- #1
cacofolius
- 30
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Homework Statement
Hi, I'm trying to familiarize with the bra-ket notation and quantum mechanics. I have to find the hamiltonian's eigenvalues and eigenstates.
##H=(S_{1z}+S_{2z})+S_{1x}S_{2x}##
Homework Equations
##S_{z} \vert+\rangle =\hbar/2\vert+\rangle##
##S_{z}\vert-\rangle =-\hbar/2\vert-\rangle ##
##S_{x} \vert+\rangle =\hbar/2\vert-\rangle##
##S_{x} \vert-\rangle =\hbar/2\vert+\rangle, ##
The states basis is ##\vert++\rangle,\vert+-\rangle, \vert-+\rangle, \vert--\rangle ##
3. The Attempt at a Solution
What I did was apply the hamiltonian to each basis ket
##H\vert++\rangle =(S_{1z}+S_{2z})\vert++\rangle + S_{1x}S_{2x}\vert++\rangle
= \hbar/2\vert++\rangle + \hbar/2\vert++\rangle + \hbar/2\vert-+\rangle . \hbar/2\vert+-\rangle = \hbar/2\vert++\rangle##
##H\vert+-\rangle = 0##
##H\vert-+\rangle = 0##
##H\vert--\rangle = -\hbar/2\vert--\rangle##
My questions:
1) Is it right to consider ##\vert-+\rangle . \vert+-\rangle = 0##, (since they're orthogonal states)? Because they're both ket vectors (unlike the more familiar ##<a|b>##).
2) In that case, is the basis also the hamiltonian's, with eigenvalues ##\hbar/2, -\hbar/2, 0## (degenerate) ?