Find the transition probability

[tgr042]

Applying an operator

to an initial state

can cause it to change into a different state

, if

is not an eigenstate of

. The probability for this transition to occur is

, where

is called the transition amplitude. Consider the case where the initial state is the

state of the harmonic oscillator,

, the final state is the

state of the harmonic oscillator,

, and the operator is

. (

is the position operator.) Find the transition probability from

to

.I really am not even sure where to start...

 

[QuasiParticle]

Look up the eigenfunctions of the harmonic oscillator, operate with the given operator on the initial state, integrate according to the definition of the transition amplitude.

 

[tgr042]

When you say on the initial state do you mean when psi(n)=1/sqrt(n) a+ psi(n-1) or do you mean psi(2)=1/sqrt(2) a+ psi(1)

 

[QuasiParticle]

I mean use the actual wave function $\psi_2$ of the harmonic oscillator. You can find it, e.g. here:

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc5.html
http://en.wikipedia.org/wiki/Quantum_harmonic_oscillator