p2-342 - the wavefunction Some possibly useful data the...

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Problem Set 2 Physics 342 Due January 21 Some abbreviations: S - Shankar. 1 . Consider a δ -function potential in two dimensions: V ( ~x ) = 2 ( ~x ). Recall that the Laplacian in two dimensions is 2 f = ± 2 r + 1 r r + 1 r 2 2 θ ² f. (i) Express the δ -function potential in polar coordinates ( r,θ ) (recall the basic property of a δ -function to check your answer) then write the Hamiltonian for a particle of mass m in terms of some purely radial terms, L z and V ( r,θ ). (ii) Consider the sector with zero angular momentum. Is there a bound-state for this system? 2 . Does this sound familiar? Consider a two-dimensional simple harmonic oscillator with Hamiltonian H = p 2 x 2 m + p 2 y 2 m + 1 2 2 ( x 2 + y 2 ) . (i) Does the angular momentum operator L z commute with H ? In terms of polar coordi- nates r and θ , what general statement can you make about the eigenstates? (ii) Consider the superselection sector with definite angular momentum n ~ . Can you find the ground state wavefunction and energy in this sector? Do not worry about normalizing
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Unformatted text preview: the wavefunction. Some possibly useful data: the form of the two-dimensional Laplacian in polar coordi-nates is given in problem (5). It might also be useful to recall that for a single SHO, the unnormalized eigenstates take the form: ψ n ( x ) = H n ( r mω ~ x ) e-mωx 2 / 2 ~ where H n ( x ) is a (Hermite) polynomial of order n in x . 3 . Show that the quantity J = X m Y * lm ( θ 1 ,φ 1 ) Y lm ( θ 2 ,φ 2 ) 1 is rotationally invariant. Use the above result to prove the spherical harmonic addition theorem P l (cos θ ) = 4 π 2 l + 1 X m Y * lm ( θ 1 ,φ 1 ) Y lm ( θ 2 ,φ 2 ) where θ is the angle between the directions specified by ( θ 1 ,φ 1 ) and ( θ 2 ,φ 2 ). 4 . Show that U = e i~σ · ˆ nφ = cos( φ ) + i~σ · ˆ n sin( φ ) where ˆ n is a unit vector. 2...
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This note was uploaded on 02/27/2010 for the course PHYSICS 342 taught by Professor Sghy during the Spring '10 term at King's College London.

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p2-342 - the wavefunction Some possibly useful data the...

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