07_751_Final

# 07_751_Final - Physics 751 Final 18 December 2007 Only hand...

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Physics 751 Final 18 December 2007 Only hand in answers to five questions! Possibly useful info: Simple Harmonic Oscillator : , | 1| 1 2 mp ax i a n n n m ω ⎛⎞ =+ = + ⎜⎟ ⎝⎠ = + Angular Momentum Operators : 22 [, ] , | ( 1 ) | | | ,| ( 1 ) ( 1 ) | , 1 ij i j k k z xy JJ i J J j m jj j m J j m mj m i J J j m j j m m j m ε ±± = ⟩= + = + ± ± == = = () ( ) ( ) 1/2 01 0 1 4, 3 / 8 s i n , 3 / 4 c o s i YY e Y ϕ π πθ = θ 2 () ( ) 2 1 0 2 15/32 sin , 15/8 sin cos , 5/16 3cos 1 ii Ye Y e Y ϕϕ ± ± = 0 1 0 ,, 10 0 0 1 i i σ ⎛− = G ______________________________________________________________________________ 1 . A particle of mass m in one dimension is in the potential () ( ) ( ) ( ) ( ) 11 Vx x x a λδ μ δ δ + + . (a) Taking , find the boundary conditions satisfied by the wave function at 0, a for states of negative or zero energy. 2 1 2/ , mm λλμμ = 2 1 = (b) Assuming all the delta functions are attractive potentials, sketch qualitatively the ground state wave function when all the potentials are very weak but of comparable strengths. (c) Find the ranges of values of λ and μ for which a second bound state exists. (Include the possibility of repulsive as well as attractive potentials.)

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07_751_Final - Physics 751 Final 18 December 2007 Only hand...

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