D.49 The gen­er­al­ized vari­a­tional prin­ci­ple

The pur­pose of this note is to ver­ify di­rectly that the vari­a­tion of the ex­pec­ta­tion en­ergy is zero at any en­ergy eigen­state, not just the ground state.

Sup­pose that you are try­ing to find some en­ergy eigen­state $\psi_n$ with eigen­value $E_n$, and that you are close to it, but no cigar. Then the wave func­tion can be writ­ten as

\varepsilon_1 \psi_1
+ \varepsilon_2 \psi_2
+ \ldo...
...varepsilon_n) \psi_n
+ \varepsilon_{n+1} \psi_{n+1}
+ \ldots

where $\psi_n$ is the one you want and the re­main­ing terms to­gether are the small er­ror in wave func­tion, writ­ten in terms of the eigen­func­tions. Their co­ef­fi­cients $\varepsilon_1,\varepsilon_2,\ldots$ are small.

The nor­mal­iza­tion con­di­tion $\langle\psi\vert\psi\rangle$ $\vphantom0\raisebox{1.5pt}{$=$}$ 1 is, us­ing or­tho­nor­mal­ity:

1 =
+ \varepsilon_2^2
+ \ldots
+ \varep...
+ (1+\varepsilon_n)^2
+ \varepsilon_{n+1}^2

The ex­pec­ta­tion en­ergy is

\left\langle{E}\right\rangle =
\varepsilon_1^2 E_1
+ \var...
...1+\varepsilon_n)^2 E_n
+ \varepsilon_{n+1}^2 E_{n+1}

or plug­ging in the nor­mal­iza­tion con­di­tion to elim­i­nate $(1+\varepsilon_n)^2$

\left\langle{E}\right\rangle & = &
\varepsilon_1^2 (E_1-E_n)...
+ E_n
+ \varepsilon_{n+1}^2 (E_{n+1}-E_n)

As­sum­ing that the en­ergy eigen­val­ues are arranged in in­creas­ing or­der, the terms be­fore $E_n$ in this sum are neg­a­tive and the ones be­hind $E_n$ pos­i­tive. So $E_n$ is nei­ther a max­i­mum nor a min­i­mum; de­pend­ing on con­di­tions $\left\langle{E}\right\rangle $ can be greater or smaller than $E_n$.

Now, if you make small changes in the wave func­tion, the val­ues of $\varepsilon_1,\varepsilon_2,\ldots$ will slightly change, by small amounts that will be in­di­cated by $\delta\varepsilon_1,\delta\varepsilon_2,\ldots$, and you get

\delta\left\langle{E}\right\rangle & = &
... 2\varepsilon_{n+1}(E_{n+1}-E_n)\delta\varepsilon_{n+1}

This is zero when $\varepsilon_1$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\varepsilon_2$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\ldots$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0, so when $\psi$ is the ex­act eigen­func­tion $\psi_n$. And it is nonzero as soon as any of $\varepsilon_1,\varepsilon_2,\ldots$ is nonzero; a change in that co­ef­fi­cient will pro­duce a nonzero change in ex­pec­ta­tion en­ergy. So the vari­a­tional con­di­tion $\delta\langle{E}\rangle$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0 is sat­is­fied at the ex­act eigen­func­tion $\psi_n$, but not at any nearby dif­fer­ent wave func­tions.

The bot­tom line is that if you lo­cate the near­est wave func­tion for which $\delta\langle{E}\rangle$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0 for all ac­cept­able small changes in that wave func­tion, well, if you are in the vicin­ity of an en­ergy eigen­func­tion, you are go­ing to find that eigen­func­tion.

One fi­nal note. If you look at the ex­pres­sion above, it seems like none of the other eigen­func­tions are eigen­func­tions. For ex­am­ple, the ground state would be the case that $\varepsilon_1$ is one, and all the other co­ef­fi­cients zero. So a small change in $\varepsilon_1$ would seem to pro­duce a change $\delta\langle{E}\rangle$ in ex­pec­ta­tion en­ergy, and the ex­pec­ta­tion en­ergy is sup­posed to be con­stant at eigen­states.

The prob­lem is the nor­mal­iza­tion con­di­tion, whose dif­fer­en­tial form says that

0 =
2 \varepsilon_1\delta\varepsilon_1
+ 2 \varepsilon_2\...
+ 2 \varepsilon_{n+1}\delta\varepsilon_{n+1}
+ \ldots

At $\varepsilon_1$ $\vphantom0\raisebox{1.5pt}{$=$}$ 1 and $\varepsilon_2$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\ldots$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\varepsilon_{n-1}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $1+\varepsilon_n$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\varepsilon_{n+1}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\ldots$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0, this im­plies that the change $\delta\varepsilon_1$ must be zero. And that means that the change in ex­pec­ta­tion en­ergy is in fact zero. You see that you re­ally need to elim­i­nate $\varepsilon_1$ from the list of co­ef­fi­cients near $\psi_1$, rather than $\varepsilon_n$ as the analy­sis for $\psi_n$ did, for the math­e­mat­ics not to blow up. A co­ef­fi­cient that is not al­lowed to change at a point in the vicin­ity of in­ter­est is a con­fus­ing co­ef­fi­cient to work with.