- #1

renec112

- 35

- 4

## Homework Statement

A particle is moving in a one-dimensional harmonic oscillator, described by the Hamilton operator:

[tex]H = \hbar \omega (a_+ a_- + \frac{1}{2})[/tex]

at t = 0 we have

[tex]\Psi(x,0) = \frac{1}{\sqrt{2}}(\psi_0(x)+i\psi_1(x))[/tex]

Find the expectation value and variance of harmonic oscillator

## Homework Equations

I want to use these equations. For varians:

[tex]\sigma_E^2 = \langle E^2\rangle - \langle E \rangle^2 [/tex]

For the energy

[tex]E_n = \hbar \omega(n+ \frac{1}{2})[/tex]

[tex]\Rightarrow \langle E \rangle^2 = (\hbar \omega(n+ \frac{1}{2}))^2[/tex]

and

[tex]\langle E^2\rangle = \langle \Psi | H^2 | \Psi \rangle[/tex]

## The Attempt at a Solution

Well i get

[tex]\ E = \hbar \omega [/tex]

[tex]\langle E \rangle^2 = \hbar^2 \omega^2 [/tex]

and by using the operators i get

[tex]\langle E^2 \rangle = \hbar^2 \omega^2 \frac{3}{4}[/tex]

wich of course means i get a bad varians

[tex]\sigma_E = \sqrt{-\frac{1}{4} hbar^2 \omega^2} [/tex]

Am i using the right method? And can you see where my calculations are wrong? It's quite a lot to write my calculations in with latex, so i would just like to hear if anyone can confirm or disagree with my method. I would love some input.