Proof of relationship between Hamiltonian and Energy

[andre220]

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.

 

[Orodruin]

Yes, your approach looks reasonable and will lead you correct. One small note on notation: I would write
$$
\frac{d\left| \psi\right>}{d\lambda}
$$
instead of
$$
\left|\frac{d\psi}{d\lambda}\right>.
$$

 

[andre220]

Okay thank you for your reply. Now for the second part I have
$$\frac{\partial E_n}{\partial a} = -\frac{n^2\pi^2\hbar^2}{m a^3}$$. Now I need the average force on the right "wall", and my thought to get that is using $-\vec{\nabla} U$, then
$$\begin{eqnarray}
\langle F_\textrm{right}\rangle &= & - \int \psi_\lambda^*\vec{\nabla}\psi_\lambda\,dq\\
& = & -\int\psi_\lambda^* \frac{dU}{dx}\psi_\lambda\, dx
\end{eqnarray}$$
So my thought here is to just use the wavefunction for the infinitely deep potential well and more less plug-and-chug, but I could see some problems with that. However, I suppose in some sense this method is using the knowledge from part a.