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Laby

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## Homework Statement

Consider an electron bound in a hydrogen atom under the influence of a homogenous

magnetic field B = zˆB . Ignore the electron spin. The Hamiltonian of the system is H = H

_{0}−ωL

_{z},where

H0 is the Hamiltonian of the hydrogen atom with the usual eigenstates nlm and eigenenergies (0) En

(we use the superscript (0) to denote the unperturbed hydrogen atom), and ω =|e |B/(2m

_{e}c).

At t = 0 the system is in the state: |ψ(t=0)>=1/[itex]\sqrt{2}([/itex](|2,1,-1> -|2,1,1>)

For each of the following states

calculate the probability of finding the system at some later time t > 0 in that state:

|ψ

_{1}(t)>=1/[itex]\sqrt{2}(|2,1,-1> - |2,1,1>)

|ψ

_{2}(t)>=1/[itex]\sqrt{2}(|2,1,-1> + |2,1,1>)

|ψ

_{3}(t)>=1/[itex]\sqrt{2}(|2,1,0>

## Homework Equations

|ψ(t)>=U(t)*|ψ(t=0)>

U(t)=exp(-i*H*t/\hbar)

## The Attempt at a Solution

So the time evolution of a system can be described by multiplying the wavefunction at t=0 by the function U(t) which in this case would be exp(-i*(E

_{n}

^{(o)}-|e |B/(2m

_{e}c)*t/\hbar). But then from there, I'm a bit confused. Do I need to transform the equation into a matrix form and then take the square of the coefficients of the superimposed wavefunctions to get the probabilities? If that was the case, then I'd be able to represent |ψ

_{1}(t)>=1/\sqrt{2}[-1,0,1], |ψ

_{2}(t)>=1/\sqrt{2}[1,0,1] and then |ψ

_{3}(t)>=[0,1,0] right?

Thanks for your help.

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