Dixanadu
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Homework Statement
Hey dudes
So here's the question:
Consider the first excited Hydrogen atom eigenstate eigenstate \psi_{2,1,1}=R_{2,1}(r)Y_{11}(\theta, \phi) with Y_{11}≈e^{i\phi}sin(\theta). You may assume that Y_{11} is correctly normalized.
(a)Show that \psi_{2,1,1} is orthogonal to the eigenstates \psi_{2,1,0}=R_{2,1}(r)Y_{1,0}(\theta,\phi) and \psi_{2,1,-1}=R_{2,1}(r)Y_{1,-1}(\theta,\phi) with Y_{1,0}≈cos(\theta) and Y_{1,-1}≈e^{-i\phi}sin(\theta).
Homework Equations
I don't think there is any...
The Attempt at a Solution
I'm completely dumbfounded here. So i have no idea...i know that orthogonality can be tested by applying the same operator on two eigenstates..for example, if we have two states \psi_{i}, \psi_{j} that correspond to two different eigenvalues a_{i}, a_{j} of an operator A, then A\psi_{i}=a_{i}\psi_{i} and A\psi_{j}=a_{j}\psi_{j}...so then (a_{i}-a_{j})<\psi_{i}|\psi_{j}>=0...right? but I don't know how to apply that here...