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 mHere 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 mHere 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 nowalmost 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 nowalmost 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 ThirdPostulate, clearly depend on appropriate normalization conditions imposed uponthe 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 thesegeneralized normalization conditions for observables with a continuous spectrum.

The second comment, is that the probabilistic predictions contained in the ThirdPostulate, clearly depend on appropriate normalization conditions imposed uponthe 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 thesegeneralized normalization conditions for observables with a continuous spectrum.

The second comment, is that the probabilistic predictions contained in the ThirdPostulate, clearly depend on appropriate normalization conditions imposed uponthe 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.