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

This preview shows pages 1–2. Sign up to view the full content.

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

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.

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...
View Full Document

## 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.

### Page1 / 2

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online