Homework Set 6 - Introduction to Quantum Mechanics I | PHYS 5250, Assignments of Quantum Mechanics

Download Homework Set 6 - Introduction to Quantum Mechanics I | PHYS 5250 and more Assignments Quantum Mechanics in PDF only on Docsity! Leo Radzihovsky, Fall 2007 PHYS 5250: Quantum Mechanics - I Homework Set 6 Issued November 5, 2007 Due November 26, 2007 Reading Assignment: Shankar, Ch. 12; Sakurai, Ch.3.1-3.6 1. Show that any function H(x) of a scalar product x = E1 · E2 of two arbitrary vector operators E1, E2 commutes with all components of the angular momentum opera- tor L. This thereby demonstrates explicitly that a rotationally-invariant Hamiltonian (i.e., one that consists only of scalar products of vector operators) commutes with the angular momentum vector. 2. Show that the z-component of the angular momentum operator Lz = −ih̄(x∂y − y∂x) can be written as Lz = −ih̄∂φ, where φ is the azimuthal angle of the spherical coordi- nate system. 3. Three identical, elementary bosonic particles of mass m are confined to move on a circle of radius r and are rigidly fixed to form an equilateral triangle, so that their only degree of freedom is the common azimuthal angle φ. Ignoring spin, compute the wavefunction and the corresponding energy spectrum. Hint: Think about the symmetry of the eigenfunctions ψE(φ). 4. Express lowest six spherical harmonics Y`,m(θ, φ) in Cartesian coordinates, x, y, z. 5. (a) A particle is described by a wavefunction ψ(ρ, φ) = Ae−ρ 2/ρ20 cos2 φ. By expressing ψ in terms of eigenstates of Lz, show that probability P (`z) of finding the particle with the z component of angular momentum equal `z is P (0) = 2/3 and P (±2h̄) = 1/6 and zero for all other values of `z. (b) Another particle is described by a wavefunction ψ(x, y, z) = A(xy+yz+xz)e−r/r0 . What is the probability of finding the particle with a particular value of its square of angular momentum? If the particle is found to have ` = 2, what are the relative probabilities of finding this particle with m = ±2,±1, 0? 6. (Shankar 12.5.14) Rotation of ` = 1 eigenstates 7. Harmonic Oscillator (a) Study a two-dimensional harmonic oscillator in polar (ρ, φ) coordinates i. Find the energy spectrum and the corresponding eigenstates. For the eigen- states it is sufficient to establish the series solution to the corresponding differ- ential equation and to determine the recursion relation for the series-solution coefficients ci. Hint: To help guide you, for the steps outlined in problem 12.3.7 of Shankar. ii. Show the spectrum En by marking (with a short horizontal dashes) energy levels for the lowest 10 states labelled by n,m, where n is the principle (radial) quantum number and m the eigenvalue of the z-component of the angular momentum Lz/h̄. Make the horizontal axis the m-axis, so that two states with the same value of n but different value of m are shifted horizontally relative to each other, with each appearing at its correct value of m along the horizontal axis. iii. Deduce the degeneracy of energy En, confirming that it agrees with that obtained solving this problem via Cartesian coordinates x, y. iv. Write down explicit wavefunctions ψn,m(ρ, φ) for the lowest eigenstates (n = 0,m = 0), (n = 1,m = ±1), (n = 2,m = 0,±2), and verify explicitly that each of these are linear combinations of corresponding degenerate eigen- states ψnx,ny(x, y) = Hnx(x)e −x2/2Hny(y)e −y2/2. Please make this connection explicit. Hint: In establishing the relation, please do not worry about any normaliza- tion constants. (b) Above Hamiltonian is a good model for a noninteracting (atoms do not see each other, only the harmonic trap potential) atomic gas confined to a harmonic trap potential. Consider what happens if one rotates such a gas at frequency Ω, as is now routinely done by our colleagues (maybe even some of your classmates) in JILA. The corresponding Hamiltonian for the rotating system is given by an addition of a single term, −ΩLz to the Hamiltonian above. Find the energy spectrum for such rotating atomic gas and plot it the way you did above, doing it for Ω < ω. What do you think happens when Ω → ω and faster rotation? Hint: Having found the spectrum for the stationary gas, you should simply be able to write down the spectrum for the rotating one without doing any calculations. Regarding Ω → ω limit, think about what happens for a corresponding classical problem of a particle trapped in a quadratic (harmonic) potential with frequency ω, and rotated with angular frequency Ω. (c) Guided by the discussion of the last section of Ch. 12 in Shankar (also see prob- lem 12.6.11) for the 3d harmonic oscillator solved in spherical coordinates and characterized by quantum numbers n, `,m, answer the following: i. Compute the degeneracy of the n-th energy level of a 3d harmonic oscillator, confirming that it agrees with the result you found on a previous homework using Cartesian coordinates D3dn = (n+ 1)(n+ 2)/2.