalue 𝑛
that does lie in the set of non-negative integers, the process naturally
terminates since, at the 𝑛th step, the vector 𝑎𝑛 |𝑛〉 is an eigenstate of 𝑁 with
eigenvalue 𝑛 − 𝑛 = 0. Application of 𝑎 to this state takes it onto the null
vector, which terminates the sequence.
In the process, states are produced with eigenvalues corresponding to all the
integers 𝑛 ≥ 𝑚 ≥ 0. Repeated application of the operator 𝑎⁺ to any one of
these will generate a sequence of eigenstates, with all integers greater than 𝑛.
Thus, all integers 𝑛 = 0,1,2, ⋯ must lie in the spectrum of the number
operator 𝑁. As a consequence of their effect on the eigenstates of 𝑁, the
operators 𝑎⁺ and 𝑎 are referred to, respectively, as raising and lowering
operators.
142
Spectrum of the Number Operator 4. The eigenvalues of 𝑁 are the non-negative integers – For any eigenvalue 𝑛
that does lie in the set of non-negative integers, the process naturally
terminates since, at the 𝑛th step, the vector 𝑎𝑛 |𝑛〉 is an eigenstate of 𝑁 with
eigenvalue 𝑛 − 𝑛 = 0. Application of 𝑎 to this state takes it onto the null
vector, which terminates the sequence.
In the process, states are produced with eigenvalues corresponding to all the
integers 𝑛 ≥ 𝑚 ≥ 0. Repeated application of the operator 𝑎⁺ to any one of
these will generate a sequence of eigenstates, with all integers greater than 𝑛.
Thus, all integers 𝑛 = 0,1,2, ⋯ must lie in the spectrum of the number
operator 𝑁. As a consequence of their effect on the eigenstates of 𝑁, the
operators 𝑎⁺ and 𝑎 are referred to, respectively, as raising and lowering
operators.
143
Spectrum of the Number Operator 4. The eigenvalues of 𝑁 are the non-negative integers – For any eigenvalue 𝑛
that does lie in the set of non-negative integers, the process naturally
terminates since, at the 𝑛th step, the vector 𝑎𝑛 |𝑛〉 is an eigenstate of 𝑁 with
eigenvalue 𝑛 − 𝑛 = 0. Application of 𝑎 to this state takes it onto the null
vector, which terminates the sequence.
In the process, states are produced with eigenvalues corresponding to all the
integers 𝑛 ≥ 𝑚 ≥ 0. Repeated application of the operator 𝑎⁺ to any one of
these will generate a sequence of eigenstates, with all integers greater than 𝑛.
Thus, all integers 𝑛 = 0,1,2, ⋯ must lie in the spectrum of the number
operator 𝑁. As a consequence of their effect on the eigenstates of 𝑁, the
operators 𝑎⁺ and 𝑎 are referred to, respectively, as raising and lowering
operators.
144
Spectrum of the Number Operator 4. The eigenvalues of 𝑁 are the non-negative integers – For any eigenvalue 𝑛
that does lie in the set of non-negative integers, the process naturally
terminates since, at the 𝑛th step, the vector 𝑎𝑛 |𝑛〉 is an eigenstate of 𝑁 with
eigenvalue 𝑛 − 𝑛 = 0. Application of 𝑎 to this state takes it onto the null
vector, which terminates the sequence.
In the process, states are produced with eigenvalues corresponding to all the
integers 𝑛 ≥ 𝑚 ≥ 0. Repeated application of the operator 𝑎⁺ to any one of
these will generate a sequence of eigenstates, with all integers greater than 𝑛.
Thus, all integers 𝑛 = 0,1,2, ⋯ must lie in the spectrum of the number
operator 𝑁. As a consequence of their effect on the eigenstates of 𝑁, the
operators 𝑎⁺ and 𝑎 are referred to, respectively, as raising and lowering
operators.
145
Spectrum of the Number Operator 4. The eigenvalues of 𝑁 are the non-negative integers – For any eigenvalue 𝑛
that does lie in the set of non-negative integers, the process naturally
terminates since, at the 𝑛th step, the vector 𝑎𝑛 |𝑛〉 is an eigenstate of 𝑁 with
eigenvalue 𝑛 − 𝑛 = 0. Application of 𝑎 to this state takes it onto the null
vector, which terminates the sequence.
In the process, states are produced with eigenvalues corresponding to all the
integers 𝑛 ≥ 𝑚 ≥ 0. Repeated application of the operator 𝑎⁺ to any one of
these will generate a sequence of eigenstates, with all integers greater than 𝑛.
Thus, all integers 𝑛 = 0,1,2, ⋯ must lie in the spectrum of the number
operator 𝑁. As a consequence of their effect on the eigenstates of 𝑁, the
operators 𝑎⁺ and 𝑎 are referred to, respectively, as raising and lowering
operators.
146
Spectrum of the Harmonic Oscillator Hamiltonian
From the arguments given above it is clear that
Spectrum 𝑁 = 𝑛 = {0,1,2, ⋯}
The eigenvalues of the 1D harmonic oscillator Hamiltonian are directly related to
those of the number operator, through the relation
which we now see form a set of equally spaced energy levels, separated from one
another by the vibrational quantum of energy ℏ𝜔, and which have a minimum or
ground state value
Sometimes referred to as the zero-point energy.
147
Spectrum of the Harmonic Oscillator Hamiltonian
From the arguments given above it is clear that
Spectrum 𝑁 = 𝑛 = {0,1,2, ⋯}
The eigenvalues of the 1D harmonic oscillator Hamiltonian are directly related to
those of the number operator, through the relation
which we now see form a set of equally spaced energy levels, separated from one
another by the vibrational quantum of energy ℏ𝜔, and which have a minimum or
ground state value
Sometimes referred to as the zero-point energy.
148
Spectrum of the Harmonic Oscillator Hamiltonian
From the arguments given above it is clear that
Spectrum 𝑁 = 𝑛 = {0,1,2, ⋯}
The eigenvalues of the 1D harmonic oscillator Hamiltonian are directly related to
those of the number operator, through the relation
which we now see form a “ladder” of equally spaced energy levels, separated by
the vibrational quantum of energy ℏ𝜔, and which have a minimum or ground
state energy value
Sometimes referred to as the zero-point energy.
149
Spectrum of the Harmonic Oscillator Hamiltonian
From the arguments given above it is clear that
Spectrum 𝑁 = 𝑛 = {0,1,2, ⋯}
The eigenvalues of the 1D harmonic oscillator Hamiltonian are directly related to
those of the number operator, through the relation
which we now see form a “ladder” of equally spaced energy levels, separated by
the vibrational quantum of energy ℏ𝜔, and which have a minimum or ground
state energy value
sometimes referred to as the zero-point energy.
150
In this lecture, we began a study of the quantum mechanical harmonic oscillator.
To treat this system using the algebraic method, we first introduced
dimensionless position and momentum operators, and then traded those and the
Hamiltonian in, for a number operator, and raising and lowering operators, the
latter of which which are not Hermitian, and are not even normal.
The product formed from the lowering and raising operator defines the number
operator, which shares eigenstates with the harmonic oscillator Hamiltonian.
Using only the commutation relations among this new set of three operators, we
deduced the spectrum of the number operator, and therefore have simply
determined the energy spectrum of the quantum harmonic oscillator without
solving any differential equations. Pretty good for a day’s work.
In the next lecture, we continue our investigation, and, among other things,
determine the degeneracy of each harmonic oscillator energy level.
151
In this lecture, we began a study of the quantum mechanical harmonic oscillator.
To treat this system using the algebraic method, we first introduced
dimensionless position and momentum operators, and then traded those and the
Hamiltonian in, for a number operator, and raising and lowering operators, the
latter two of which are not Hermitian, and are not even normal.
The product formed from the lowering and raising operator defines the number
operator, which shares eigenstates with the harmonic oscillator Hamiltonian.
Using only the commutation relations among this new set of three operators, we
deduced the spectrum of the number operator, and therefore have simply
determined the energy spectrum of the quantum harmonic oscillator without
solving any differential equations. Pretty good for a day’s work.
In the next lecture, we continue our investigation, and, among other things,
determine the degeneracy of each harmonic oscillator energy level.
152
In this lecture, we began a study of the quantum mechanical harmonic oscillator.
To treat this system using the algebraic method, we first introduced
dimensionless position and momentum operators, and then traded those and the
Hamiltonian in, for a number operator, and raising and lowering operators, the
latter two of which are not Hermitian, and are not even normal.
The product formed from the raising and lowering operator defines the number
operator, which shares eigenstates with the harmonic oscillator Hamiltonian.
Using only the commutation relations among this new set of three operators, we
deduced the spectrum of the number operator, and therefore have simply
determined the energy spectrum of the quantum harmonic oscillator without
solving any differential equations. Pretty good for a day’s work.
In the next lecture, we continue our investigation, and, among other things,
determine the degeneracy of each harmonic oscillator energy level.
153
In this lecture, we began a study of the quantum mechanical harmonic oscillator.
To treat this system using the algebraic method, we first introduced
dimensionless position and momentum operators, and then traded those and the
Hamiltonian in, for a number operator, and raising and lowering operators, the
latter two of which are not Hermitian, and are not even normal.
The product formed from the lowering and raising operator defines the number
operator, which shares eigenstates with the harmonic oscillator Hamiltonian.
Using only the commutation relations among this new set of three operators, we
deduced the spectrum of the number operator, and therefore completely
determined the energy spectrum of the quantum harmonic oscillator without
ever having solved any differential equations. Pretty good for a day’s work.
In the next lecture, we continue our investigation, and, among other things,
determine the degeneracy of each harmonic oscillator energy level.
154
In this lecture, we began a study of the quantum mechanical harmonic oscillator.
To treat this system using the algebraic method, we first introduced
dimensionless position and momentum operators, and then traded those and the
Hamiltonian in, for a number operator, and raising and lowering operators, the
latter two of which are not Hermitian, and are not even normal.
The product formed from the lowering and raising operator defines the number
operator, which shares eigenstates with the harmonic oscillator Hamiltonian.
Using only the commutation relations among this new set of three operators, we
deduced the spectrum of the number operator, and therefore completely
determined the energy spectrum of the quantum harmonic oscillator without
ever having solved any differential equations. Pretty good for a day’s work.
In the next lecture, we continue our investigation, and, among other things,
determine the degeneracy of each harmonic oscillator energy level.
155
In this lecture, we began a study of the quantum mechanical harmonic oscillator.
To treat this system using the algebraic method, we first introduced
dimensionless position and momentum operators, and then traded those and the
Hamiltonian in, for a number operator, and raising and lowering operators, the
latter two of which are not Hermitian, and are not even normal.
The product formed from the lowering and raising operator defines the number
operator, which shares eigenstates with the harmonic oscillator Hamiltonian.
Using only the commutation relations among this new set of three operators, we
deduced the spectrum of the number operator, and therefore completely
determined the energy spectrum of the quantum harmonic oscillator without
ever having solved any differential equations. Pretty good for a day’s work.
In the next lecture, we continue our investigation, and, among other things,
determine the degeneracy of each harmonic oscillator energy level.
156
157