Quantum One: Lecture 6

The Initial Value Problem for Free Particles, and the Emergence of Fourier Transforms.

Having determined a complete, appropriately normalized set of free particle eigenstates

and the associated energy eigenvalues

we have completed the first step in solving the initial value problem for a free particle.

To proceed, we need to carry out the second step: find the amplitudes that allow us to expand the initial state as a linear superposition of energy or momentum eigenfunctions (which, as we have seen, are the same thing for a free particle).

Recall, that if we had a set of energy eigenfunctions characterized by a discrete index n, we would write the expansion of an arbitrary state as a discrete sum over the index n, i.e.

For our free particle eigenfunctions which are characterized by a continuous (vector) index, the expansion takes the form of an integral over all wavevectors:

It will be useful going forward to alter our notation slightly, so that we denote the amplitude function in this last expression by the symbol

Recall, that if we had a set of energy eigenfunctions characterized by a discrete index n, we would write the expansion of an arbitrary state as a discrete sum over the index n, i.e.

For our free particle eigenfunctions which are characterized by a continuous (vector) index, the expansion takes the form of an integral over all wavevectors:

It will be useful going forward to alter our notation slightly, so that we denote the amplitude function in this last expression by the symbol

Recall, that if we had a set of energy eigenfunctions characterized by a discrete index n, we would write the expansion of an arbitrary state as a discrete sum over the index n, i.e.

For our free particle eigenfunctions which are characterized by a continuous (vector) index, the expansion takes the form of an integral over all wavevectors:

It will be useful going forward to alter our notation slightly, so that we denote the amplitude function in this last expression by the symbol

Recall, that if we had a set of energy eigenfunctions characterized by a discrete index n, we would write the expansion of an arbitrary state as a discrete sum over the index n, i.e.

For our free particle eigenfunctions which are characterized by a continuous (vector) index, the expansion takes the form of an integral over all wavevectors:

It will be useful going forward to alter our notation slightly, so that we denote the amplitude function in this last expression by the symbol

Recall, that if we had a set of energy eigenfunctions characterized by a discrete index n, we would write the expansion of an arbitrary state as a discrete sum over the index n, i.e.

For our free particle eigenfunctions which are characterized by a continuous (vector) index, the expansion takes the form of an integral over all wavevectors:

It will be useful going forward to alter our notation slightly, so that we denote the amplitude function in this last expression by the symbol

With this convention, the continuous plane-wave expansion of an arbitrary wave function then takes the form

Written in this form, the function is clearly the amplitude that a measurement of the wavevector of the particle will yield the vector , or that a measurement of momentum will yield the value .

The corresponding probability density is

With this convention, the continuous plane-wave expansion of an arbitrary wave function then takes the form

Written in this form, the function is clearly the amplitude that a measurement of the wavevector of the particle will yield the vector , or that a measurement of momentum will yield the value .

The corresponding probability density is

With this convention, the continuous plane-wave expansion of an arbitrary wave function then takes the form

Written in this form, the function is clearly the amplitude that a measurement of the wavevector of the particle will yield the vector , or that a measurement of momentum will yield the value .

The corresponding probability density is

With this convention, the continuous plane-wave expansion of an arbitrary wave function then takes the form

Written in this form, the function is clearly the amplitude that a measurement of the wavevector of the particle will yield the vector , or that a measurement of momentum will yield the value .

The corresponding probability density is

With this convention, the continuous plane-wave expansion of an arbitrary wave function then takes the form

Written in this form, the function is clearly the amplitude that a measurement of the wavevector of the particle will yield the vector , or that a measurement of momentum will yield the value .

The corresponding probability density is

With this convention, the continuous plane-wave expansion of an arbitrary wave function then takes the form

Written in this form, the function is clearly the amplitude that a measurement of the wavevector of the particle will yield the vector , or that a measurement of momentum will yield the value .

The corresponding probability density is

Because of the form of this relation, the function is often called thewave function in k-space, or in momentum space.

Returning to the initial value problem for the free particle, we see that the task at hand is to determine the momentum space wave function for an arbitrary state

Presumably, this is also a useful thing to do if want to consider making measurements of wave vector, momentum, or kinetic energy on an arbitrary state of the particle.

It turns out that there is a straightforward way of doing this, that underscores the advantages of having chosen our free particle states to be Dirac normalized.

Returning to the initial value problem for the free particle, we see that the task at hand is to determine the momentum space wave function for an arbitrary state

Presumably, this is also a useful thing to do if want to consider making measurements of wave vector, momentum, or kinetic energy on an arbitrary state of the particle.

It turns out that there is a straightforward way of doing this, that underscores the advantages of having chosen our free particle states to be Dirac normalized.

Returning to the initial value problem for the free particle, we see that the task at hand is to determine the momentum space wave function for an arbitrary state

Presumably, this is also a useful thing to do if want to consider making measurements of wave vector, momentum, or kinetic energy on an arbitrary state of the particle.

It turns out that there is a straightforward way of doing this, that underscores the advantages of having chosen our free particle states to be Dirac normalized.

In the class notes I work this out using the explicit form of the plane waves, and you should work through that derivation so that you are comfortable with it, but the level of detail in that derivation obscures the general nature of the result.

So we proceed as follows. First, let’s put a prime on the integration variable in our expansion and write

Next, (and this is not an obvious step, but trust me for the moment!) Multiply this last expression by to obtain

So now, we just follow our temptation to integrate over all space

In the class notes I work this out using the explicit form of the plane waves, and you should work through that derivation so that you are comfortable with it, but the level of detail in that derivation obscures the general nature of the result.

So we proceed as follows. First, let’s put a prime on the integration variable in our expansion and write

Next, (and this is not an obvious step, but trust me for the moment!) Multiply this last expression by to obtain

So now, we just follow our temptation to integrate over all space

In the class notes I work this out using the explicit form of the plane waves, and you should work through that derivation so that you are comfortable with it, but the level of detail in that derivation obscures the general nature of the result.

So we proceed as follows. First, let’s put a prime on the integration variable in our expansion and write

Next, (and this is not an obvious step, but trust me for the moment!) Multiply this last expression by to obtain

So now, we just follow our temptation to integrate over all space

In the class notes I work this out using the explicit form of the plane waves, and you should work through that derivation so that you are comfortable with it, but the level of detail in that derivation obscures the general nature of the result.

So we proceed as follows. First, let’s put a prime on the integration variable in our expansion and write

Next, (and this is not an obvious step, but trust me for the moment!) Multiply this last expression by to obtain

So now, we just follow our temptation to integrate over all space

In the class notes I work this out using the explicit form of the plane waves, and you should work through that derivation so that you are comfortable with it, but the level of detail in that derivation obscures the general nature of the result.

So we proceed as follows. First, let’s put a prime on the integration variable in our expansion and write

Next, (and this is not an obvious step, but trust me for the moment!) Multiply this last expression by to obtain

So now, we just follow our temptation to integrate over all space

The integral of this last expression over all space is

We conclude that, to find the momentum space wave function we just have to perform the integral

The integral of this last expression over all space is

We conclude that, to find the momentum space wave function we just have to perform the integral

The integral of this last expression over all space is

We conclude that, to find the momentum space wave function we just have to perform the integral

The integral of this last expression over all space is

We conclude that, to find the momentum space wave function we just have to perform the integral

The integral of this last expression over all space is

We conclude that, to find the momentum space wave function we just have to perform the integral

If we now put in the explicit form of our kinetic energy or momentum eigenfunctions, and re-arrange things a bit we find that our general expression

Once the momentum space wave function has been evaluated, then our expansion for the real space wave function

If we now put in the explicit form of our kinetic energy or momentum eigenfunctions, and re-arrange things a bit we find that our general expression

takes the form

Once the momentum space wave function has been evaluated, then our expansion for the real space wave function

If we now put in the explicit form of our kinetic energy or momentum eigenfunctions, and re-arrange things a bit we find that our general expression

takes the form

Once the momentum space wave function has been evaluated, then our expansion for the real space wave function

If we now put in the explicit form of our kinetic energy or momentum eigenfunctions, and re-arrange things a bit we find that our general expression

takes the form

Once the momentum space wave function has been evaluated, then our expansion for the real space wave function can be expressed similarly, we find that

So we find that the momentum space wave function and the real space wave function (what we were formerly calling “the wave function”), are actually Fourier transform pairs:

Thus, our ability to find the momentum space wave function for an arbitrary state is limited only by our ability to perform the three-dimensional integral indicated.

So we find that the momentum space wave function and the real space wave function (what we were formerly calling “the wave function”), are actually Fourier transform pairs:

Thus, our ability to find the momentum space wave function for an arbitrary state is limited only by our ability to perform the three-dimensional integral indicated.

So we find that the momentum space wave function and the real space wave function (what we were formerly calling “the wave function”), are actually Fourier transform pairs:

Thus, our ability to find the momentum space wave function for an arbitrary state is limited only by our ability to perform the three-dimensional integral indicated.

So, we have made good progress toward solving the initial value problem for the free particle. In terms of our three-step prescription:

1. Solve the energy eigenvalue equation:

Done!

2. Find the amplitudes for the initial state when expanded in energy eigenfunctions:

Done!

3. Evolve: Yet to do! In the next lecture, we finish up this last step, and make a few final comments, before going on to introduce postulates of the general formalism of quantum mechanics as it applies to arbitrary systems.

So, we have made good progress toward solving the initial value problem for the free particle. In terms of our three-step prescription:

1. Solve the energy eigenvalue equation:

Done!

2. Find the amplitudes for the initial state when expanded in energy eigenfunctions:

Done!

3. Evolve: Yet to do! In the next lecture, we finish up this last step, and make a few final comments, before going on to introduce postulates of the general formalism of quantum mechanics as it applies to arbitrary systems.

So, we have made good progress toward solving the initial value problem for the free particle. In terms of our three-step prescription:

1. Solve the energy eigenvalue equation:

Done!

2. Find the amplitudes for the initial state when expanded in energy eigenfunctions:

Done!

3. Evolve: Yet to do! In the next lecture, we finish up this last step, and make a few final comments, before going on to introduce postulates of the general formalism of quantum mechanics as it applies to arbitrary systems.

So, we have made good progress toward solving the initial value problem for the free particle. In terms of our three-step prescription:

1. Solve the energy eigenvalue equation:

Done!

2. Find the amplitudes for the initial state when expanded in energy eigenfunctions:

Done!

3. Evolve: Yet to do! In the next lecture, we finish up this last step, and make a few final comments, before going on to introduce postulates of the general formalism of quantum mechanics as it applies to arbitrary systems.

So, we have made good progress toward solving the initial value problem for the free particle. In terms of our three-step prescription:

1. Solve the energy eigenvalue equation:

Done!

2. Find the amplitudes for the initial state when expanded in energy eigenfunctions:

Done!

3. Evolve: Yet to do! In the next lecture, we finish up this last step, and make a few final comments, before going on to introduce postulates of the general formalism of quantum mechanics as it applies to arbitrary systems.

So, we have made good progress toward solving the initial value problem for the free particle. In terms of our three-step prescription:

1. Solve the energy eigenvalue equation:

Done!

2. Find the amplitudes for the initial state when expanded in energy eigenfunctions:

Done!

3. Evolve: Still needs to be done.

So, we have made good progress toward solving the initial value problem for the free particle. In terms of our three-step prescription:

1. Solve the energy eigenvalue equation:

Done!

2. Find the amplitudes for the initial state when expanded in energy eigenfunctions:

Done!

3. Evolve: Still needs to be done.But this last step is the easiest of the three, we just put in the time dependence associated with each of the energy eigenfunctions in the expansion.

3. Evolve:

Done!

Note that this can be written in the form

where

3. Evolve:

Done!

Note that this can be written in the form

where

3. Evolve:

Done!

Note that this can be written in the form

where

3. Evolve:

Done!

Note that this can be written in the form

where

3. Evolve:

Done!

Note that this can be written in the form

where

3. Evolve:

Done!

Note that this can be written in the form

where

This last form shows that we can think of the momentum space wave function as evolving in time also, just as the real space wave function does.

For the free particle, the evolution of the momentum space wave function is very simple, at each wavevector , it just acquires an oscillating phase factor.

But even when the particle is moving in a more complicated potential energy field, whatever the state of the system is at time t, we can always expand it in eigenfunctions of momentum / kinetic energy. These don’t change just because the particle now feels a force! So at any instant, we can always write

In the presence of the force, the momentum space wave function will not take the simple form that it does for a free particle. It will have a more complicated evolution

This last form shows that we can think of the momentum space wave function as evolving in time also, just as the real space wave function does.

For the free particle, the evolution of the momentum space wave function is very simple, at each wavevector , it just acquires an oscillating phase factor.

But even when the particle is moving in a more complicated potential energy field, whatever the state of the system is at time t, we can always expand it in eigenfunctions of momentum / kinetic energy. These don’t change just because the particle now feels a force! So at any instant, we can always write

In the presence of the force, the momentum space wave function will not take the simple form that it does for a free particle. It will have a more complicated evolution

This last form shows that we can think of the momentum space wave function as evolving in time also, just as the real space wave function does.

For the free particle, the evolution of the momentum space wave function is very simple, at each wavevector , it just acquires an oscillating phase factor.

But even when the particle is moving in a more complicated potential energy field, whatever the state of the system is at time t, we can always expand it in eigenfunctions of momentum / kinetic energy. These don’t change just because the particle now feels a force! So at any instant, we can always write

In the presence of the force, the momentum space wave function will not take the simple form that it does for a free particle. It will have a more complicated evolution

This last form shows that we can think of the momentum space wave function as evolving in time also, just as the real space wave function does.

For the free particle, the evolution of the momentum space wave function is very simple, at each wavevector , it just acquires an oscillating phase factor.

But even when the particle is moving in a more complicated potential energy field, whatever the state of the system is at time t, we can always expand it in eigenfunctions of momentum / kinetic energy. These don’t change just because the particle now feels a force! So at any instant, we can always write

In the presence of the force, the momentum space wave function will not take the simple form that it does for a free particle. It will have a more complicated evolution

But no matter how complicated its evolution becomes, we can always relate it each instant to the real space wave function, i.e., at any time t we can expand the wave function in momentum eigenstates

where

Even in the more general situation in which the particle feels a force, if we know the momentum space wave function at some instant of time, we can predict the probability density that a momentum measurement will yield the valuei.e.,

But no matter how complicated its evolution becomes, we can always relate it each instant to the real space wave function, i.e., at any time t we can expand the wave function in momentum eigenstates

where

Even in the more general situation in which the particle feels a force, if we know the momentum space wave function at some instant of time, we can predict the probability density that a momentum measurement will yield the valuei.e.,

But no matter how complicated its evolution becomes, we can always relate it each instant to the real space wave function, i.e., at any time t we can expand the wave function in momentum eigenstates

where

Even in the more general situation in which the particle feels a force, if we know the momentum space wave function at some instant of time, we can predict the probability density that a momentum measurement will yield the valuei.e.,

This ends our brief review of Schrödinger's wave mechanics for a single quantum mechanical particle. In studying the free particle, we noticed again the possibility of representing the dynamical state by something other than a wave function in real space, as given to us in the postulates.

Clearly, at any instant of time, the state of the system is equally well represented by the wave function in momentum space. We can go back and forth between the two representations.

A little reflection, makes us recognize that there are actually an infinite number of different ways of representing the dynamical state of the particle.

This should not surprise us, since it’s true classically as well!

This ends our brief review of Schrödinger's wave mechanics for a single quantum mechanical particle. In studying the free particle, we noticed again the possibility of representing the dynamical state by something other than a wave function in real space, as given to us in the postulates.

Clearly, at any instant of time, the state of the system is equally well represented by the wave function in momentum space. We can go back and forth between the two representations.

A little reflection, makes us recognize that there are actually an infinite number of different ways of representing the dynamical state of the particle.

This should not surprise us, since it’s true classically as well!

This ends our brief review of Schrödinger's wave mechanics for a single quantum mechanical particle. In studying the free particle, we noticed again the possibility of representing the dynamical state by something other than a wave function in real space, as given to us in the postulates.

Clearly, at any instant of time, the state of the system is equally well represented by the wave function in momentum space. We can go back and forth between the two representations.

A little reflection, makes us recognize that there are actually an infinite number of different ways of representing the dynamical state of the particle.

This should not surprise us, since it’s true classically as well!

This ends our brief review of Schrödinger's wave mechanics for a single quantum mechanical particle. In studying the free particle, we noticed again the possibility of representing the dynamical state by something other than a wave function in real space, as given to us in the postulates.

Clearly, at any instant of time, the state of the system is equally well represented by the wave function in momentum space. We can go back and forth between the two representations.

A little reflection, makes us recognize that there are actually an infinite number of different ways of representing the dynamical state of the particle.

This should not surprise us, since it’s true classically as well!

Consider, at any instant a classical particle is located at a point in space that can be associated with a position vectorBut even if we agree on the origin there different ways of specifying that same position vector, using different coordinate systems, i.e.,

Consider, at any instant a classical particle is located at a point in space that can be associated with a position vectorBut even if we agree on the origin there different ways of specifying that same position vector, using different coordinate systems, i.e.,

Consider, at any instant a classical particle is located at a point in space that can be associated with a position vectorBut even if we agree on the origin there different ways of specifying that same position vector, using different coordinate systems, i.e.,

Consider, at any instant a classical particle is located at a point in space that can be associated with a position vectorBut even if we agree on the origin there different ways of specifying that same position vector, using different coordinate systems, i.e.,

Consider, at any instant a classical particle is located at a point in space that can be associated with a position vectorBut even if we agree on the origin there different ways of specifying that same position vector, using different coordinate systems, i.e.,

Consider, at any instant a classical particle is located at a point in space that can be associated with a position vectorBut even if we agree on the origin there different ways of specifying that same position vector, using different coordinate systems, i.e.,

Consider, at any instant a classical particle is located at a point in space that can be associated with a position vectorBut even if we agree on the origin there different ways of specifying that same position vector, using different coordinate systems, i.e.,

Consider, at any instant a classical particle is located at a point in space that can be associated with a position vectorBut even if we agree on the origin there different ways of specifying that same position vector, using different coordinate systems, i.e.,

Consider, at any instant a classical particle is located at a point in space that can be associated with a position vectorBut even if we agree on the origin there different ways of specifying that same position vector, using different coordinate systems, i.e.,

Note: to communicate the position vector, we don’t just need the numbers, we need to specify the coordinate system, or the unit vectors associated with it.

What we are seeing in quantum mechanics is exactly the same thing. There is an underlying dynamical state, but there are many ways to communicate what it is

What we are seeing in quantum mechanics is exactly the same thing. There is an underlying dynamical state, but there are many ways to communicate what it is

What we are seeing in quantum mechanics is exactly the same thing. There is an underlying dynamical state, but there are many ways to communicate what it is

What we are seeing in quantum mechanics is exactly the same thing. There is an underlying dynamical state, but there are many ways to communicate what it is

What we are seeing in quantum mechanics is exactly the same thing. There is an underlying dynamical state, but there are many ways to communicate what it is

What we are seeing in quantum mechanics is exactly the same thing. There is an underlying dynamical state, but there are many ways to communicate what it is

What we are seeing in quantum mechanics is exactly the same thing. There is an underlying dynamical state, but there are many ways to communicate what it is

What we are seeing in quantum mechanics is exactly the same thing. There is an underlying dynamical state, but there are many ways to communicate what it is

So starting off with a postulate, which sort of tells us what the coordinate system is, takes away our freedom to choose a representation that is better suited to the specific problem at hand.

So, in the next lecture, we begin to present a version of the postulates of quantum mechanics that is

1. Manifestly representation independent (but which allows for the natural emergence of appropriate numerical representations), and which

2. In principle applies to arbitrary quantum mechanical systems.

So starting off with a postulate, which sort of tells us what the coordinate system is, takes away our freedom to choose a representation that is better suited to the specific problem at hand.

So, in the next lecture, we begin to present a version of the postulates of quantum mechanics that is

1. Manifestly representation independent (but which allows for the natural emergence of appropriate numerical representations), and which

2. In principle applies to arbitrary quantum mechanical systems.

So starting off with a postulate, which sort of tells us what the coordinate system is, takes away our freedom to choose a representation that is better suited to the specific problem at hand.

So, in the next lecture, we begin to present a version of the postulates of quantum mechanics that is

1. Manifestly representation independent (but which allows for the natural emergence of appropriate numerical representations), and which

2. In principle applies to arbitrary quantum mechanical systems.

So starting off with a postulate, which sort of tells us what the coordinate system is, takes away our freedom to choose a representation that is better suited to the specific problem at hand.

So, in the next lecture, we begin to present a version of the postulates of quantum mechanics that is

1. Manifestly representation independent (but which allows for the natural emergence of appropriate numerical representations), and

2. In principle applies to arbitrary quantum mechanical systems.