Practice Problems in Quantum Mechanics: Orbital Momentum and Harmonic Oscillator, Exams of Quantum Mechanics

Download Practice Problems in Quantum Mechanics: Orbital Momentum and Harmonic Oscillator and more Exams Quantum Mechanics in PDF only on Docsity! PRACTICE PROBLEMS 1. Find the commutators [l̂i, p̂2]; [l̂i, (p̂ · r̂)2]; [l̂i, (p̂ · r̂)r]; [l̂i, â†k]; [l̂i, x̂j p̂k]. (1) Here l̂i is the orbital momentum component and â † k is the creation operator for the oscillator quanta along the axis k. 2. Find the commutator [l̂i, l̂′j ] where l̂ and l̂ ′ are the orbital momentum operators with respect to the origin and the point a, respectively. 3. Decompose the orbital momentum of a two-body system into the relative motion orbital momentum and that of the center-of-mass motion. 4. Write down a normalized wave function with orbital momentum l and its z-projection m for a particle bound in a very thin spherical shell at a distance r = R from the center. 5. Write down the orbital momentum operator in the momentum representa- tion and the angular part of the momentum representation wave function for a particle in a state with orbital momentum quantum numbers l and m. 6. The wave function of a particle is ψ(r) = Axye−r 2/a2 . (2) Find the probabilities of various values of l, m and parity. 7. For a particle with orbital momentum l find the projection operator P̂ (m) which selects the state with a certain value m of l̂z. 8. a. For a particle with orbital momentum l = 1 construct the wave function ψ(θ, φ) for the state with the zero projection of the orbital momentum vector onto the axis defined by the polar angle α and asimuthal angle β. b. The same for the state with projection lx = 1 onto the x-axis. 9. For a particle in the state with the orbital momentum quantum numbers l = 1 and m find a. the probabilities of various values m′ of the orbital momentum projec- tion on the axis z′ which has an angle α with the respect to the z-axis; b. the expectation values 〈l̂3x〉 and 〈l̂3y〉. 1 10. A particle is placed into a two-dimensional isotropic (ωx = ωy) harmonic oscillator field. a. Find the energy levels and the wave functions of the stationary states. b. Determine the degeneracy of the stationary states. c. For the state with the Cartesian quantum numbers nx = ny = 1 find the probabilities of various values of the orbital momentum projection lz. d. Determine the approximate value of the ground state energy using the trial function ψ(ρ) = A exp(−βρ) and compare the result with the exact value. 11. For a spinless particle in a three-dimensional isotropic potential U(r), is it possible to have a level with the degeneracies d = 3, d = 4? 12. Which l values are possible for the states of the three-dimensional isotropic harmonic oscillator with N = 2? Find the combination of the states which corresponds to the s-wave. 13. Consider the set of operators ax̂ix̂j + bp̂ip̂j with some constants a and b. Show that, for a specific choice of those constants, these operators are constants of motion for a particle in the three-dimensional isotropic harmonic oscillator field; explain the underlying physics. 14. Find the expectation value 〈rs〉 for the ground state of the hydrogen-like ion with the charge Z of the nucleus. 15. Find the average electric field created at large distances by the hydrogen atom in the 2p-state with the projection lz = m. 16. Determine the approximate value of the ground state energy for the hy- drogen atom using the trial wave function ψ(r) = A exp(−β2r2). 17. For a particle in a ground state of the spherically symmetric potential well, U(r) = { 0, r ≤ R, ∞, r > R, (3) find the momentum distribution. 18. Consider a rotor, a particle with the rotational Hamiltonian Ĥ = h̄2 l̂2 2I , (4) where I is the moment of inertia. At the initial moment the wave function is given by Ψ(t = 0) = A cos2 θ. (5) Find the wave function as a function of time. 2