Time-Dependence of Eigenvalues:

When dealing with time-dependent systems, one may want to ascertain the rate at which eigenvalues change with respect to time. Why any normal person would want to do this, one can't be sure. There is a means of determining the change in these values. To find an easy relation to work with, one begins with the definition of an eigenvalue:

eq.1

The sub-script N indicates a normalized wave function. Taking the time-derivative of both sides:

eq.2

Applying the differential operator through, keeping in mind the change in time of the operator W is zero:

eq.3

To get anywhere at this point, substitutions are used which originate from the time-dependent form of the Schrödinger equation:

eq.4

Using these:

eq.5

Simplifying:

eq.6

Knowing -1 / i = i and consolidating:

eq.7

Rewritting the observable as a commutator produces a generalized equation for determining the rate of change of an eigenvalue with time:

eq.8

Equation 8 can be applied to operators such as the linear momentum operator:

eq.9

The linear momentum operator is of the form:

eq.10

Using this:

eq.11

Expanding, retaining the order in which the operators act on the arbitrary function:

eq.12

Simplifying produces the final result:

eq.13

The negative gradient of the potential in which the electron exists is equal to the change in momentum of the electron with respect to time. This quantity represents the mean force that the electron experiences while in the potential. The change in the mean position of the electron with respect to time can also be determined through the same procedure. Starting with equation 8:

eq.14

Writing out the commutator explicitly:

eq.15

Expanding, retaining the order in which the operators act on the arbitrary function:

eq.16

Simplifying:

eq.17

Further simplifying produces the final result:

eq.18

The change in the mean position of the electron with respect to time is equal to the average linear momentum divided by the mass of the electron. One can assert that this is also equal to the mean velocity of the electron which isnt all that surprising.