- #1

etotheipi

- Homework Statement
- Show that $$\langle H \rangle = \sum_{n=1}^{\infty} E_n |c_n|^2$$

- Relevant Equations
- N/A

If we can identify ##|c_n|^2## as the probability of having an energy ##E_n##, then that equation is just the bog standard one for expectation. But the book has not proved this yet, so I assumed it wants a derivation from the start.

I tried $$

\begin{align*}

\Psi(x,t) = \sum_n c_n \psi_n(x)e^{-\frac{iE_n}{\hbar}t} \implies |\Psi|^2 &= \left(\sum_n c_n \psi_n(x)e^{-\frac{iE_n}{\hbar}t}\right)\left(\sum_n c_n^* \psi^*_n(x)e^{\frac{iE_n}{\hbar}t}\right)

\end{align*}$$And since the ##\psi_n##'s are orthogonal e.g. ##\int \psi_n \psi^*_m dx= \delta_{mn}##. Then I did$$\int |\Psi|^2 dx = \sum_n |c_n|^2 \int |\psi_n(x)|^2 dx = \sum_n |c_n|^2 = 1$$using the fact that all of the ##\psi_n(x)##s are normalised. That's consistent with ##|c_n|^2## being a probability, but it's still not a proof of anything. I wondered if someone could point me in the right direction? I thought to try $$\langle H \rangle = -\frac{\hbar^2}{2m}\int \Psi^* \left(\frac{\partial^2}{\partial x^2}\right) \Psi dx$$but that doesn't seem too helpful here... thanks!

I tried $$

\begin{align*}

\Psi(x,t) = \sum_n c_n \psi_n(x)e^{-\frac{iE_n}{\hbar}t} \implies |\Psi|^2 &= \left(\sum_n c_n \psi_n(x)e^{-\frac{iE_n}{\hbar}t}\right)\left(\sum_n c_n^* \psi^*_n(x)e^{\frac{iE_n}{\hbar}t}\right)

\end{align*}$$And since the ##\psi_n##'s are orthogonal e.g. ##\int \psi_n \psi^*_m dx= \delta_{mn}##. Then I did$$\int |\Psi|^2 dx = \sum_n |c_n|^2 \int |\psi_n(x)|^2 dx = \sum_n |c_n|^2 = 1$$using the fact that all of the ##\psi_n(x)##s are normalised. That's consistent with ##|c_n|^2## being a probability, but it's still not a proof of anything. I wondered if someone could point me in the right direction? I thought to try $$\langle H \rangle = -\frac{\hbar^2}{2m}\int \Psi^* \left(\frac{\partial^2}{\partial x^2}\right) \Psi dx$$but that doesn't seem too helpful here... thanks!

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