# Quantum Mechanics Homework Solutions Review Of Related Literature In Educational Research (a) The coupling strength, is slowly increased over a duration T1 , to the value ) , with 1 . 0 (b) The detuning is then decreased to zero, over a very short duration T2 , while holding the coupling strength fixed, i.e. What condition on T2 sufficient to permit one to use the Sudden Approximation (Hint: it is the opposite of the adiabatic condition)?Assuming that your condition is satisfied, and keeping only the term in your previous expression for )i, what is T2 )i? What is the mean (d) Lastly, the detuning is adiabatically decreased to , over a duration, T4 .This problem can be solved directly in Mathematica: The possible results are a1 , a2 , a3 , a4 , with corresponding probabilities P (a1 ) 0.501511 , P (a2 ) 0.498068, P (a3 ) and P (a4 ) If result aj is obtained, the system collapses into state i.4 PHYS851 Quantum Mechanics I, Fall 2009 HOMEWORK ASSIGNMENT 4 1.  The course introduces quantum-mechanical operators, wave functions, Hilbert spaces, Heisenberg uncertainty principle, Heisenberg and Schrödinger formulations of quantum mechanics and their interpretation in terms of physical observations.Label the state corresponding to as and the other state as Using Dirac notation, express the Full Hamiltonian as an operator in terms of the kets and and the corresponding bras, and then again using the kets and and the corresponding bras. What are the limiting values of and , and their corresponding eigenvectors, in the limits and What do you expect to be different for the case 0? Adiabatic and Sudden Approximations A quantum system is prepared initially in the of H0 with a large, negative detuning, 0, and the coupling strength is initially zero, 0.(d) Sketch the energy spectrum versus for the case of fixed 0. In the following, when a state is requested, give two expressions for one using the basis and the other using where the later always refers to the instantaneous values of the system parameters at the specified time. Expand your results for the energy and the state to in .(chapters 1 - 15) Homework: There will normally be a homework assignment each week.Late homeworks will not be accepted without special permission, generally requested in advance. i i hf2 i R R 2 i dx hf2 i dx x2 , so that i i i i i i 2 R 2 i dx x2 i 0 parity. Numerically determine the Assume that the observable A is measured at t What are the probabilities P (an ) to obtain each possible result? What should c be so that han Solution: i i i can i an i) an i. Write an expression for the probability to obtain the result an . The Hamiltonian for this system is given in this basis H 4 X ihan (1) The system is prepared initially in the state state of the system at time t i 2 i). Now assume that the system is in an arbitrary state when A is measured.The Rabi Model: The standard Rabi Model consists of a bare Hamiltonian H0 and a coupling term V 2 (a) What is the energy, degeneracy, and state vector of the bare ground state for 0, 0, and 0? Write down the 2x2 Hamiltonian matrix in the basis and then compute the energy levels for the case 0.Use for the lowest eigenvalue, and for the highest (in energy). treating positive and negative detunings separately, and matching the limiting values of the dressed and bare eigenstates in the limits determine the normalized eigenvectors.(c) The parameters are then held fixed for duration T3 energy as a function of time during this duration? Give the adiabaticity condition on T4 , and give the state T2 T3 T4 )i.(e) Now we switch to a completely new system, whose Hamiltonian is also H0 .

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