- #1

Dixanadu

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## Homework Statement

Hey everyone!

The question is this:

Consider a two-state system with normalized energy eigenstates [itex]\psi_{1}(x)[/itex] and [itex]\psi_{2}(x)[/itex], and corresponding energy eigenvalues [itex]E_{1}[/itex] and [itex]E_{2} = E_{1}+\Delta E; \Delta E>0[/itex]

(a) There is another linear operator [itex]\hat{S}[/itex] that acts by swapping the two energy eigenstates: [itex]\hat{S}\psi_{1}=\psi_{2}[/itex] and [itex]\hat{S}\psi_{2}=\psi_{1}[/itex]. Show that the corresponding normalized eigenfunctions of [itex]\hat{S}[/itex] are [itex]\phi_{1}(x)=\frac{1}{\sqrt{2}}(\psi_{1}(x)+\psi_{2}(x))[/itex] and [itex]\phi_{2}(x)=\frac{1}{\sqrt{2}}(\psi_{1}(x)-\psi_{2}(x))[/itex], with eigenvalues [itex]\lambda_{1}=+1[/itex] and [itex]\lambda_{2}=-1[/itex]

## Homework Equations

- The regular eigenvalue/eigenvector stuff.

- In matrix mechanics, the wavefunction [itex]\psi_{n}(x)=C_{1}(1,0...n)^{T}+C_{2}(0,1,0...n)^{T}+...C_{n}(0....n)^{T}[/itex]

## The Attempt at a Solution

So I've found the matrix associated with the linear operator [itex]\hat{S}[/itex] in the natural basis [itex]U_{1}=(1,0)^{T}, U_{2}=(0,1)^{T}[/itex]; which is:

http://imageshack.com/a/img266/5713/bfo6.jpg [Broken]

This gives the correct eigenvalues of +1 and -1. But I dont know how to find the eigenfunctions that the question is talking about. I'm not even sure if the matrix is right...

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