HomeworkSetIII

HomeworkSetIII - PHY4221 Quantum Mechanics (Fall Term Of...

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PHY4221 Quantum Mechanics (Fall Term Of 2005) Problem Set 3 (Suggested Solutions) Question 2: The Spherical Harmonic Functions (SHFs) are the common eigen functions of L 2 and L Z : () 22 1 lm lm Zl m l m LY l l Y m Y =+ = = = (1) Or Equation (1) is sometimes written as: ,1 , ,, Z Llm ll lm Llm mlm = = = (2) The raising angular operator L + , as defined as xy LL i L + = + , has the following property: ( ) , 1 Llm l ml m + =−+ + + = (3) So we have: ( ) 2,1 2 1 2 1 1 2,2 , 2 L + =− + + = = = (4) As a result: 1 2,2 2,1 2 L + = = (5) Or equally: 22 21 21 21 , 1 , 2 1 , 2 1 sin cot cos cos cot sin , 2 15 sin cot cos cos cot sin sin cos 8 2 15 sin 32 i i Y Li iY ie e φ θφ φθ θ θθ π + = ⎡⎤ ⎛⎞ ∂∂ + ⎢⎥ ⎜⎟ ⎝⎠ ⎣⎦ ++ = = = = = (6) Question 3:
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This is a typical question of central force field, for which the angular momentum is conserved. This is easy to show: Consider a particle moving in a central potential V ( r ), then we have: [] () 2 2 ,, 2 1 2 0 p LH L V r m Lp LV r m ⎡⎤ =+ ⎢⎥ ⎣⎦ = (1) In above we have used the property of L that it does not act on the radial component. Thus the time independent Schrodinger Equation could be written as: 22 2 1 L rV r E mr r mr ψ −+ + = = (2) Separating variables gives: ( ) ( ) 0,1,. .. , , 1,. ..,0,1,. .., ll m l rR r Y ml l l θφ = =← =− − + (3) Then Equation (2) leads to: () () () 2 1 1 r R r E R r + + = = = (4a) Or Equally Equation (2) could be rewritten as: ( ) 2 2 1 0 l dd m Rr EVr dr r dr r + ++ = = (4b) In the case of infinite spherical well we have, in the well, V =0, and thus Equation (4b) leads to: ( ) 2 2 1 2 0 l k dr r dr r + = (5) In Equation (5) we have defined: 2 2 2 m kE = (6) Equation (5) has the solution as the combination of the Spherical Bessel Functions: () ( ) ( ) l l l R rA j k rB n k r (7) But we know that when r 0 the second SBF n l diverges, so it should be excluded for physical consideration and we finally have:
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() ( ) ll l R rA j k r = (8) Furthermore, because the wave functions should vanish at the boundary of the spherical well,
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This note was uploaded on 12/10/2011 for the course PHYS 4221 taught by Professor Cflo during the Spring '11 term at CUHK.

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HomeworkSetIII - PHY4221 Quantum Mechanics (Fall Term Of...

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