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Institut für Theoretische Physik Institut für Experimentelle und Angewandte Physik Prof. Dr. Christian Back, Prof. Dr. Klaus Richter SS 2009 Übungen zum Integrierten Kurs: Quantenmechanik Blatt 12 Assistent: Dr. Juan-Diego Urbina (Raum: 4.1.31, Tel: 943-2027) Di., 07.07.2008 Aufgabe 1: Variational method: fundamentals and the Helium atom Ω Remember that the variational method finds the best possible approximate solution |ψop i of the Schrödinger equation for a given subset Ω of the full Hilbert space H. The approximate ground Ω state |ψop i is optimal in the sense that minimizes the expectation value of the Hamiltonian among all members of Ω, namely, Ω Ω hψop |Ĥ|ψop i ≤ hψ Ω |Ĥ|ψ Ω i for any |ψ Ω i ∈ Ω a) Show that, if the optimal solution is sought over the whole Hilbert space (that is, Ω = H), the variational method leads to the Schrödinger equation. b) The Hamiltonian of the He (in Born-Oppenheimer approximation where the nucleus located at the origin does not move) is Ĥ = Ĥ (1) + Ĥ (2) + V̂ where Ĥ (i) = p̂2i 2e2 − 2me |r̂i | describe the two (i = 1, 2) non-interacting electrons, and V̂ = e2 |r̂1 − r̂2 | is the Coulomb interaction between the electrons. Z Z Z Take as variatonal ansatz |ψ Z >= |ψ100 >(1) |ψ100 >(2) > where |ψ100 > is the ground state of the ”Hydrogen” with nuclear charge Ze. Use this Z as variatonal parameter, and find the approximate ground state and energy of the Helium atom. Aufgabe 2: ”Exact” perturbation theory Consider a one-dimensional harmonic oscilator with hamiltonian Ĥ = 1 2 mw2 2 p̂ + x̂ 2m 2 perturbed by a linear potential V̂ = bx̂. a) Using perturbation theory, calculate the energies of Ĥ + V̂ up to first non-vanishing order in b. b) Solve the problem exactly, and compare. 1 Aufgabe 3: Sodium triple line In the emission spectrum of sodium, the following three neighboring lines have been found, 3 2 D → 2 2 P1/2 , 3 2 D → 2 2 P3/2 , 3 2 D → 2 2 P3/2 Take into account the transition rules and give the quantum numbers of the total angular momentum of the initial D-state. Aufgabe 4: Fine structure of Hydrogen-like Ions The fine structure of hydrogen-like ions (ions with only one electron) is described by the additional term hcRZ 4 α2 3 1 EF S = − , − n3 j + 1/2 4n where α = 1/137 is Sommerfeld’s fine structure constant, R is the Rydberg constant and Z is the atomic number. The expression takes into account relativistic effects and spin-orbital coupling. Thus the total energy is given by Enlj = Enl + EF S . a) Can n and j be chosen in a way so that the additional term disappears? What effect has this on the total energy? b) In how many different energy levels do the levels n = 3 and n = 4 of simply ionized helium split due to the fine structure interaction? c) Give the amount of the displacement of the energies relative to the “original” energies. d) Determine the allowed transitions taking into account the selection rules. Aufgabe 5: Zeeman splitting Sodium vapor with a temperature of T = 573 K is enclosed in a glass container. It emits light due to electron impact. In the visible range, the two Na-D-lines are observed at λ1 = 588.995 nm and λ2 = 589.592 nm. a) Sodium has 11 electrons. In the ground state, how are these electrons distributed to the individual shells s, p, d, f (notation 1s2 , . . . )? b) What is the spectroscopic denotation multiplicity XJ for the ground state of sodium? c) From what excited states do the Na-D-lines emanate? d) Calculate the energetic splitting in cm−1 from the wavelength. What determines this splitting? e) In a magnetic field B, the energy of a state J, L, S is described by an additional energy term. The g-factor is J(J + 1) + S(S + 1) − L(L + 1) gJLS = 1 + . 2J(J + 1) Calculate the g-factor for the three levels that take part in the formation of the Na-D-lines. f) Sketch the energy levels with the calculated Zeeman splitting. Denote the relevant quantum numbers unambiguously in the sketch. g) Calculate the Zeeman splitting in cm−1 for B = 2 T for the three terms. h) In this case, do we observe the normal or the anomalous Zeeman splitting? What is the difference between the two? 2 Aufgabe 6: Two dimensional square well with perturbation Consider a particle in a two dimensional infinite square well ( 0, 0 ≤ x ≤ a, 0 ≤ y ≤ a, V = ∞, otherwise. a) What are the energy eigenvalues of the three lowest states? Is it there any degeneracy? b) We now add a weak perturbing potential Vi = λxy, 0 ≤ x ≤ a, 0 ≤ y ≤ a. Obtain the energy shifts of the three lowest states to order λ. c) Draw an energy diagram with or without the perturbation and specify which unperturbed state is connected to which perturbed state. Aufgabe 7: Spin-spin interaction Consider a system consisting of two spin-1/2 objects. For t < 0, the Hamiltonian does not depend on spin and can be taken to be zero. For t > 0, the Hamiltonian is given by H= 4∆ S1 · S2. ~2 Suppose that the system is in the state | + −i for t ≤ 0. Find, as a function of time, the probability for being found in each of the following states | + +i, | + −i, | − +i, | − −i: a) By solving the problem exactly. b) By solving the problem assuming the validity of first-order time dependent perturbation theory with H as a perturbation switched on at t = 0. Under which condition does it give the accurate result in part (a)? Return in written form exercices 1) and 3) before Di. 14.07.2009 at 12:00. Do not write all the details, but the main intermediate steps. 3