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Fund Quantum Mechanics Lect &amp; HW Solutions 59

# Fund Quantum Mechanics Lect &amp; HW Solutions 59 - 4.1...

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4.1. ANGULAR MOMENTUM 41 4.1.3.1 Solution anguc-a Question: The general wave function of a state with azimuthal quantum number l and magnetic quantum number m is Ψ = R ( r ) Y m l ( θ,φ ), where R ( r ) is some further arbitrary function of r . Show that the condition for this wave function to be normalized, so that the total probability of finding the particle integrated over all possible positions is one, is that integraldisplay r =0 R ( r ) R ( r ) r 2 d r = 1 . Answer: You need to have ( Ψ | Ψ ) = integraltext Ψ Ψ d 3 vectorr = 1 for the wave function to be normalized. Now the volume element d 3 vectorr is in spherical coordinates given by r 2 sin θ d r d θ d φ , so you must have integraldisplay r =0 integraldisplay π θ =0 integraldisplay 2 π φ =0 R ( r ) Y m l ( θ,φ ) R ( r ) Y m l ( θ,φ ) r 2 sin θ d r d θ d φ = 1 . Taking this apart into two separate integrals: integraldisplay r =0 R ( r ) R ( r ) r 2 d r integraldisplay π θ =0 integraldisplay 2 π φ =0 Y m l ( θ,φ ) Y m l ( θ,φ ) sin θ d θ d φ = 1 .
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