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

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

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:

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

The expec­tation energy is

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

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

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

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 $\big\langle E\big\rangle$ 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

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

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

$$0 = 2 \varepsilon_1\delta\varepsilon_1 + 2 \varepsilon... ...n_n + 2 \varepsilon_{n+1}\delta\varepsilon_{n+1} + \ldots$$

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.