CH5 - central potential and angular momentum

CH5 - central potential and angular momentum - Chapter 5...

This preview shows pages 1–10. Sign up to view the full content.

Central potential problems and angular momentum operator Chapter 5

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Schrödinger equation of central potential problems 2 2 () (,,) ( ,,) 2 Vr r E r λλ λ ψ θϕ μ ⎡⎤ −∇ + = ⎢⎥ ⎣⎦ = A particle interacts of mass in a central potential V ( r ). The energy eigenvalues and eigenvectors are defined by, 22 2 2 1 L r rr μμ ∂∂ ⎛⎞ = ⎜⎟ ⎝⎠ == = In spherical coordinate: 2 2222 2 11 sin sin sin xyz LLLL θ θθ =++= + = (,) ( 1 ) (,) lm lm z lm lm LY ll Y m Y ϕθϕ =+ = = = = × Lr p G GG where is the angular momentum operator, and for which spherical harmonics are eigenfunctions , 1, . ..., 1, mll l l = −− + l = non-negative integer z Li ϕ =− =
Spherical harmonics are common eigenfunctions of 2 22 2 11 (,) s i n ( 1 ) (,) sin sin lm lm lm LY Y ll Y θ ϕθ ϕ θθ θϕ ⎡⎤ ∂∂ ⎛⎞ =− + = + ⎢⎥ ⎜⎟ ⎝⎠ ⎣⎦ == zlm lm i Y m Y ϕθϕ = 2 * '' ' ' 00 sin ( , ) ( , ) l l mm dd Y Y π δ = ∫∫ Orthonormality and completeness: () ( ) * 0 1 (' ,' ' ' sin l lm lm lml YY ∑∑ 00 1 4 Y = 11 3 sin 8 i Ye φ 10 3 cos 4 Y = * , (1 ) m l m 2 , z LL

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Projection on spherical harmonics 2 * '' ' ' 00 sin ( , ) ( , ) l m lm ll mm dd Y Y π ϕθθ θ ϕ δ = ∫∫ 0 (,) l lm lm ml l Fc Y ϕθφ = −= = ∑∑ A general function of angular variables F can be expanded as By 2 * sin ( , ) ( , ) lm lm cd d F Y θϕ = Similarly a general wave function 0 (, , ) () (, ) l lm lm l rr cY = Ψ= 2 * () s in (, , ) (, ) lm lm cr d d r Y
0 1 2 3 0 2 4 6 0 0.05 0.1 0.15 0.2 0 1 2 3 0 2 4 6 0 0 1 2 3 0 2 4 6 0 0 1 2 3 0 2 4 6 0 2 55 || Y 2 54 Y 2 53 Y 2 52 Y θ φ Examples of spherical harmonics

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
0 1 2 3 0 2 4 6 0 2 4 5 2 5 5 || m m Y =− Examples of superposition of spherical harmonics θ φ Angular wave packet is formed by superposition of spherical harmonics. 0 1 2 3 0 2 4 6 0 2.5 5 7.5 10 8 2 8 8 m m Y
22 2 2( 1 ) () 2 2 E ll k l dd l l Vr R r ER r rdr dr r μ ⎡⎤ ⎛⎞ + −+ + + = ⎢⎥ ⎜⎟ ⎝⎠ ⎣⎦ == Let then the angular part is separated. For each l , the radial equation reads, (, , ) () (, ) Elm El lm rR r Y λ ψ ψθ ϕ θ = = There is a 2 l +1 degeneracy, because same energy for states , 1,. ..., 1, ml l = −−+ 2 , 0 HL = • Eigenfunctions of are eigenfunctions of H 2 L • Eigenfunctions of are eigenfunctions of H z L z Li =− = [ ] 0 z = (,) zlm lm LY i Y m Y ϕθϕ =

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Example: Free particle in 3D 2 2 2 E λλ ψ μ −∇ = = (, , ) () (, ) Elm El lm rR r Y θϕ = 22 2 2( 1 ) () 2 2 El El dd l l R rE R r rdr dr r ⎡⎤ ⎛⎞ + −+ + = ⎢⎥ ⎜⎟ ⎝⎠ ⎣⎦ == 2 k E = = ( ) E ll R rC j k r = Use The solution of the radial equation satisfying the boundary condition at r = 0 is Spherical Bessel function ( ) (, ) Elm l lm j k r Y = C is a normalization constant Free-particle energy eigenfunctions can be labeled by two discrete indices l and m and the continuous index E (or k )
Spherical Bessel differential equation and its solutions 2 22 2( 1 ) 1( ) 0 l dd l l f d d ρ ρρ ⎡⎤ + ++ = ⎢⎥ ⎣⎦ Spherical Bessel differential equation There are two kinds of solutions: 1s i n () ( ) l l l d j d ⎛⎞ =− ⎜⎟ ⎝⎠ Spherical Bessel function 1c o s l l l d n d =− − Spherical Neumann function Behavior at small ρ : () 1 3 5 . .. (2 1) l l j l ρ→ ⋅⋅ ⋅ + 0 1 1 3 5 . .. (2 l l l n +

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 12/10/2011 for the course PHYS 4221 taught by Professor Cflo during the Spring '11 term at CUHK.

Page1 / 46

CH5 - central potential and angular momentum - Chapter 5...

This preview shows document pages 1 - 10. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online