L14. H atom

# L14. H atom - Schroedinger equation in spherical...

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Ψ ( r , θ , φ ) = Ε Ψ ( r , θ , φ ) Schroedinger equation in spherical coordinates Ψ ( r , θ , φ ) V(r)
E n proportional to 1/n 2 E n n 2 n 2 n

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V(r) V(r) Use of spherical coordinates, separation of variables and some algebraic manipulations yield two separate equations for radial and angular part β β β = positive constant
Separation of angular variables Equation for the rigid rotor β β

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Equation for the rigid rotor “split” into two separate equations β m 2 m 2 -m 2 -m 2
Single valued function Equation for the rigid rotor “split” into two separate equations : SOLVE Eq. I -m 2 Quantization of m

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Equation for the rigid rotor “split” into two separate equations : SOLVE Eq. II m 2 β quantization of b

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We have solved the rigid rotor equation phase

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We have solved the rigid rotor equation
Some Spherical Harmonics

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The Hamiltonian of the rigid rotor and the operator L 2 eigenvalues
we have seen a couple of weeks ago that:

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Angular momentum components in spherical coordinates

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V(r) V(r) Use of spherical coordinates, separation of

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L14. H atom - Schroedinger equation in spherical...

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