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Homework Statement
The operator Q satisfies the two equations
[tex]Q^{\dagger}Q^{\dagger}=0[/tex] , [tex]QQ^{\dagger}+Q^{\dagger}Q=1[/tex]
The hamiltonian for a system is
[tex] H= \alpha*QQ^{\dagger}[/tex],
Show that H is self-adjoint
b) find an expression for [tex]H^2[/tex] , the square of H , in terms of H.
c)Find the eigenvalues of H allowed by the result from part(b) .
where [tex]\alpha[/tex] is a real constant
Homework Equations
The Attempt at a Solution
[tex]QQ^{\dagger}=1-Q^{\dagger}Q[/tex]
[tex]H=\alpha*QQ^{\dagger}=\alpha*(1-Q^{\dagger}Q)[/tex]
self adjoint of an operator
[tex](Q^{\dagger}\varphi,\phi)=(\varphi,Q\phi)[/tex] Should I take the conjugate of the operator H?
b)[tex]H^2=(\alpha)*(1-Q^{\dagger}Q)(\alpha)*(1-Q^{\dagger}Q)=(\alpha)^2*(1-Q^{\dagger}Q-Q^{\dagger}Q+Q^{\dagger}Q^{\dagger}QQ.) [/tex] since [tex]Q^{\dagger}Q^{\dagger}=0 [/tex] then the expression for H^2 is : [tex]H^2=(\alpha)^2*(1-Q^{\dagger}Q-Q^{\dagger}Q=(\alpha)^2(1-2*Q^{\dagger}Q)[/tex]. Now what?
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