- #1
Dixanadu
- 254
- 2
Homework Statement
Hey dudes
So here's the question:
Consider the first excited Hydrogen atom eigenstate eigenstate [itex]\psi_{2,1,1}=R_{2,1}(r)Y_{11}(\theta, \phi)[/itex] with [itex]Y_{11}≈e^{i\phi}sin(\theta)[/itex]. You may assume that [itex]Y_{11}[/itex] is correctly normalized.
(a)Show that [itex]\psi_{2,1,1}[/itex] is orthogonal to the eigenstates [itex]\psi_{2,1,0}=R_{2,1}(r)Y_{1,0}(\theta,\phi)[/itex] and [itex]\psi_{2,1,-1}=R_{2,1}(r)Y_{1,-1}(\theta,\phi)[/itex] with [itex]Y_{1,0}≈cos(\theta)[/itex] and [itex]Y_{1,-1}≈e^{-i\phi}sin(\theta)[/itex].
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 [itex]\psi_{i}, \psi_{j}[/itex] that correspond to two different eigenvalues [itex]a_{i}, a_{j}[/itex] of an operator [itex]A[/itex], then [itex]A\psi_{i}=a_{i}\psi_{i}[/itex] and [itex]A\psi_{j}=a_{j}\psi_{j}[/itex]...so then [itex](a_{i}-a_{j})<\psi_{i}|\psi_{j}>=0[/itex]...right? but I don't know how to apply that here...