- #1

milkism

- 118

- 15

- Homework Statement
- Finding P(x, t>0) and <x>.

- Relevant Equations
- See under.

I have this 1D LHO problem.

https://gyazo.com/4cd913d9da3a743443ef7dc2d1c2ab1e

For ##\psi_n (x)## I get

$$\left( \frac{\alpha}{\sqrt{\pi} 2^n n!} \right) ^{\frac{1}{2}} e^{\frac{- \alpha^2 x^2}{2}} H_n(x)$$

with ##E_n = (n+ \frac{1}{2}) \hat{h} \omega##. where ##\hat{h}## is hbar.

For ##\psi_{n+1}(x)## I get

$$ \left( \frac{ \alpha }{ \sqrt{\pi} 2^{n+1} (n+1)!} \right) ^{\frac{1}{2}} e^{ \frac{- \alpha^2 x^2}{2}} H_{n+1}(x)$$

with ##E_{n+1} = ((n+1)+ \frac{1}{2}) \hat{h} \omega##

We can find ##\Psi(x,t>0)## by multiplying the eigenfunctions with their corresponding factors and eigenenergies in the form of ##\e^{-\frac{i}{\hat{h} E_n t}}##, to find ##P(x, t>0)## we basically take ##|\Psi(x, t>0)|^2## which I think will be a long expression.

But how can we find <x>, if we don't know the actual expressions for the Hermite polynomials? How can we compute the integral?

https://gyazo.com/4cd913d9da3a743443ef7dc2d1c2ab1e

For ##\psi_n (x)## I get

$$\left( \frac{\alpha}{\sqrt{\pi} 2^n n!} \right) ^{\frac{1}{2}} e^{\frac{- \alpha^2 x^2}{2}} H_n(x)$$

with ##E_n = (n+ \frac{1}{2}) \hat{h} \omega##. where ##\hat{h}## is hbar.

For ##\psi_{n+1}(x)## I get

$$ \left( \frac{ \alpha }{ \sqrt{\pi} 2^{n+1} (n+1)!} \right) ^{\frac{1}{2}} e^{ \frac{- \alpha^2 x^2}{2}} H_{n+1}(x)$$

with ##E_{n+1} = ((n+1)+ \frac{1}{2}) \hat{h} \omega##

We can find ##\Psi(x,t>0)## by multiplying the eigenfunctions with their corresponding factors and eigenenergies in the form of ##\e^{-\frac{i}{\hat{h} E_n t}}##, to find ##P(x, t>0)## we basically take ##|\Psi(x, t>0)|^2## which I think will be a long expression.

But how can we find <x>, if we don't know the actual expressions for the Hermite polynomials? How can we compute the integral?

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