Transcript powerpoint file

Quantum One
1
2
Different Formulations of Quantum Mechanics
3
As it turns out, we have developed our postulates of the general formulation of
quantum mechanics by working implicitly in what is called the Schrödinger
picture. In this picture, the state of the system evolves in time
while fundamental observables of the system are usually associated with timeindependent Hermitian operators
except, for example, explicitly time-dependent perturbations
usually constructed out of time-independent observables.
, which are
4
As it turns out, we have developed our postulates of the general formulation of
quantum mechanics by working implicitly in what is called the Schrödinger
picture. In this picture, the state of the system evolves in time
while fundamental observables of the system are usually associated with timeindependent Hermitian operators
except, for example, explicitly time-dependent perturbations
usually constructed out of time-independent observables.
, which are
5
As it turns out, we have developed our postulates of the general formulation of
quantum mechanics by working implicitly in what is called the Schrödinger
picture. In this picture, the state of the system evolves in time
while fundamental observables of the system are usually associated with timeindependent Hermitian operators
except, for example, explicitly time-dependent perturbations
usually constructed out of time-independent observables.
, which are
6
As it turns out, we have developed our postulates of the general formulation of
quantum mechanics by working implicitly in what is called the Schrödinger
picture. In this picture, the state of the system evolves in time
while fundamental observables of the system are usually associated with timeindependent Hermitian operators
Exceptions to this general rule include, for example, explicitly time-dependent
perturbations
, but these are usually constructed as time-dependent
operator-valued functions of the more basic time-independent observables.
7
On the other hand, there are a number of other pictures of quantum mechanics,
all of which are ultimately equivalent to the Schrödinger picture, but which
sometimes provide a more useful starting point for certain classes of problems.
These different pictures or formulations can usually be related to the standard
Schrödinger picture through consideration of the evolution operator 𝑈(𝑡, 𝑡0 ).
It is useful therefore to review some of the main features associated with this
very important operator.
8
On the other hand, there are a number of other pictures of quantum mechanics,
all of which are ultimately equivalent to the Schrödinger picture, but which
sometimes provide a more useful starting point for certain classes of problems.
These different pictures or formulations can usually be related to the standard
Schrödinger picture through consideration of the evolution operator 𝑈(𝑡, 𝑡0 ).
It is useful therefore to review some of the main features associated with this
very important operator.
9
On the other hand, there are a number of other pictures of quantum mechanics,
all of which are ultimately equivalent to the Schrödinger picture, but which
sometimes provide a more useful starting point for certain classes of problems.
These different pictures or formulations can usually be related to the standard
Schrödinger picture through consideration of the evolution operator 𝑈(𝑡, 𝑡0 ).
It is useful therefore to review some of the main features associated with this
very important operator.
10
1. It is unitary, i.e.,
2. It obeys a simple composition rule
3. It is smoothly connected to the identity operator
4. It obeys an operator form of the Schrödinger equation
5. The evolution operator for a time-independent Hamiltonian H₀ is
11
1. It is unitary, i.e.,
2. It obeys a simple composition rule
3. It is smoothly connected to the identity operator
4. It obeys an operator form of the Schrödinger equation
5. The evolution operator for a time-independent Hamiltonian H₀ is
12
1. It is unitary, i.e.,
2. It obeys a simple composition rule
3. It is smoothly connected to the identity operator
4. It obeys an operator form of the Schrödinger equation
5. The evolution operator for a time-independent Hamiltonian H₀ is
13
1. It is unitary, i.e.,
2. It obeys a simple composition rule
3. It is smoothly connected to the identity operator
4. It obeys an operator form of the Schrödinger equation
5. The evolution operator for a time-independent Hamiltonian H₀ is
14
1. It is unitary, i.e.,
2. It obeys a simple composition rule
3. It is smoothly connected to the identity operator
4. It obeys an operator form of the Schrödinger equation
5. The evolution operator for a time-independent Hamiltonian H₀ is
15
Now, in the Schrödinger picture mean values of even time-independent
observables can evolve in time
due to the evolution of the state itself.
16
Now, in the Schrödinger picture mean values of even time-independent
observables can evolve in time
due to the evolution of the state itself.
But we can always express these in terms of expectation values taken with
respect to the initial state by relating the state at time t to the initial state using
the evolution operator.
17
Now, in the Schrödinger picture mean values of even time-independent
observables can evolve in time
due to the evolution of the state itself.
But we can always express these in terms of expectation values taken with
respect to the initial state by relating the state at time t to the initial state using
the evolution operator.
18
By contrast, it is possible to develop a different formulation of quantum
mechanics, referred to as the Heisenberg picture, in which the Heisenberg state
of the system
remains fixed in time, and equal to its initial state, which is the same as in the
Schrödinger picture.
But in the Heisenberg picture fundamental observables are now associated with
Heisenberg operators that evolve in time, and which are defined in terms of the
corresponding Schrödinger operators through the relation
.
19
By contrast, it is possible to develop a different formulation of quantum
mechanics, referred to as the Heisenberg picture, in which the Heisenberg state
of the system
remains fixed in time, and equal to its initial state, which is the same as in the
Schrödinger picture.
But in the Heisenberg picture fundamental observables are now associated with
Heisenberg operators that evolve in time, and which are defined in terms of the
corresponding Schrödinger operators through the relation
.
20
By contrast, it is possible to develop a different formulation of quantum
mechanics, referred to as the Heisenberg picture, in which the Heisenberg state
of the system
remains fixed in time, and equal to its initial state, which is the same as in the
Schrödinger picture.
But in the Heisenberg picture, fundamental observables are now associated with
Heisenberg operators that evolve in time, and which are defined in terms of the
corresponding Schrödinger operators through the relation
.
21
By contrast, it is possible to develop a different formulation of quantum
mechanics, referred to as the Heisenberg picture, in which the Heisenberg state
of the system
remains fixed in time, and equal to its initial state, which is the same as in the
Schrödinger picture.
But in the Heisenberg picture, fundamental observables are now associated with
Heisenberg operators that evolve in time, and which are defined in terms of the
corresponding Schrödinger operators through the relation
.
22
The kets and operators of one picture are related to those of the other through
the unitary transformation induced by the evolution operator 𝑈(𝑡, 𝑡0 ) and its
adjoint, which preserve all mean values, i.e.,
is exactly the same at each instant as in the Schrödinger picture.
Hence the predictions of quantum mechanics are exactly the same in these two
different pictures.
23
The kets and operators of one picture are related to those of the other through
the unitary transformation induced by the evolution operator 𝑈(𝑡, 𝑡0 ) and its
adjoint, which preserve all mean values, i.e.,
is exactly the same at each instant as in the Schrödinger picture.
Hence the predictions of quantum mechanics are exactly the same in these two
different pictures.
24
The kets and operators of one picture are related to those of the other through
the unitary transformation induced by the evolution operator 𝑈(𝑡, 𝑡0 ) and its
adjoint, which preserve all mean values, i.e.,
is exactly the same at each instant as in the Schrödinger picture.
Hence the predictions of quantum mechanics are exactly the same in these two
different pictures.
25
The kets and operators of one picture are related to those of the other through
the unitary transformation induced by the evolution operator 𝑈(𝑡, 𝑡0 ) and its
adjoint, which preserve all mean values, i.e.,
is exactly the same at each instant as in the Schrödinger picture.
Hence the predictions of quantum mechanics are exactly the same in these two
different pictures.
26
The kets and operators of one picture are related to those of the other through
the unitary transformation induced by the evolution operator 𝑈(𝑡, 𝑡0 ) and its
adjoint, which preserve all mean values, i.e.,
is exactly the same at each instant as in the Schrödinger picture.
Hence the predictions of quantum mechanics are exactly the same in these two
different pictures.
27
In this same spirit, it is possible to develop a formulation in which some of the
time evolution is associated with the kets of the system and some of it associated
with the operators of interest.
An interaction picture of this sort can be defined for any system in which the
Hamiltonian can be written in the form
with the state vector of
this picture
being defined relative to that of the Schrödinger picture through the inverse of
the unitary transformation
which evolves the system in the absence of the perturbation.
28
In this same spirit, it is possible to develop a formulation in which some of the
time evolution is associated with the kets of the system and some of it associated
with the operators of interest.
An interaction picture of this sort can be defined for any system in which the
Hamiltonian can be written in the form
with the state vector of
this picture
being defined relative to that of the Schrödinger picture through the inverse of
the unitary transformation
which evolves the system in the absence of the perturbation.
29
In this same spirit, it is possible to develop a formulation in which some of the
time evolution is associated with the kets of the system and some of it associated
with the operators of interest.
An interaction picture of this sort can be defined for any system in which the
Hamiltonian can be written in the form
with the state vector of
this picture
being defined relative to that of the Schrödinger picture through the inverse of
the unitary transformation
which evolves the system in the absence of the perturbation.
30
In this same spirit, it is possible to develop a formulation in which some of the
time evolution is associated with the kets of the system and some of it associated
with the operators of interest.
An interaction picture of this sort can be defined for any system in which the
Hamiltonian can be written in the form
with the state vector of
this picture
obtained from that of the Schrödinger picture through the inverse of the unitary
transformation
which evolves the system in the absence of the perturbation.
31
This form for the state vector suggests that the inverse (or adjoint) operator 𝑈0+
acts on the fully-evolving state vector of the Schrödinger picture to "back out" or
undo the fast evolution associated with the unperturbed part of the Hamiltonian.
In a similar fashion, the operators
of the interaction picture are related to those of the Schrödinger picture through
the same corresponding unitary transformation, but as applies to operators,
rather than states.
Note that we have included a time dependence in the Schrodinger operator A(t)
on the right to take into account any intrinsic time dependence exhibited, e.g., by
a time dependent perturbing field
.
32
This form for the state vector suggests that the inverse (or adjoint) operator 𝑈0+
acts on the fully-evolving state vector of the Schrödinger picture to "back out" or
undo the fast evolution associated with the unperturbed part of the Hamiltonian.
In a similar fashion, the operators of the interaction picture
are related to those of the Schrödinger picture through the same corresponding
unitary transformation, but as it applies to operators, rather than states.
Note that we have included a time dependence in the Schrodinger operator A(t)
on the right to take into account any intrinsic time dependence exhibited, e.g., by
a time dependent perturbing field
.
33
This form for the state vector suggests that the inverse (or adjoint) operator 𝑈0+
acts on the fully-evolving state vector of the Schrödinger picture to "back out" or
undo the fast evolution associated with the unperturbed part of the Hamiltonian.
In a similar fashion, the operators of the interaction picture
are related to those of the Schrödinger picture through the same corresponding
unitary transformation, but as it applies to operators, rather than states.
Note that we have included a time dependence in the Schrodinger operator A(t)
on the right to take into account any intrinsic time dependence exhibited, e.g., by
a weak time dependent perturbing field
.
34
Naturally, we can define an evolution operator
for the interaction
picture that evolves the state vector
in time, according to the relation
where
is the same as
since
.
Using the definitions given above we deduce that since
which makes it clear that
35
Naturally, we can define an evolution operator
for the interaction
picture that evolves the state vector
in time, according to the relation
where
is the same as
since
.
Using the definitions given above we deduce that since
which makes it clear that
36
Naturally, we can define an evolution operator
for the interaction
picture that evolves the state vector
in time, according to the relation
where
is the same as
since
.
Using the definitions given above we deduce that since
which makes it clear that
37
Naturally, we can define an evolution operator
for the interaction
picture that evolves the state vector
in time, according to the relation
where
is the same as
since
.
Using the definitions given above we deduce that since
which makes it clear that
38
Naturally, we can define an evolution operator
for the interaction
picture that evolves the state vector
in time, according to the relation
where
is the same as
since
.
Using the definitions given above we deduce that since
which makes it clear that
39
Naturally, we can define an evolution operator
for the interaction
picture that evolves the state vector
in time, according to the relation
where
is the same as
since
.
Using the definitions given above we deduce that since
which makes it clear that
40
Multiplying this last equation through by U₀(t,t₀), we obtain a result for the full
evolution operator
in terms of the evolution operators
and
.
To obtain information about transitions between the unperturbed eigenstates of
H₀, then, we need transition amplitudes
and transition probabilities
41
Multiplying this last equation through by
evolution operator
in terms of the evolution operators
and
we obtain a result for the full
.
To obtain information about transitions between the unperturbed eigenstates of
H₀, then, we need transition amplitudes
and transition probabilities
42
The evolution equation obeyed by
is also straightforward to obtain.
By taking derivatives of
we establish (with t₀ fixed) that
But clearly
and
Making these substitutions,
the parts involving
disappear, and we obtain . . .
43
The evolution equation obeyed by
is also straightforward to obtain.
By taking derivatives of
we establish (with t₀ fixed) that
But clearly
and
Making these substitutions,
the parts involving
disappear, and we obtain . . .
44
The evolution equation obeyed by
is also straightforward to obtain.
By taking derivatives of
we establish (with t₀ fixed) that
But clearly
and
Making these substitutions,
the parts involving
disappear, and we obtain . . .
45
The evolution equation obeyed by
is also straightforward to obtain.
By taking derivatives of
we establish (with t₀ fixed) that
But clearly
and
Making these substitutions,
the parts involving
disappear, and we obtain . . .
46
The evolution equation obeyed by
is also straightforward to obtain.
By taking derivatives of
we establish (with t₀ fixed) that
But clearly
and
Making these substitutions,
the parts involving
disappear, and we obtain . . .
47
The evolution equation obeyed by
is also straightforward to obtain.
By taking derivatives of
we establish (with t₀ fixed) that
But clearly
and
Making these substitutions,
the parts involving
disappear, and we obtain . . .
48
The evolution equation obeyed by
is also straightforward to obtain.
By taking derivatives of
we establish (with t₀ fixed) that
But clearly
and
Making these substitutions,
the parts involving
disappear, and we obtain . . .
49
The evolution equation obeyed by
is also straightforward to obtain.
By taking derivatives of
we establish (with t₀ fixed) that
But clearly
and
Making these substitutions,
the parts involving
disappear, and we obtain . . .
50
which we can write as
Thus, the evolution operator in the interaction picture evolves under a
Schrödinger equation that is governed by a Hamiltonian
that only includes the perturbing part of the Hamiltonian (the interaction), as it is
naturally represented in this interaction picture.
51
which we can write as
Thus, the evolution operator in the interaction picture evolves under a
Schrödinger equation that is governed by a Hamiltonian
that only includes the perturbing part of the Hamiltonian (the interaction), as it is
naturally represented in this interaction picture.
52
which we can write as
Thus, the evolution operator in the interaction picture evolves under a
Schrödinger equation that is governed by a Hamiltonian
that only includes the perturbing part of the Hamiltonian (the interaction), as it is
naturally represented in this interaction picture.
53
which we can write as
Thus, the evolution operator in the interaction picture evolves under a
Schrödinger equation that is governed by a Hamiltonian
that only includes the perturbing part of the total Hamiltonian (i.e., the
interaction), as it is naturally represented in this interaction picture.
54
Since U_{I}(t,t₀) shares the limiting behavior
of any evolution operator, it obeys the integral equation that we derived earlier
for evolution operators governed by a time-dependent Hamiltonian, i.e.,
and hence can be iterated to obtain an expansion in (integrated) powers of the
perturbation. For example, iterating one time, gives
55
Since U_{I}(t,t₀) shares the limiting behavior
of any evolution operator, it obeys the integral equation that we derived earlier
for evolution operators governed by a time-dependent Hamiltonian, i.e.,
and hence can be iterated to obtain an expansion in (integrated) powers of the
perturbation. For example, iterating one time, gives
56
Since U_{I}(t,t₀) shares the limiting behavior
of any evolution operator, it obeys the integral equation that we derived earlier
for evolution operators governed by a time-dependent Hamiltonian, i.e.,
and hence can be iterated to obtain an expansion in (integrated) powers of the
perturbation. For example, iterating one time, gives
57
Since U_{I}(t,t₀) shares the limiting behavior
of any evolution operator, it obeys the integral equation that we derived earlier
for evolution operators governed by a time-dependent Hamiltonian, i.e.,
and hence can be iterated to obtain an expansion in (integrated) powers of the
perturbation. For example, iterating one time, gives
58
If the perturbation is small enough, this formal expansion for the evolution
operator can be truncated after the first order term, i.e.,
and as such allows us to address in a perturbative sense the problem originally
posed.
Thus we can substitute into our previous expression for the transition amplitude
and for m ≠ n the part associated with the identity operator vanishes and we
obtain the result . . .
59
If the perturbation is small enough, this formal expansion for the evolution
operator can be truncated after the first order term, i.e.,
which is the main starting point for time-dependent perturbation theory.
60
So in this lecture we saw that there are a number of different formulations of
quantum mechanics, all of which make the same predictions regarding the
possible outcomes of experiment, but adopt different viewpoints regarding
1) how a quantum system evolves from one moment to the next, and
2) exactly what the objects are that actually evolve in the first place.
In the next lecture, we consider another very different picture of quantum
mechanics referred to as the path integral formulation developed by Feynman.
This approach, once properly formulated, seems to suggest that the answer to
the Question: “How does a quantum particle get from one point to another?”
is
“It somehow seems to simultaneously take all possible paths
connecting those two points.”
61
So in this lecture we saw that there are a number of different formulations of
quantum mechanics, all of which make the same predictions regarding the
possible outcomes of experiment, but adopt different viewpoints regarding
1) how a quantum system evolves from one moment to the next, and
2) exactly what the objects are that actually evolve in the first place.
In the next lecture, we consider another very different picture of quantum
mechanics referred to as the path integral formulation developed by Feynman.
This approach, once properly formulated, seems to suggest that the answer to
the Question: “How does a quantum particle get from one point to another?”
is
“It somehow seems to simultaneously take all possible paths
connecting those two points.”
62
So in this lecture we saw that there are a number of different formulations of
quantum mechanics, all of which make the same predictions regarding the
possible outcomes of experiment, but adopt different viewpoints regarding
1) how a quantum system evolves from one moment to the next, and
2) exactly what the objects are that actually evolve in the first place.
In the next lecture, we consider another very different picture of quantum
mechanics referred to as the path integral formulation developed by Feynman.
This approach, once properly formulated, seems to suggest that the answer to
the Question: “How does a quantum particle get from one point to another?”
is
“It somehow seems to simultaneously take all possible paths
connecting those two points.”
63
So in this lecture we saw that there are a number of different formulations of
quantum mechanics, all of which make the same predictions regarding the
possible outcomes of experiment, but adopt different viewpoints regarding
1) how a quantum system evolves from one moment to the next, and
2) exactly what the objects are that actually evolve in the first place.
In the next lecture, we consider another very different picture of quantum
mechanics referred to as the path integral formulation developed by Feynman.
This approach, once properly formulated, seems to suggest that the answer to
the Question: “How does a quantum particle get from one point to another?”
is
“It somehow seems to simultaneously take all possible paths
connecting those two points.”
64
So in this lecture we saw that there are a number of different formulations of
quantum mechanics, all of which make the same predictions regarding the
possible outcomes of experiment, but adopt different viewpoints regarding
1) how a quantum system evolves from one moment to the next, and
2) exactly what the objects are that actually evolve in the first place.
In the next lecture, we consider another very different picture of quantum
mechanics referred to as the path integral formulation developed by Feynman.
This approach, once properly formulated, seems to suggest that the answer to
the Question: “How does a quantum particle get from one point to another?”
is
“It somehow seems to simultaneously take all possible paths
connecting those two points.”
65
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