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