- #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.