- #1

knut-o

- 17

- 0

## Homework Statement

1. I got the wavefunctions:[tex]\psi _0=(\frac{m\omega }{\pi\hbar})^{\frac{1}{4}}\cdot e^{-\frac{m\omega}{2\hbar}\cdot x^2}[/tex],

and [tex]\psi _1=(\frac{m\omega }{\pi\hbar})^{\frac{1}{4}}\cdot \sqrt{\frac{2m\omega }{\hbar}}\cdot e^{-\frac{m\omega}{2\hbar}\cdot x^2}[/tex].

Also recomended by the task given to introduce two new variables:

[tex]\xi =\sqrt{\frac{m\omega }{\hbar}}\cdot x\\ \alpha=(\frac{m\omega}{\pi\hbar})^{\frac{1}{4}}[/tex]

Now, I am tolk to find <x>, <p>, <x

^{2}> and <p

^{2}>.

## Homework Equations

I am informed how to find <x> and <p>:

[tex]<x>=\int _{-\inf} ^{\inf} x|\psi(x,t)|^2dx=\inf\psi*(x)\cdot \psi dx[/tex]

[tex]<p>=-i\hbar \int \psi* \cdot\frac{\partial\psi}{\partial x}dx[/tex]

I also wonder what the * stands for, it's not a normal multiplication-sign is it?

## The Attempt at a Solution

What I am mostly curious about, is how do I find <x

^{2}> and <p

^{2}>?

I have also found:

[tex]\psi _0=\alpha\cdot e^{-\frac{\xi ^2}{2}}\\ \psi _1=\alpha\cdot\xi\cdot e^{-\frac{\xi ^2}{2}}[/tex].

Do I, when I calculate

[tex]<x>=\int _{-\inf} ^{\inf} x|\psi(x,t)|^2dx=\inf\psi*(x)\cdot \psi dx[/tex] get insterted for x [tex]x=\xi \cdot\sqrt{\frac{\hbar}{m\omega}}[/tex] and [tex]\frac{d\xi}{dx}=\sqrt{\frac{m\omega}{\hbar}}\Rightarrow dx=d\xi\cdot\sqrt{\frac{\hbar}{m\omega}[/tex]? Giving even more variables to work with in thei ntegratian/calculation?

And to find <x^

^{2}>, do I simply just square the function standing inside there, giving me [/tex]|\psi |^4[/tex] and the function I calculate for <p> and just square it?

I am so not getting this thing..