Quali cation Exam: Quantum Mechanics

Qualification Exam: Quantum MechanicsName:, QEID#43228029:July, 2019 Qualification ExamQEID#4322802921 Undergraduate levelProblem :QM-U-2 Consider two spin 1/2 particles interacting with one another and with an externaluniform magnetic field~Bdirected along thez-axis. The Hamiltonian is given byH= A~S1 ~S2 B(g1~S1+g2~S2) ~Bwhere Bis the Bohr magneton,g1andg2are theg-factors, andAis a In the large field limit, what are the eigenvectors and eigenvalues ofHin the spin-space in the basis of eigenstates ofS1zandS2z?2. In the limit when|~B| 0, what are the eigenvectors and eigenvalues ofHinthe same basis?3. In the Intermediate regime, what are the eigenvectors and eigenvalues ofHinthe spin space? Show that you obtain the results of the previous two parts inthe appropriate :QM-U-201. Show that, for an arbitrary normalized function| , |H| > E0, whereE0is the lowest eigenvalue A particle of massmmoves in a potentialV(x) ={12kx2, x 0+ , x <0(1)Find the trial state of the lowest energy among those parameterized by (x) =Axe x22 does the first part tell you aboutE0?}

tem, where the only condition imposed on is that it satis es the time-dependent Schr odinger equation. (This is part of Ehrenfest’s theorem. For simplicity as-sume that goes to zero exponentially as r!1.) 3.Consider an in nite well of width 2a, V(x) = ˆ 1; for jxj a 0; for a<x<a: At time t= 0 the wavefunction of a particle of mass min this ...

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  Time, Dependent, Equations, The time, Eroding, Hsrc, Dependent schr odinger equation

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