Transcript powerpoint file

Quantum One: Lecture 4
Schrödinger's Wave Mechanics
for a Free Quantum Particle
In the last lecture we explored implications of Schrödinger's Mechanics
for the case in which the Hamiltonian
is independent of time.
Using the method of separation of variables we obtained separable solutions
to the Schrödinger equation that arise when the initial wave function is itself an
energy eigenfunction:
We discussed appropriate boundary conditions for bound states and
continuum states, and excluded those solutions that diverge at infinity.
Then, using the fact that the Schrödinger equation is first order in time,
and linear, we deduced a 3-step prescription for solving the initial value
problem:
Given: the Hamiltonian H = T + V (i.e., given V, independent of time)
and an arbitrary initial state
To find the wave function for times t > 0
1) Solve:
2) Find the initial amplitudes λn
3) Evolve:
In this lecture we try to make these ideas more concrete by considering the
simplest possible scalar potential energy field the particle could move in, i.e.,
which corresponds to a free quantum mechanical particle of mass m
Here free means "force free", since the classical force
on the particle obviously vanishes if V = 0.
In this limit the total energy operator H consists only of the kinetic energy operator
so the energy eigenvalue equation reduces for a free particle to the explicit form
In this lecture we try to make these ideas more concrete by considering the
simplest possible scalar potential energy field the particle could move in, i.e.,
which corresponds to a free quantum mechanical particle of mass m
Here free means "force free", since the classical force
on the particle obviously vanishes if V = 0.
In this limit the total energy operator H consists only of the kinetic energy operator
so the energy eigenvalue equation reduces for a free particle to the explicit form
In this lecture we try to make these ideas more concrete by considering the
simplest possible scalar potential energy field the particle could move in, i.e.,
which corresponds to a free quantum mechanical particle of mass m.
Here free means "force free", since the classical force
on the particle obviously vanishes if V = 0.
In this limit the total energy operator H consists only of the kinetic energy operator
so the energy eigenvalue equation reduces for a free particle to the explicit form
In this lecture we try to make these ideas more concrete by considering the
simplest possible scalar potential energy field the particle could move in, i.e.,
which corresponds to a free quantum mechanical particle of mass m.
Here free means "force free", since the classical force
on the particle obviously vanishes if V = 0.
In this limit the total energy operator H consists only of the kinetic energy operator
so the energy eigenvalue equation reduces for a free particle to the explicit form
In this lecture we try to make these ideas more concrete by considering the
simplest possible scalar potential energy field the particle could move in, i.e.,
which corresponds to a free quantum mechanical particle of mass m.
Here free means "force free", since the classical force
on the particle obviously vanishes if V = 0.
In this limit the total energy operator H consists only of the kinetic energy operator
so the energy eigenvalue equation reduces for a free particle to the explicit form
In this lecture we try to make these ideas more concrete by considering the
simplest possible scalar potential energy field the particle could move in, i.e.,
which corresponds to a free quantum mechanical particle of mass m.
Here free means "force free", since the classical force
on the particle obviously vanishes if V = 0.
In this limit the total energy operator H consists only of the kinetic energy operator
so the energy eigenvalue equation
In this lecture we try to make these ideas more concrete by considering the
simplest possible scalar potential energy field the particle could move in, i.e.,
which corresponds to a free quantum mechanical particle of mass m.
Here free means "force free", since the classical force
on the particle obviously vanishes if V = 0.
In this limit the total energy operator H consists only of the kinetic energy operator
so the energy eigenvalue equation reduces for a free particle to the explicit form
To simplify this energy eigenvalue equation for the free particle
multiply through by
. Then, introducing the constant,
so that
the energy eigenvalue equation above takes the form
Thus, the free particle energy eigenvalue equation reduces to
what is called Helmholtz equation.
To simplify this energy eigenvalue equation for the free particle
multiply through by
. Then, introducing the constant,
so that
the energy eigenvalue equation above takes the form
Thus, the free particle energy eigenvalue equation reduces to
what is called Helmholtz equation.
To simplify this energy eigenvalue equation for the free particle
multiply through by
. Then, introducing the constant,
in terms of which
the energy eigenvalue equation above takes the form
Thus, the free particle energy eigenvalue equation reduces to
what is called Helmholtz equation.
To simplify this energy eigenvalue equation for the free particle
multiply through by
. Then, introducing the constant,
in terms of which
the energy eigenvalue equation above takes the form
Thus, the free particle energy eigenvalue equation reduces to
what is called Helmholtz equation.
To simplify this energy eigenvalue equation for the free particle
multiply through by
. Then, introducing the constant,
in terms of which
the energy eigenvalue equation above takes the form
Thus, the free particle energy eigenvalue equation reduces to
what is called the Helmholtz equation.
The Helmholtz equation
can be separated in many different coordinates systems.
We will focus on applying the method of separation of variables using Cartesian
coordinates where it takes the form:
and assume separable solutions of the form
As before, we substitute into the Helmholtz and divide by φ=XYZ to obtain
The Helmholtz equation
can be separated in many different coordinates systems.
We will focus on applying the method of separation of variables using Cartesian
coordinates where it takes the form:
and assume separable solutions of the form
As before, we substitute into the Helmholtz and divide by φ=XYZ to obtain
The Helmholtz equation
can be separated in many different coordinates systems.
We will focus on applying the method of separation of variables using Cartesian
coordinates where it takes the form:
We then assume separable solutions of the form
As before, we substitute into the Helmholtz and divide by φ=XYZ to obtain
The Helmholtz equation
can be separated in many different coordinates systems.
We will focus on applying the method of separation of variables in Cartesian
coordinates where it takes the form:
We then assume separable solutions of the form
Substituting in, and dividing by φ=XYZ we find that
The Helmholtz equation
can be separated in many different coordinates systems.
We will focus on applying the method of separation of variables using Cartesian
coordinates where it takes the form:
We then assume separable solutions of the form
Substituting in, and dividing by φ=XYZ we find that
The Helmholtz equation
can be separated in many different coordinates systems.
We will focus on applying the method of separation of variables using Cartesian
coordinates where it takes the form:
and assume separable solutions of the form
Substituting in, and dividing by φ=XYZ we find that
The individual terms on the left must each equal a constant that add up to -k²
Thus we end up with three ordinary differential equations
which have the solutions
The product of these gives possible free particle energy eigenfunctions:
where
The individual terms on the left must each equal a constant that add up to -k²
Thus we end up with three ordinary differential equations
which have the solutions
The product of these gives possible free particle energy eigenfunctions:
where
The individual terms on the left must each equal a constant that add up to -k²
Thus we end up with three ordinary differential equations
which have the solutions
The product of these gives possible free particle energy eigenfunctions:
where
The individual terms on the left must each equal a constant that add up to -k²
Thus we end up with three ordinary differential equations
which have the solutions
The product of these gives possible free particle energy eigenfunctions:
where
The individual terms on the left must each equal a constant that add up to -k²
Thus we end up with three ordinary differential equations
which have the solutions
The product of these gives possible free particle energy eigenfunctions:
where
So we have found functional forms
that satisfy the eigenvalue equation. Next step?
We have to ask: for what values of
are these solutions acceptable?
Recall: We just need to exclude solutions that diverge in any direction.
But if the constant
has a nonzero imaginary part,
then the solution will diverge in some direction. You should verify that
So we have found functional forms
that satisfy the eigenvalue equation. Next step?
We have to ask: for what values of
are these solutions acceptable?
Recall: We just need to exclude solutions that diverge in any direction.
But if the constant
has a nonzero imaginary part,
then the solution will diverge in some direction. You should verify that
So we have found functional forms
that satisfy the eigenvalue equation. Next step?
We have to ask: for what values of
are these solutions acceptable?
Recall: We just need to exclude solutions that diverge in any direction.
But if the constant
has a nonzero imaginary part,
then the solution will diverge in some direction. You should verify that
So we have found functional forms
that satisfy the eigenvalue equation. Next step?
We have to ask: for what values of
are these solutions acceptable?
Recall: We just need to exclude solutions that diverge in any direction.
But if the constant
has a nonzero imaginary part,
then the solution will diverge in some direction.
You should verify that
So we have found functional forms
that satisfy the eigenvalue equation. Next step?
We have to ask: for what values of
are these solutions acceptable?
Recall: We just need to exclude solutions that diverge in any direction.
But if the constant
has a nonzero imaginary part,
then the solution will diverge in some direction.
You should verify that
If all three components are real (positive or negative), then
remains bounded. Thus the complete set of solutions is obtained by considering
all possible wavevectors
The corresponding energy eigenvalues for the free particle (or for the kinetic
energy operator, which is the same thing here) take the form
The continuous spectrum includes all positive energies, as in the classical theory.
If all three components are real (positive or negative), then
remains bounded. Thus the complete set of solutions is obtained by considering
all possible wavevectors
The corresponding energy eigenvalues for the free particle (or for the kinetic
energy operator, which is the same thing here) take the form
which includes all positive energies, as in the classical theory.
Combining each of our energy eigenfunctions with its corresponding time dependence,
we obtain the stationary solutions for the free particle, namely
where
Thus, the free particle energy eigenstates are plane waves traveling in the
direction associated with the wavevector
So having solved the energy eigenvalue problem for the free particle we are now
almost ready to solve the initial value problem for the free particle.
But we still have some work to do, which we will motivate with a few preliminary
comments.
Combining each of our energy eigenfunctions with its corresponding time dependence,
we obtain the stationary solutions for the free particle, namely
where
Thus, the free particle energy eigenstates are plane waves traveling in the
direction associated with the wavevector
So having solved the energy eigenvalue problem for the free particle we are now
almost ready to solve the initial value problem for the free particle.
But we still have some work to do, which we will motivate with a few preliminary
comments.
Combining each of our energy eigenfunctions with its corresponding time dependence,
we obtain the stationary solutions for the free particle, namely
where
Thus, the free particle energy eigenstates are plane waves traveling in the
direction associated with the wavevector
So, as usual, before proceeding we make a few comments on these results.
Combining each of our energy eigenfunctions with its corresponding time dependence,
we obtain the stationary solutions for the free particle, namely
where
Thus, the free particle energy eigenstates are plane waves traveling in the
direction associated with the wavevector
So, as usual, before proceeding we make a few comments on these results.
Combining each of our energy eigenfunctions with its corresponding time dependence,
we obtain the stationary solutions for the free particle, namely
where
Thus, the free particle energy eigenstates are plane waves traveling in the
direction associated with the wavevector
So, as usual, before proceeding we make a few comments on these results.
First, we observe that these free-particle energy eigenstates (or eigenstates of
the kinetic energy operator) are also eigenstates of the momentum operator,
which is a vector operator with components
The eigenvalue equation for the momentum operator takes the form
where for this vector operator the eigenvalue itself is also a vector.
First, we observe that these free-particle energy eigenstates (or eigenstates of
the kinetic energy operator) are also eigenstates of the momentum operator,
which is a vector operator with components
The eigenvalue equation for the momentum operator takes the form
where for this vector operator the eigenvalue itself is also a vector.
First, we observe that these free-particle energy eigenstates (or eigenstates of
the kinetic energy operator) are also eigenstates of the momentum operator,
which is a vector operator with components
The eigenvalue equation for the momentum operator takes the form
where for this vector operator the eigenvalue itself is also a vector.
First, we observe that these free-particle energy eigenstates (or eigenstates of
the kinetic energy operator) are also eigenstates of the momentum operator,
which is a vector operator with components
The eigenvalue equation for the momentum operator takes the form
where the eigenvalue itself is a vector.
Our plane wave solutions satisfy the momentum eigenvalue equation, since, e.g.,
For all three components, this implies that
or
where
Our plane wave solutions satisfy the momentum eigenvalue equation, since, e.g.,
For all three components, this implies that
or
where
Our plane wave solutions satisfy the momentum eigenvalue equation, since, e.g.,
For all three components, this implies that
or
where
Our plane wave solutions satisfy the momentum eigenvalue equation, since, e.g.,
For all three components, this implies that
or
where
Thus, Schrödinger's wave mechanics recovers, as a special case, the hypothesis
of deBroglie:
With every free material particle
of momentum
and energy
we can associate a plane wave of
wavevector
wavelength
and frequency
Thus, Schrödinger's wave mechanics recovers, as a special case, the hypothesis
of deBroglie:
With every free material particle
of momentum
and energy
we can associate a plane wave of
wavevector
wavelength
and frequency
Thus, Schrödinger's wave mechanics recovers, as a special case, the hypothesis
of deBroglie:
With every free material particle
of momentum
and energy
we can associate a plane wave of
wavevector
wavelength
and frequency
Thus, Schrödinger's wave mechanics recovers, as a special case, the hypothesis
of deBroglie:
With every free material particle
of momentum
and energy
we can associate a plane wave of
wavevector
wavelength
and frequency
The second comment, is that the probabilistic predictions contained in the Third
Postulate, clearly depend on appropriate normalization conditions imposed upon
the wave function (e.g, that it be square normalized) and upon the eigenfunctions
of the observable of interest.
We have also asserted that eigenfunctions associated with continuous eigenvalues
are not square normalizable, so we will need a mathematically appropriate
normalization convention to deal with that situation.
The energy eigenfunctions of the free particle, which has a positive, continuous
energy spectrum, clearly fall into this second class.
To proceed further, which we will do in the next lecture, we need to address these
generalized normalization conditions for observables with a continuous spectrum.
The second comment, is that the probabilistic predictions contained in the Third
Postulate, clearly depend on appropriate normalization conditions imposed upon
the wave function (e.g, that it be square normalized) and upon the eigenfunctions
of the observable of interest.
We have also asserted that eigenfunctions associated with continuous eigenvalues
are not square normalizable, so we will need a mathematically appropriate
normalization convention to deal with that situation.
The energy eigenfunctions of the free particle, which has a positive, continuous
energy spectrum, clearly fall into this second class.
To proceed further, which we will do in the next lecture, we need to address these
generalized normalization conditions for observables with a continuous spectrum.
The second comment, is that the probabilistic predictions contained in the Third
Postulate, clearly depend on appropriate normalization conditions imposed upon
the wave function (e.g, that it be square normalized) and upon the eigenfunctions
of the observable of interest.
We have also asserted that eigenfunctions associated with continuous eigenvalues
are not square normalizable, so we will need a mathematically appropriate
normalization convention to deal with that situation.
The energy eigenfunctions of the free particle, which has a positive, continuous
energy spectrum, clearly fall into this class.
To proceed, therefore, we need to consider
normalization conventions for free particle eigenfunctions.
To proceed, therefore, we need to consider
normalization conventions for free particle eigenfunctions.
Once we do so, we will have all the mathematical tools we will need to treat
the initial value problem for the free particle.
To proceed, therefore, we need to consider
normalization conventions for free particle eigenfunctions.
Once we do so, we will have all the mathematical tools we will need to treat
the initial value problem for the free particle.
This critical extension is covered in the next lecture.