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Homework Statement
Given the a|n> = α|n-1>, show that α = √n :
Homework Equations
The Attempt at a Solution
[tex]<n|a^{+}\hat {a}|n> = \alpha <n|a^{+}|n-1> = | \hat a|n>|^2 [/tex]
[tex] \alpha = \frac{<n|a^{+}\hat {a}|n>}{<n|a^{+}|n-1>}[/tex]
Taking the complex conjugate of both sides:
[tex] \alpha* = \frac{<n|a^{+}\hat {a}|n>}{\alpha<n-1|n-1>} [/tex]
[tex] |\alpha^2| = \frac{<n|a^{+}\hat {a}|n>}{<n-1|n-1>} [/tex]
Where ##\hat {a} a^{+} = \frac{1}{2m\omega} \left( (m\omega x)^2 + p^2 + im\omega [\hat{x},\hat{p}] \right) ##
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