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Finding normalized eigenfunctions of a linear operator in Matrix QM

  • Thread starter Dixanadu
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  • #1
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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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Answers and Replies

  • #2
dextercioby
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I think the hint I can give is to use that S is linear and consider a linear combination of the 2 vectors psi1 and psi2.
 
  • #3
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But is my matrix correct for S..?
 
  • #4
dextercioby
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Who cares, you needn't use any matrix to solve the problem.
 
  • #5
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what other ways are there that I can solve this? is it the bra-kets? And yea that matrix is wrong, it should be 0 along the diagonal and 1 otherwise. That's the only way it can be linear always and hermitian, which is the requirement.
 

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