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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 = H0 −ωLz ,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/(2mec).
At t = 0 the system is in the state: |ψ(t=0)>=1/\sqrt{2}((|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:
|ψ1(t)>=1/\sqrt{2}(|2,1,-1> - |2,1,1>)<br /> <br /> |ψ<sub>2</sub>(t)>=1/\sqrt{2}(|2,1,-1&gt; + |2,1,1&gt;)<br /> <br /> |ψ<sub>3</sub>(t)&gt;=1/\sqrt{2}(|2,1,0&amp;gt;&lt;br /&gt; &lt;br /&gt; &lt;h2&gt;Homework Equations&lt;/h2&gt;&lt;br /&gt; &lt;br /&gt; |ψ(t)&amp;gt;=U(t)*|ψ(t=0)&amp;gt;&lt;br /&gt; U(t)=exp(-i*H*t/\hbar)&lt;br /&gt; &lt;br /&gt; &lt;h2&gt;The Attempt at a Solution&lt;/h2&gt;&lt;br /&gt; &lt;br /&gt; 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&lt;sub&gt;n&lt;/sub&gt;&lt;sup&gt;(o)&lt;/sup&gt;-|e |B/(2m&lt;sub&gt;e&lt;/sub&gt;c)*t/\hbar). But then from there, I&amp;#039;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&amp;#039;d be able to represent |ψ&lt;sub&gt;1&lt;/sub&gt;(t)&amp;gt;=1/\sqrt{2}[-1,0,1], |ψ&lt;sub&gt;2&lt;/sub&gt;(t)&amp;gt;=1/\sqrt{2}[1,0,1] and then |ψ&lt;sub&gt;3&lt;/sub&gt;(t)&amp;gt;=[0,1,0] right? &lt;br /&gt; &lt;br /&gt; Thanks for your help.
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