D.84 Expectation powers of r for hydrogen

This note derives the expectation values of the powers of for the
hydrogen energy eigenfunctions . The various values
to be be derived are:

The trickiest to derive is the expectation value of ,
and that one will be done first. First recall the hydrogen
Hamiltonian from chapter 4.3,

Its energy eigenfunctions of given square and angular momentum and their energy are

where the are called the spherical harmonics.

When this Hamiltonian is applied to an eigenfunction
, it produces the exact same result as the following
dirty trick Hamiltonian

in which the angular
derivatives have been replaced by :

The reason is that the angular derivatives are essentially the square angular momentum operator of chapter 4.2.3. Now, while in the hydrogen Hamiltonian the quantum number has to be an integer because of its origin, in the dirty trick one can be allowed to assume any value. That means that you can differentiate the Hamiltonian and its eigenvalues with respect to . And that allows you to apply the Hellmann-Feynman theorem of section A.37.1:

(Yes, the eigenfunctions are good, because the purely radial commutes with both and , which are angular derivatives.) Substituting in the dirty trick Hamiltonian,

So, if you can figure out how the dirty trick energy changes with near some desired integer value , the desired expectation value of at that integer value of follows. Note that the eigenfunctions of can still be taken to be of the form , where can be divided out of the eigenvalue problem to give . If you skim back through chapter 4.3 and its note, you see that that eigenvalue problem was solved in derivation {D.15}. Now, of course, is no longer an integer, but if you skim through the note, it really makes almost no difference. The energy eigenvalues are still . If you look near the end of the note, you see that the requirement on is that where must remain an integer for valid solutions, hence must stay constant under small changes. So 1, and then according to the chain rule the derivative of is . Substitute it in and there you have that nasty expectation value as given in (D.60).

All other expectation values of for integer values of
may be found from the “Kramers relation,” or “(second) Pasternack relation:”

(D.61) |

Substituting 0 into the Kramers-Pasternack relation produces the expectation value of as in (D.60). It may be noted that this can instead be derived from the virial theorem of chapter 7.2, or from the Hellmann-Feynman theorem by differentiating the hydrogen Hamiltonian with respect to the charge . Substituting in 1, 2, ...produces the expectation values for , , .... Substituting in 1 and the expectation value for from the Hellmann-Feynman theorem gives the expectation value for . The remaining negative integer values 2, 3, ...produce the remaining expectation values for the negative integer powers of as the term in the equation.

Note that for a sufficiently negative powers of , the expectation value becomes infinite. Specifically, since is proportional to , {D.15}, it can be seen that becomes infinite when . When that happens, the coefficient of the expectation value in the Kramers-Pasternack relation becomes zero, making it impossible to compute the expectation value. The relationship can be used until it crashes and then the remaining expectation values are all infinite.

The remainder of this note derives the Kramers-Pasternack relation.
First note that the expectation values are defined as

When this integral is written in spherical coordinates, the integration of the square spherical harmonic over the angular coordinates produces one. So, the expectation value simplifies to

To simplify the notations, a nondimensional radial coordinate will be used. Also, a new radial function will be defined. In those terms, the expression above for the expectation value shortens to

To further shorten the notations, from now on the limits of integration and will be omitted throughout. In those notations, the expectation value of is

Also note that the integrals are improper. It is to be assumed that the integrations are from a very small value of to a very large one, and that only at the end of the derivation, the limit is taken that the integration limits become zero and infinity.

According to derivation {D.15}, the function satisfies
in terms of the ordinary differential equation.

where primes indicate derivatives with respect to . Substituting in , you get in terms of the new unknown function that

Since this makes proportional to , forming the integral produces a combination of terms of the form , hence of expectation values of powers of :

The idea is now to apply integration by parts on to produce a different combination of expectation values. The fact that the two combinations must be equal will then give the Kramers-Pasternack relation.

Before embarking on this, first note that since

the latter from integration by parts, it follows that

This result will be used routinely in the manipulations below to reduce integrals of that form.

Now an obvious first integration by parts on produces

The first of the two integrals reduces to an expectation value of using (D.64). For the final integral, use another integration by parts, but make sure you do not run around in a circle because if you do you will get a trivial expression. What works is integrating and differentiating :

In the final integral, according to the differential equation (D.62), the factor can be replaced by powers of times :

and each of the terms is of the form (D.64), so you get

Plugging this into (D.65) and then equating that to
(D.63) produces the Kramers-Pasternack relation. It
also gives an additional right hand side

but that term becomes zero when the integration limits take their final values zero and infinity. In particular, the upper limit values always become zero in the limit of the upper bound going to infinity; and its derivative go to zero exponentially then, beating out any power of . The lower limit values also become zero in the region of applicability that exists, because that requires that is for small proportional to a power of greater than zero.

The above analysis is not valid when 1, since then the final integration by parts would produce a logarithm, but since the expression is valid for any other , not just integer ones you can just take a limit to cover that case.