CH5 - central potential and angular momentum

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

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Central potential problems and angular momentum operator Chapter 5
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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 =− =
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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
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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
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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
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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
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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 =
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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 )
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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 +
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CH5 - central potential and angular momentum - Chapter 5...

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