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## Homework Statement

Prove the relationship

$$\left(\frac{\partial H}{\partial\lambda}\right)_{nn} = \frac{\partial E_{nn}}{\partial\lambda},$$

where ##\lambda## is a parameter in the Hamiltonian. Using this relationship, show that the average force exerted by a particle in an infinitely deep potential well ##(0\le x\le a)##, on the right "wall" can be written as $$\langle F_\textrm{right}\rangle_n = \frac{\partial E_n}{\partial a},$$ where ##n## is the energy level. Calculate the force and compare it with the classical expression.

## Homework Equations

For the first part: ##\hat{H}_\lambda|\psi_\lambda\rangle = E_\lambda|\psi_\lambda\rangle##

##\langle \psi_\lambda|\psi_\lambda\rangle = 1##

##\frac{d}{d\lambda}\langle \psi_\lambda|\psi_\lambda\rangle = 0##

For the well: ##E_n = \frac{n^2\pi^2\hbar^2}{2ma^2}##

## The Attempt at a Solution

Okay so here is what I am thinking for the first part:

$$\begin{eqnarray}

\frac{d E_\lambda}{d\lambda} & = & \frac{d}{d\lambda}\langle\psi_\lambda|\hat{H}_\lambda|\psi_\lambda\rangle \\

& = & \langle \frac{d\psi_\lambda}{d\lambda}|\hat{H}_\lambda|\psi_\lambda\rangle + \langle\psi_\lambda |\hat{H}_\lambda|\frac{d\psi_\lambda}{d\lambda}\rangle + \langle\psi_\lambda|\frac{d\hat{H}_\lambda}{d\lambda}|\psi_\lambda\rangle \\

& = & E_\lambda \langle\frac{d\psi_\lambda}{d\lambda}|\psi_\lambda\rangle + E_\lambda\langle\psi_\lambda|\frac{d\psi_\lambda}{d\lambda}\rangle + \langle\psi_\lambda|\frac{d\hat{H}_\lambda}{d\lambda}|\psi_\lambda\rangle\\

& = & \langle \psi_\lambda |\frac{d\hat{H}_\lambda}{d\lambda}|\psi_\lambda\rangle

\end{eqnarray}$$

I am not sure if this is the correct direction to go but I figured it was worth a try. Any help or comments are appreciated.