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Unformatted text preview: 5.61 Spherical Harmonics page 1 ANGULAR MOMENTUM Now that we have obtained the general eigenvalue relations for angular momentum directly from the operators, we want to learn about the associated wave functions. Returning to spherical polar coordinates, we recall that the angular momentum operators are given by: ⎛ ∂ ∂ ⎞ L ˆ x = − i ⎜ − sin φ − cot θ cos φ ⎟ ⎝ ∂ θ ∂ φ ⎠ ⎛ ∂ ∂ ⎞ L ˆ y = − i ⎜ cos φ − cot θ sin φ ⎟ ⎝ ∂ θ ∂ φ ⎠ ∂ L ˆ = − i z ∂ φ ˆ 2 ˆ 2 ˆ 2 ˆ 2 ˆ 2 2 ⎡ 1 ∂ ⎛ ∂ ⎞ 1 ∂ 2 ⎤ L = L x + L Y + L z ⇒ L = − ⎢ ⎜ sin θ ⎟ + 2 2 ⎥ ⎣ sin θ ∂ θ ⎝ ∂ θ ⎠ sin θ ∂ φ ⎦ In terms of these, our original Schrödinger Equation for rigid rotations was m L ˆ 2 m m HY ˆ = Y = E Y l l l l 2 I − 2 ⎡ 1 ∂ ⎛ ∂ ⎞ 1 ∂ 2 ⎤ m m + Y , = E Y l , ⇒ ⎢ ⎜ sin θ ⎟ 2 2 ⎥ l ( θ φ ) l ( θ φ ) 2 I ⎣ sin θ ∂ θ ⎝ ∂ θ ⎠ sin θ ∂ φ ⎦ where l was the quantum number for L ˆ 2 and m was the quantum number for L ˆ z . Taking what we learned in the last section about the eigenvalues of L ˆ 2 and L ˆ z we can say that at most we can have l = 0, 1 2 ,1, 2 3 ,2,... m = − l , − l + 1,..., l We will see that there is an additional restriction on the possible values of l in the present case, but these are the possible values for the quantum numbers. In terms of the quantum numbers, we have the eigenvalue relations ˆ 2 m 2 m 2 m L Y l = L Y l = l ( l + 1 ) Y l ˆ L Y m = mY m z l l Now, the functions, Y l m , that satisfy these relations for rigid rotations are called Spherical Harmonics . It is possible to derive the spherical harmonics by solving the 2D differential equation above. McQuarrie goes through a fairly complete derivation and we outline that solution in the appendix to these notes (below). The result is that: 5.61 Spherical Harmonics page 2 Y l m ( θ φ ) lm P l m im φ , = A ( cos θ ) e where A lm is a normalization constant and P l m ( x ) is an associated Legendre Polynomial. The first few Associated Legendre Polynomials are: P 0 ( cos θ ) = 1 P 1 0 ( cos θ ) = cos θ 0 P 1 ( cos θ ) = sin θ P 0 ( cos θ ) = 1 ( 3cos 2 θ − 1 ) 1 2 2 P 1 ( cos θ ) = 3cos θ sin θ P 2 ( cos θ ) = 3sin 2 θ 2 2 There are a number of important features of the Spherical Harmonics we can recognize simply by inspecting these solutions: • The wavefunctions factorize into a product of a function of θ and a function of φ ....
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This note was uploaded on 09/24/2011 for the course MATH 1090 taught by Professor Greenwood during the Spring '08 term at MIT.
 Spring '08
 greenwood
 Calculus

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