How Do I Calculate Probability Amplitudes for a Perturbed Quantum State?

[Jumbuck]

For my homework, I have a problem in which a (harmonic oscillator) system is prepared in state n=2 for t<0.

For time t>0, there is a perturbation given by

V(t) = sqrt(3/4)*h_bar*omega* (|2><1| + |1><2|)

After this I need to compute the probability amplitudes. However, my background is in engineering, so I'm unsure how to work with these outer products of two state vectors, or even how this mixing works. If anyone has any hints or links on how to work with these, I would appreciate it very much.

Also, for future reference, do these forums automatically generate LaTeX, or do you import the LaTeX equations I've seen in other posts?

 

[Hurkyl]

In these forums, you can get the $ ... $ environment by using the [ itex ] ... [ /itex] tags. And you can get the \[ ... \] environment with the [ tex ] ... [ /tex ] tags. (Remove the spaces to use those tags) (note the direction of the slash)

 

[Hurkyl]

As to doing the algebra, just manipulate it formally. If you were faced with the product of |1><2| with |2>, that's given by |1><2|2> = |1>. Just remember that the distributive rules work (i.e. (A+B)C = AC + BC), but the commutative rule only works for scalars (i.e. for most S and T: $ST \neq TS$, but rS = Sr)

 

[Jumbuck]

As to doing the algebra, just manipulate it formally. If you were faced with the product of |1><2| with |2>, that's given by |1><2|2> = |1>. Just remember that the distributive rules work (i.e. (A+B)C = AC + BC), but the commutative rule only works for scalars (i.e. for most S and T: $ST \neq TS$, but rS = Sr)

Thanks!

Since these are state vectors, would

(|2><1| + |1><2|) * |2>) = |2><1|2> + |1><2|2> = |1> ?

I believe these state vectors are orthogonal, so the <1|2> term is 0, but my textbooks isn't very clear.

 

[Hurkyl]

Here's a useful little calculation: suppose that v and w are eigenstates of a hermetian operator T, with different associated eigenvalues. Then, compute:

$$\langle v | T | w \rangle$$

and

$$\langle w | T | v \rangle$$

Since these two expressions are complex conjugates of each other, it tells you something about $\langle v | w \rangle = \langle w | v \rangle^*$.