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## Homework Statement

The problem is to show that,

[tex]\hat{a_{+}}|\alpha>=A_{\alpha}|\alpha+1>[/tex]

using

[tex]\hat{a_{+}}\hat{a_{-}}|\alpha>=\alpha|\alpha>[/tex]

It's not hard to manipulate [tex]\hat{a_{+}}\hat{a_{-}}|\alpha>=\alpha|\alpha>[/tex] into the form,

[tex]\hat{a_{+}}\hat{a_{-}}[{\hat{a_{+}}|\alpha>}]=(1+\alpha)[\hat{a_{+}}|\alpha>][/tex]

But I am unable to make the connection from this to,

[tex]\hat{a_{+}}|\alpha>=A_{\alpha}|\alpha+1>[/tex]

I know it's just using the properties of the eigenfunctions/values of a Hermatian operator at this point, but I seem to be missing exactly what that is.

What am I missing?