Quantum Mechanics II HW 9: Time-Dependent Perturbations & Electromagnetic Fields, Study notes of Quantum Mechanics

Download Quantum Mechanics II HW 9: Time-Dependent Perturbations & Electromagnetic Fields and more Study notes Quantum Mechanics in PDF only on Docsity! PHY662 - Quantum Mechanics II HWK #9, Due Tues., Mar. 30, at the start of class • Reading:  Read up to (but not including) the higher orders (Sec. 18.3) in Shankar and start to read Sec. 18.4 on electromagnetism.  Sections 9.1 and 10.1 of Griths might be useful for review and a dierent perspective. 1. Practice with creation and annihilation operators. [2 pts] Consider a par- ticle in a 1D harmonic well. (a) [This is just Ex. 18.2.1 in Shankar] Let your particle be in the ground state |0〉 at large negative times t. Expose the particle to the time- dependent perturbation H1(t) = −eEX/[1 + (t/τ)2]. This corre- sponds to an electric eld of time varying strength. To rst order in the time-dependent part, compute the probability that the particle will be in the rst excited state of the well at large positive times t. (b) Let your particle be in the third excited state |3〉 at large negative time t. For the perturbation of part (a), what are the possible states the particle can be in at large positve t, if you compute transition probabilities to rst order in the perturbation? What are the prob- abilities for the particle to be in these other states, to rst order in E? 2. Dierent types of changes in the Hamiltonian. [3 pts] Again, consider a particle in a 1D harmonic well, with Hamiltonian H− = k2X 2 + 12mP 2 at large negative time t. Let the particle initially be in the ground state, that is |ψ(t = −∞)〉 = |0〉, in each part below. (a) If H is equal to H− for all t < 0 and H = H+ = k ′ 2 X 2 + 12mP 2 for t > 0, what is the probability that the particle is in the ground state of the new Hamiltonian H+ at positive t? (b) If H(t) = H− for t < 0, H(t) = (1− tT )H − + ( tT )H + for 0 < t < T , H(t) = H+ for t > T , what is the probability that the particle is in the ground state of H+ at time T , in the limit of large T? (c) If H(t) = H− + uX2δ(t), what is the probability that the parti- cle is in an excited state of H− at large positive t? (Note that H(t) =Compute your answer using time-dependent perturbation the- ory to rst order in u and the notes from class. (Please note that, to rst order, there is only one excited state that can be reached by this perturbation - during your calculation, you should determine what state the particle can be excited to.) 1 3. Lenz's Law via quantum mechanics. [5 pts] Classically, when you apply a time-changing magnetic eld to a loop of conducting material, you create a current in the loop that opposes the change in the magnetic eld. This is Lenz's law. Based on last week's homework (#8), consider the quantum dynamics of a spinless electron in a circular ring. (Spinless means we neglect the spin degree of freedom, so that there is no −~µ · ~B part of the Hamiltonian). You will compute how a changing magnetic eld might induce currents in a ring that connes a single electron. First, for review of notation and concepts, consider the general problem of a charged particle in the presence of electromagnetic elds. The vector potential ~A is a eld such that the magnetic eld can be written as ~B = ∇ × ~A. It is possible to write ~B in this fashion, using ∇ · ~B = 0 and vector calculus. The electric eld is then ~E = −∇φ − 1c ∂ ~A ∂t , where φ is the electrostatic potential. Remember also that ~A can be transformed by adding a gradient of a scalar eld, ~A → ~A + ∇χ(~r), without aecting ~B or any other physically measurable quantity. This insensitivity is called gauge invariance. If ~A = 0, the kinetic part of the Hamiltonian is 12µ~p 2 and the current of a particle is given by ~j(~r) = (ψ∗(~r)∇ψ(~r)−ψ(~r)∇ψ∗(~r))/2µ (here current is not electric current - this is just the velocity times the density - to get the electric current, you need to multiply this particle current by the charge q). For the Hamiltonian with ~A 6= 0, the kinetic part of the Hamiltonian is 1 2µ ( ~p− q ~A c )2 . (When φ 6= 0, the full Hamiltonian is 12µ (~p − q ~A c ) 2 + qφ, but for this problem, we will take φ = 0.) This part of the Hamiltonian can be derived from a correspondence with classical mechanics or can be viewed as the simplest gauge invariant Hamiltonian for a particle interacting with the electromagnatic eld. In the presence of a vector potential ~A, the locally conserved particle current is ~j(~r) = h̄ 2iµ (ψ∗∇ψ − ψ∇ψ∗)− q µc ψ∗ψ ~A . It is easily shown from the Hamiltonian in a presence of a magnetic eld that ∂ρ ∂t = −∇ ·~j , where the particle density ρ(~r) = ψ∗(~r)ψ(~r). This justies the denition of the current ~j(~r). (You can try to derive this yourself, looking up dis- cussions of this topic for hints. We will also review how to do this in class.) 2