Angular Momentum. Classically the orbital angular momentum with respect to a fixed origin is. L = r p. = yp z. L x. zp y L y. = zp x. xpz L z.

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1 Angular momentum is an important concept in quantum theory, necessary for analyzing motion in 3D as well as intrinsic properties such as spin Classically the orbital angular momentum with respect to a fixed origin is L = r p L x = yp z zp y L y = zp x xpz = xp y yp x The vector L is perpendicular to the plane containing r and p

2 The quantum mechanical operators for orbital angular momentum are obtained by replacing the position and momentum variables by the corresponding operators obeying the canonical commutation relations: L x = yp z zp y = i y z z y L y = zp x xpz = i z x x z = xp y yp x = i x y y z

3 Commutation relations: L x, L y = yp zp, zp xp z y x z = yp z, zp x zp y, zp x yp, xp z z + zp, xp y z yp z,zp x zp y, zp x = yp z zp x zp x yp z = yp x [ p z, z] = i yp x = zp y zp x zp x zp y = zp x [ p y, z] = 0 yp z, xp z = yp zxp z xp z yp z = yp z [ p z, x] = 0 zp y, xp z = zp y xp z xp z zp y = xp y [z, p z ] = i xp y L x, L y = i (xp y yp x ) = i

4 Commutation relations: L x, L y = i L y, = i L x, L x = i L y The angular momentum components L x, L y and, do not commute with each other and are incompatible observables. Applying the generalized uncertainty principle: ΔL x ΔL y 2 ΔL x Δ 2 L y ΔL y Δ 2 L x

5 Orbital angular momentum: Commutation relations: Consider the square of the total angular momentum: L 2 = L 2 x + L 2 2 y + L 2, L x = L 2 x, L x = L y L y, L x = L y i = 0 + L 2 y, L x + L, L y x + L 2 z, L x L y +, L x ( ) + ( i ) L y + i L y + L, x ( ) + ( i L ) y

6 Commutation relations: L 2, L x = 0 L 2, L y = 0 L2, = 0 L 2 commutes with all the components of angular momentum. However, the components do not commute with each other. We can construct simultaneous eigenfunctions of L 2 and any one component of the angular momentum. Here we pick We seek to find eigenstates and eigenvalues such that L 2 l, m = λ l l, m l, m = µ m l, m

7 Spherical polar coordinates: x = r sinθ cosφ y = r sinθ sinφ z = r cosθ L x = yp z zp y = i 1 sinθ sinφ cosθ cosφz sinθ θ φ L y = zp x xpz = i 1 sinθ cosφ cosθ sinφ sinθ θ φ = xp y yp x = i ϕ

8 Eigenvalues and eigenfunctions of = xp y yp x = i ϕ Note that the operator is similar to the momentum operator for the angular coordinate φ. Eigenvalue equation: Solution: Φ m (φ) = i φ Φ m(φ) = µ m Φ m (φ) i Φ m (φ) = Ne µ mφ

9 Eigenvalues and eigenfunctions of : i Φ m (φ) = Ne µ mφ φ ranges from 0 to 2π. We require the function to be single valued, hence Φ m (φ + 2π ) = Φ m (φ) i Ne µ i m (φ + 2π ) = Ne µ i m (φ) e µ m (2π ) = 1 µ m = integer Eigenvalues of : µ m = 0, ±, ±2, ±3...

10 Eigenvalues and eigenfunctions of : i Φ m (φ) = Ne µ mφ µ m = 0, ±, ±2, ±3... Normalization: 2π Φ m (φ) 2 dφ = N 2 e 0 2π 0 i µ mφ e i µ mφ dφ = N 2 2π N = 1 2π Eigenfunctions of : Φ m (φ) = 1 2π eimφ m = 0,±1,±2,±3...

11 Eigenvalues and eigenfunctions of L 2 : In spherical coordinates, Eigenvalue equation: 1 L 2 = 2 sinθ θ sinθ 1 2 θ + sin 2 θ φ 2 L 2 (θ,φ) = λ l (θ,φ) We want to obtain common eigenfunctions so that (θ,φ) = µ m (θ,φ) ( θ,φ) = P lm ( θ )Φ m (φ) = P lm ( θ ) 1 2π eimφ m = 0,±1,±2,±3...

12 Eigenvalues and eigenfunctions of L 2 : 1 L 2 = 2 sinθ θ sinθ θ + 1 sin 2 θ 2 φ 2 Eigenvalue equation: L 2 (θ,φ) = λ l (θ,φ) Solution: ( θ,φ) = P lm ( θ )Φ m (φ) = P lm ( θ ) 1 2π eimφ Inserting this solution into the eigenvalue equation for L 2 : 1 sinθ d dθ sinθ d dθ + λ l 2 m2 sin 2 θ P lm ( θ) = 0

13 Eigenvalues and eigenfunctions of L 2 : 1 sinθ d dθ sinθ d dθ + λ l 2 m2 sin 2 θ P lm ( θ) = 0 Making the substitution w = cosθ The above equation becomes d dw 1 w2 ( ) d dw + λ l 2 m2 1 w 2 P lm ( w) = 0

14 Eigenvalues equation for L 2 : d dw ( 1 ) d w2 dw P (w) + λ lm l m2 1 w 2 P lm ( w) = 0 Consider the simplest case of m=0: d ( dw 1 ) d w2 dw P (w) + λ l lm P 2 lm This is the Legendre equation. The solutions are Legendre polynomials: ( w) = 0 P l (w) = 1 l d 2 l l! dw (w 2 1) l λ l = l(l + 1) 2 λ l 0

15 Orbital angular momentum: Eigenvalues and eigenfunctions of L 2 : Legendre polynomials P l (w) = 1 2 l l! d dw l (w 2 1) l P 0 (w) = 1 P 1 (w) = 1 2 P 2 (w) = 1 8 d dw d dw (w 2 1) = w 2 (w 2 1) 2 = 1 2 (3w2 1) (l + 1)P l+1 (w) = (2l + 1)wP l (w) lp l 1 (w) Recurrence relations

16 Orbital angular momentum: Eigenvalues and eigenfunctions of L 2 : Legendre polynomials P l (w) = 1 2 l l! d dw l (w 2 1) l Associated Legendre polynomials: P l m (w) = (1 w 2 ) m 2 d dw m P l (w)

17 Orbital angular momentum: Eigenvalues and eigenfunctions of L 2 : Associated Legendre polynomials: P l m (w) = (1 w 2 ) m 2 d dw m P l (w) Associated Legendre polynomials are solutions of the general eigenvalue equation for L 2 : d ( dw 1 ) dp (w) w2 lm dw + λ l 2 m2 1 w 2 P lm ( w) = 0 λ l = l(l + 1) 2, l m l

18 Orbital angular momentum: Eigenvalues and eigenfunctions of L2 and Lz: The solutions to the two eigenvalue equations L 2 (θ,φ) = λ l (θ,φ) (θ,φ) = µ m (θ,φ) are thus (θ,φ) = N lm P l m (cosθ)e imφ With l and m being integers such that l 0, l m l l : azimuthal quantum number m :magnetic quantum number Quantized angular momentum eigenvalues: λ l = l(l + 1) 2, µ m = m

19 Eigenvalues and eigenfunctions of L 2 and : are called the spherical harmonics Normalization: 2π 0 π (θ,φ) = N lm P l m (cosθ)e imφ (θ,φ) 2 sinθdθdφ = 1 N lm = 0 (2l + 1)(l m )! ( 1) m, m 0 4π(l + m)! (2l + 1)(l m )!, m 0 4π(l + m)! The spherical harmonics form a complete, orthonormal basis on a sphere.

20 Spherical Harmonics: Y 00 = 1 2 π Y 10 = Y 2,0 = 3 4π cosθ, Y = 3 1,±1 8π sinθe±φ 5 16π 3cos2 θ 1 ( ) Y 2,±1 = 15 sinθ cosθe±φ 8π Y 2,±2 = 15 32π sin2 θe ±2φ l=0: s states, l=1: p states, l=2: d states, l=3: f states