7.9 Position and Linear Momentum

The subsequent sections will be looking at the time evolution of various quantum systems, as predicted by the Schrödinger equation. However, before that can be done, first the eigenfunctions of position and linear momentum must be found. That is something that the book has been studiously avoiding so far. The problem is that the position and linear momentum eigenfunctions have awkward issues with normalizing them.

These normalization problems have consequences for the coefficients of the eigenfunctions. In the orthodox interpretation, the square magnitudes of the coefficients should give the probabilities of getting the corresponding values of position and linear momentum. But this statement will have to be modified a bit.

One good thing is that unlike the Hamiltonian, which is specific to a
given system, the position operator

and the linear momentum operator

are the same for all systems. So, you only need to find their eigenfunctions once.

7.9.1 The position eigenfunction

The eigenfunction that corresponds to the particle being at a precise

(Note the need in this analysis to use

To solve this eigenvalue problem, try again separation of variables, where it is assumed that

This equation implies that at all points

To resolve the ambiguity, the function Dirac delta function,

The delta function is, loosely speaking, sufficiently strongly infinite at the single point

The fact that the integral is one leads to a very useful mathematical
property of delta functions: they are able to pick out one specific
value of any arbitrary given function

(7.49) |

The problems for the position eigenfunctions

According to the orthodox interpretation, the probability of finding
the particle at

It can be simplified to

(7.51) |

However, the apparent conclusion that

Instead, according to Born's statistical interpretation of chapter
3.1, the expression

gives the probability of finding the particle in an infinitesimal volume

Besides the normalization issue, another idea that needs to be
somewhat modified is a strict collapse of the wave function. Any
position measurement that can be done will leave some uncertainty
about the precise location of the particle: it will leave

Key Points

- Position eigenfunctions are delta functions.

- They are not properly normalized.

- The coefficient of the position eigenfunction for a position
is the good old wave function .

- Because of the fact that the delta functions are not normalized, the square magnitude of
does not give the probability that the particle is at position .

- Instead the square magnitude of
gives the probability that the particle is near position per unit volume.

- Position eigenfunctions are not stationary, so localized particle wave functions will spread out over time.

7.9.2 The linear momentum eigenfunction

Turning now to linear momentum, the eigenfunction that corresponds to
a precise linear momentum

The solution is a complex exponential:

where

Just like the position eigenfunction earlier, the linear momentum
eigenfunction has a normalization problem. In particular, since it
does not become small at large

(However, other books, in particular nonquantum ones, are likely to make a different choice.)

The problems for the

The coefficient

(7.53) |

The momentum space wave function does not quite give the probability
for the momentum to be

gives the probability of finding the linear momentum within a small momentum range

momentum space volume.That is much like the square magnitude

There is even an inverse relationship to recover

(7.54) |

If this inner product is written out, it reads:

Key Points

- The linear momentum eigenfunctions are complex exponentials of the form:

- They are not properly normalized.

- The coefficient of the linear momentum eigenfunction for a momentum
is indicated by . It is called the momentum space wave function.

- Because of the fact that the momentum eigenfunctions are not normalized, the square magnitude of
does not give the probability that the particle has momentum .

- Instead the square magnitude of
gives the probability that the particle has a momentum close to per unit momentum space volume.

- In writing the complete wave function in terms of the momentum eigenfunctions, you must integrate over the momentum instead of sum.

- The transformation between the physical space wave function
and the momentum space wave function is called the Fourier transform. It is invertible.