Quantum One: Lecture 5a

Normalization Conditions for Free Particle Eigenstates

In the last lecture we began to explore Schrödinger's Mechanics as it applies to a free quantum mechanical particle, for which the Hamiltonian is simply the kinetic energy operator

The energy eigenvalue equation for the free particle reduces to the Helmoltz

equation where

Using separation of variables we found energy eigenfunctions of the form

with energies

where the wavevector has real components.

The free particle energy eigenvalues are thus positive and continuous, and lead to stationary solutions of the form

where

The free particle energy eigenstates are also eigenstates of

the momentum operator

with momentum eigenvalues

In this lecture we proceed towards a general solution of the initial value problem for the free particle, by exploring appropriate normalization conditions for the free particle energy eigenfunctions.

Clearly, the free particle energy eigenfunctions are not square normalizable.

Indeed, for these states the probability density

of finding the particle at any point is independent of position, reflecting the fact that a classical free particle moving, e.g., along the x axis spends equal time in any given interval of unit length.

In this lecture we proceed towards a general solution of the initial value problem for the free particle, by exploring appropriate normalization conditions for the free particle energy eigenfunctions.

Clearly, the free particle energy eigenfunctions are not square normalizable.

Indeed, for these states the probability density

of finding the particle at any point is independent of position, reflecting the fact that a classical free particle moving, e.g., along the x axis spends equal time in any given interval of unit length.

In this lecture we proceed towards a general solution of the initial value problem for the free particle, by exploring appropriate normalization conditions for the free particle energy eigenfunctions.

Clearly, the free particle energy eigenfunctions are not square normalizable.

Indeed, for these states the probability density of finding the particle at any point is independent of position, i.e.,

Reflecting the fact that a classical free particle moving, e.g., along the x axis spends equal time in any given interval of unit length.

In this lecture we proceed towards a general solution of the initial value problem for the free particle, by exploring appropriate normalization conditions for the free particle energy eigenfunctions.

Clearly, the free particle energy eigenfunctions are not square normalizable.

Indeed, for these states the probability density of finding the particle at any point is independent of position, i.e.,

Presumably, this reflects the fact that a classical free particle moving, e.g., along the x axis spends equal time in any given interval of unit length.

But in this situation, for any value of the normalization constant A, the integral over all space diverges, i.e., since it is proportional to the (presumed infinite) volume of the universe, i.e.,

We are forced, therefore, to find a mathematical convention that allows us to sensibly expand arbitrary dynamical states in terms of these free particle energy eigenfunctions.

Two different conventions are commonly employed for this purpose.

But in this situation, for any value of the normalization constant A, the integral over all space diverges, i.e., since it is proportional to the (presumed infinite) volume of the universe, i.e.,

We are forced, therefore, to find a mathematical convention that allows us to sensibly expand arbitrary dynamical states in terms of these free particle energy eigenfunctions.

Two different conventions are commonly employed for this purpose.

But in this situation, for any value of the normalization constant A, the integral over all space diverges, i.e., since it is proportional to the (presumed infinite) volume of the universe, i.e.,

We are forced, therefore, to find a mathematical convention that allows us to sensibly expand arbitrary dynamical states in terms of these free particle energy eigenfunctions.

Two different conventions are commonly employed for this purpose.

But in this situation, for any value of the normalization constant A, the integral over all space diverges, i.e., since it is proportional to the (presumed infinite) volume of the universe, i.e.,

We are forced, therefore, to find a mathematical convention that allows us to sensibly expand arbitrary dynamical states in terms of these free particle energy eigenfunctions.

Two different conventions are commonly employed for this purpose.

The first such convention (which we will not use) is called

Box Normalization – In this convention the universe is assumed to be a very large cubic box of volume V in which the particle is confined, with the value of the wave function required to be equal on opposite faces (referred to as periodic boundary conditions). Thus, for this box, one sets

Thus, although the box is large, the wave function is square-normalized.

Unfortunately, this makes the spectrum of both energy and momentum discrete, since only wavelengths that just fit within the box are allowed.

This unfortunate quantization of the free particle spectrum can be avoided by employing an alternative convention.

The first such convention (which we will not use) is called

Box Normalization – In this convention the universe is modeled as a very large cubic box of volume V in which the particle is confined, with the value of the wave function required to be equal on opposite faces of the box (what are referred to as periodic boundary conditions). For this box universe, one sets

Thus, although the box is large, the wave function is square-normalized.

Unfortunately, this makes the spectrum of both energy and momentum discrete, since only wavelengths that just fit within the box are allowed.

This unfortunate quantization of the free particle spectrum can be avoided by employing an alternative convention.

The first such convention (which we will not use) is called

Box Normalization – In this convention the universe is modeled as a very large cubic box of volume V in which the particle is confined, with the value of the wave function required to be equal on opposite faces of the box (what are referred to as periodic boundary conditions). For this box universe, one sets

Thus, although the box is large, the wave function is square-normalized.

Unfortunately, this makes the spectrum of both energy and momentum discrete, since only wavelengths that just fit within the box are allowed.

This unfortunate quantization of the free particle spectrum can be avoided by employing an alternative convention.

The first such convention (which we will not use) is called

Box Normalization – In this convention the universe is modeled as a very large cubic box of volume V in which the particle is confined, with the value of the wave function required to be equal on opposite faces of the box (what are referred to as periodic boundary conditions). For this box universe, one sets

Thus, although the box is large, the wave function is square-normalized. Unfortunately, this makes the spectrum of both energy and momentum discrete, since only wavelengths that just fit within the box are allowed.

This unfortunate quantization of the free particle spectrum can be avoided by employing an alternative convention, referred to as . . .

The first such convention (which we will not use) is called

Box Normalization – In this convention the universe is modeled as a very large cubic box of volume V in which the particle is confined, with the value of the wave function required to be equal on opposite faces of the box (what are referred to as periodic boundary conditions). For this box universe, one sets

Thus, although the box is large, the wave function is square-normalized. Unfortunately, this makes the spectrum of both energy and momentum discrete, since only wavelengths that just fit within the box are allowed.

This unfortunate quantization of the free particle spectrum can be avoided by employing an alternative convention, referred to as . . .

The first such convention (which we will not use) is called

Box Normalization – In this convention the universe is modeled as a very large cubic box of volume V in which the particle is confined, with the value of the wave function required to be equal on opposite faces of the box (what are referred to as periodic boundary conditions). For this box universe, one sets

Thus, although the box is large, the wave function is square-normalized. Unfortunately, this makes the spectrum of both energy and momentum discrete, since only wavelengths that just fit within the box are allowed.

This unfortunate quantization of the free particle spectrum can be avoided by employing an alternative convention, referred to as . . .

Dirac or Delta Function Normalization – In this approach one chooses the normalization constant A simply for mathematical convenience, namely, so that the complete set of functions forms a generalized orthonormal set.

Definition: Orthonormal Set of Functions A set of functions labeled by a discrete index n forms an orthonormal set of functions on R³ if

So this describes a set of square normalized functions, different members of which are said to be orthogonal, rather like the dot product of a set of orthogonal unit vectors.

Dirac or Delta Function Normalization – In this approach one chooses the normalization constant A simply for mathematical convenience, namely, so that the complete set of functions forms a generalized orthonormal set.

Definition: Orthonormal Set of Functions A set of functions labeled by a discrete index n forms an orthonormal set of functions on R³ if

So this describes a set of square normalized functions, different members of which are said to be orthogonal, rather like the dot product of a set of orthogonal unit vectors.

Dirac or Delta Function Normalization – In this approach one chooses the normalization constant A simply for mathematical convenience, namely, so that the complete set of functions forms a generalized orthonormal set.

Definition: Orthonormal Set of Functions A set of functions labeled by a discrete index n forms an orthonormal set of functions on R³ if

So this describes a set of square normalized functions, different members of which are said to be orthogonal, rather like the dot product of a set of orthogonal unit vectors.

One can define orthonormal sets of functions on other domains as well, such as the real axis, or some finite subset thereof.

For our free particle eigenfunctions, we need a generalization of this idea of orthonormality to include continuously-indexed sets of functions.

Note: the continuous analog of the Kronecker delta function is the Dirac delta function. This suggests the following natural extension of this idea:

Generalized Definition: Orthonormal Set of Functions A set of functions labeled by a continuous index a forms an orthonormal set of functions on R³ if

Such a set is said to be Dirac normalized, or delta function normalized.

One can define orthonormal sets of functions on other domains as well, such as the real axis, or some finite subset thereof.

For our free particle eigenfunctions, we need a generalization of this idea of orthonormality to include continuously-indexed sets of functions.

Note: the continuous analog of the Kronecker delta function is the Dirac delta function. This suggests the following natural extension of this idea:

Generalized Definition: Orthonormal Set of Functions A set of functions labeled by a continuous index a forms an orthonormal set of functions on R³ if

Such a set is said to be Dirac normalized, or delta function normalized.

One can define orthonormal sets of functions on other domains as well, such as the real axis, or some finite subset thereof.

For our free particle eigenfunctions, we need a generalization of this idea of orthonormality to include continuously-indexed sets of functions.

Note: the continuous analog of the Kronecker delta function is the Dirac delta function. This suggests the following natural extension of this idea:

Generalized Definition: Orthonormal Set of Functions A set of functions labeled by a continuous index a forms an orthonormal set of functions on R³ if

Such a set is said to be Dirac normalized, or delta function normalized.

One can define orthonormal sets of functions on other domains as well, such as the real axis, or some finite subset thereof.

For our free particle eigenfunctions, we need a generalization of this idea of orthonormality to include continuously-indexed sets of functions.

Note: the continuous analog of the Kronecker delta function is the Dirac delta function. This suggests the following natural extension of this idea:

Generalized Definition: Orthonormal Set of Functions A set of functions labeled by a continuous index a forms an orthonormal set of functions on R³ if

Such a set is said to be Dirac normalized, or delta function normalized.

One can define orthonormal sets of functions on other domains as well, such as the real axis, or some finite subset thereof.

For our free particle eigenfunctions, we need a generalization of this idea of orthonormality to include continuously-indexed sets of functions.

Note: the continuous analog of the Kronecker delta function is the Dirac delta function. This suggests the following natural extension of this idea:

Generalized Definition: Orthonormal Set of Functions A set of functions labeled by a continuous index a forms an orthonormal set of functions on R³ if

Such a set is said to be Dirac normalized, or delta function normalized.

As before, functions corresponding to different values of the continuous index are said to be orthogonal, since the integral vanishes for this situation.

But the functions in this set are not square normalized to unity: when the value of the two continuous indices are the same, the argument of the delta function vanishes, and the integral of the squared magnitude is then found to be infinite (as occurs for the free particle energy eigenstates).

Thus, in dealing with eigenfunctions of observables with continuous eigenvalues, we will choose the normalization so the associated set of eigenfunctions forms a generalized orthonormal set of functions, properly Dirac normalized, as defined above.

As before, functions corresponding to different values of the continuous index are said to be orthogonal, since the integral vanishes for this situation.

But the functions in this set are not square normalized to unity: When the value of the two continuous indices are the same, the argument of the delta function vanishes, and the integral of the squared magnitude is then found to be infinite (as occurs for the free particle energy eigenstates).

Thus, in dealing with eigenfunctions of observables with continuous eigenvalues, we will choose the normalization so the associated set of eigenfunctions forms a generalized orthonormal set of functions, properly Dirac normalized, as defined above.

As before, functions corresponding to different values of the continuous index are said to be orthogonal, since the integral vanishes for this situation.

But the functions in this set are not square normalized to unity: when the value of the two continuous indices are the same, the argument of the delta function vanishes, and the integral of the squared magnitude is then found to be infinite (as occurs for the free particle energy eigenstates).

Therefore, to deal with eigenfunctions of observables with continuous eigenvalues, we will choose the normalization so the associated set of eigenfunctions forms a generalized orthonormal set of functions, properly Dirac normalized, as defined above.

Thus, for the free particle, we need to choose the value of the normalization constant A so that the eigenfunctions of energy and momentum form a generalized orthonormal set, i.e., so that

which implies

A change of variable in the plane wave representation of the 1D delta function

leads to

Thus, for the free particle, we need to choose the value of the normalization constant A so that the eigenfunctions of energy and momentum form a generalized orthonormal set, i.e., so that

which implies

A change of variable in the plane wave representation of the 1D delta function

leads to

Thus, for the free particle, we need to choose the value of the normalization constant A so that the eigenfunctions of energy and momentum form a generalized orthonormal set, i.e., so that

which implies

A change of variable in the plane wave representation of the 1D delta function

leads to

Thus, for the free particle, we need to choose the value of the normalization constant A so that the eigenfunctions of energy and momentum form a generalized orthonormal set, i.e., so that

which implies

A change of variable in the plane wave representation of the 1D delta function

leads to

Thus, for the free particle, we need to choose the value of the normalization constant A so that the eigenfunctions of energy and momentum form a generalized orthonormal set, i.e., so that

which implies

A change of variable in the plane wave representation of the 1D delta function

leads to

Thus, for the free particle, we need to choose the value of the normalization constant A so that the eigenfunctions of energy and momentum form a generalized orthonormal set, i.e., so that

which implies

A change of variable in the plane wave representation of the 1D delta function

leads to

A similar integral in y and z leads to the 3D version:

Comparing this to our previous expression

we deduce that to have Dirac normalized plane waves we must set

giving, finally

A similar integral in y and z leads to the 3D version:

Comparing this to our previous expression

we deduce that to have Dirac normalized plane waves we must set

giving, finally

A similar integral in y and z leads to the 3D version:

Comparing this to our previous expression

we deduce that to have Dirac normalized plane waves we must set

giving, finally

A similar integral in y and z leads to the 3D version:

Comparing this to our previous expression

we deduce that to have Dirac normalized plane waves we must set

giving, finally

A similar integral in y and z leads to the 3D version:

Comparing this to our previous expression

we deduce that to have Dirac normalized plane waves we can set,

giving, finally

A similar integral in y and z leads to the 3D version:

Comparing this to our previous expression

we deduce that to have Dirac normalized plane waves we can set,

for Dirac normalized plane waves, or free energy eigenstates.

Having obtained normalized plane waves, we note in passing, that the eigenfunctions

that we deduced earlier for the position operator

are actually already Dirac normalized, because

which is readily observed to be an example of the Dirac normalization condition.

Having obtained normalized plane waves, we note in passing, that the eigenfunctions

that we deduced earlier for the position operator

are actually already Dirac normalized, because

which is readily observed to be an example of the Dirac normalization condition.

Having obtained normalized plane waves, we note in passing, that the eigenfunctions

that we deduced earlier for the position operator

are actually already Dirac normalized, because

which is readily observed to be an example of the Dirac normalization condition.

Having obtained normalized plane waves, we note in passing, that the eigenfunctions

that we deduced earlier for the position operator

are actually already Dirac normalized, because

which is readily observed to be an example of the Dirac normalization condition.

Having obtained normalized plane waves, we note in passing, that the eigenfunctions

that we deduced earlier for the position operator

are actually already Dirac normalized, because

which is readily observed to be an example of the Dirac normalization condition.

Having obtained normalized plane waves, we note in passing, that the eigenfunctions

that we deduced earlier for the position operator

are actually already Dirac normalized, because

which is readily observed to be an example of the Dirac normalization condition.

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).

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).

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).

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).

This important 2nd step is carried out in the next lecture