Lunds Universitet Andreas Wacker, Johannes Bjerlin Spring 2015 Exercises to Quantum Mechanics FYSN17/FMFN01, Week 5 Homework to be handed in on February 24 Exercise 1: First-order correction of energy Aim: Learning how to apply the formalism of stationary perturbation theory ˆp = Consider a one-dimensional harmonic oscillator with frequency ω which is perturbed by H −αˆ x2 λe . Calculate the first order correction to the ground state energy and to the energy of the first excited state. Exercise 2 (Homework): Molecule vibration Aim: Applying stationary perturbation theory; understanding basic features of molecular vibrations A diatomic molecule can rotate and vibrate. To describe vibrations one can use a potential as in the figure where r denotes the distance between the nuclei. As a first approximation we may consider this as a one-dimensional problem (in x direction) with the parabolic potential 1 V0 (x) = mω 2 x2 − De 2 where m is the reduced mass for the system of both atoms and x is the deviation from the equilibrium position re . For higher excitation energies the potential deviates from the parabolic approximation and one can add a third order term: 3 ˆ p = α~ω 2mω 2 xˆ3 . H ~ ˆ p as a We shall in this exercise study the effect of H Graphical depiction of the Morse potential with a harperturbation. monic potential for comparison, CC-BY-SA Mark Soˆ p in terms of a and a† of the harmonic moza (2006) a) Express H oscillator corresponding to V0 (x). ˆ p |ni between unperturbed harmonic oscillator states. b) Determine all matrix elements hm|H c) Consider the second excited state. Calculate the change in energy in second order perturbation theory and its state in the first order. d) Calculate hxi with the new state and discuss its relation to thermal expansion of a crystal Exercise 3: Perturbation of two dimensional oscillator Aim: Getting familiar with degeneracies in perturbation theory ˆ p = λmω 2 xˆyˆ. A two dimensional harmonic oscillator is perturbed by H a) Use first order perturbation theory to calculate the energy shift for the ground state and the first excited state. b) Calculate the ground state energy to second order. c) Solve the full problem exactly and compare the result with the approximation you obtained in a) and b). Exercise 4 (Homework): Level crossing Aim: Applying stationary perturbation theory; understanding a generic scenario There is a rule in quantum mechanics that says: ’Perturbations remove level-crossings’. A level-crossing is when the Hamiltonian depends on a parameter and the eigenvalues of it cross if plotted as functions of this parameter. We shall investigate this in a very simple model. ˆ 0 and the perConsider a two dimensional state space where the unperturbed Hamiltonian H ˆ p are given by: turbation H kλ 0 0 c ˆ 0 (λ) = ˆp = H ,H (1) 0 −kλ c∗ 0 Here k is a positive real constant, c a constant, and λ a parameter (−1 < λ < 1). ˆ 0 as a function of λ. a) Plot the eigenvalues of H ˆ0 + H ˆ p and plot those in the same diagram. b) Calculate the eigenvalues of H c) Treat the system in perturbation theory and show that the second order-approximation becomes bad, when λ approaches 0. Exercise 5: 3-dimensional harmonic oscillator in magnetic field Aim: Training degenerate perturbation theory A charged particle with mass m and charge q is confined by a three dimensional harmonic ˆ 0 with frequency ω. A magnetic field is added which gives rise to a oscillator potential H perturbation ˆz. ˆ p = − q BL H 2m ˆ z in step operators. a) Express L b) Determine the splitting of the first excited state in first order in B. Exercise 6: Optical transition Aim: Learning how to apply the time-dependent perturbation theory Consider a one-dimensional harmonic oscillator with frequency ω0 and mass m. For t → −∞ the oscillator is in its ground state. Calculate the transition probability to the first excited 2 2 state for the perturbation Vˆ (t) = F xˆe−t /τ . Hint: replace the integration limits in Eq. (5.20) of the compendium by ±∞. Exercise 7 (Homework): Rabi Oscillation in a quantum dot Aim: Training time-dependent perturbation theory; comparing with exact result A quantum dot shall have two energy levels |1i with E10 = 0 and |2i with E20 = ∆E > 0. A laser field provides a periodic perturbation Vˆ (t) = 2Fˆ cos(ωt), which is in resonance with the level spacing, i.e., ∆E = ~ω. We assume the matrix elements 0 F hi|Fˆ |ji = F 0 with F ∈ R. Let the system be at t = 0 in the state |1i, which means that there is an electron in the ground level. a) Determine the transition probability P1→2 (t) as a function of time in lowest order in F using Eq. (5.20) of the compendium. b) Let |ΨD (t)i = a(t)|1i + b(t)|2i hold in the interaction picture. Provide the equations of motion for the amplitudes a(t) and b(t)! c) Solve the equation of motion with the approximation that the rapidly oscillating terms e±2iωt are negligible (which is called Rotating Wave Approximation). Compare |b(t)|2 with P1→2 (t) from part (a).