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

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

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:

$$1 = \varepsilon_1^2 + \varepsilon_2^2 + \ldots + \varep... ...{n-1}^2 + (1+\varepsilon_n)^2 + \varepsilon_{n+1}^2 +\ldots$$

The ex­pec­ta­tion en­ergy is

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

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

$$\begin{eqnarray*} \left\langle{E}\right\rangle & = & \varepsilon_1^2 (E_1-E_n)... ..._{n-1}-E_n) + E_n + \varepsilon_{n+1}^2 (E_{n+1}-E_n) +\ldots \end{eqnarray*}$$

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

$$\begin{eqnarray*} \delta\left\langle{E}\right\rangle & = & 2\varepsilon_1(E_1-... ... 2\varepsilon_{n+1}(E_{n+1}-E_n)\delta\varepsilon_{n+1} +\ldots \end{eqnarray*}$$

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

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

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.