Adiabatic Time Evolution and Hamiltonian Perturbations - Prof. Thomas D. Cohen, Assignments of Quantum Physics

Download Adiabatic Time Evolution and Hamiltonian Perturbations - Prof. Thomas D. Cohen and more Assignments Quantum Physics in PDF only on Docsity! PHYS 402 Homework---Due Friday April 15 1. Consider a Hamiltonian which explicitly depends on time. At t=0 the Hamiltonian is 0Ĥ and at t=T it is fĤ . Suppose that at t=0 the system is in the ground state of 0Ĥ . We have argued in that if the time variation of the Hamiltonian is very slow (adiabatic), then at t=T it will be in the ground state of fĤ . In general, this does not mean, however, that there is no probability of finding the state in the ground state of 0Ĥ . Similarly we have argued that if the time variation is very fast, then at t=T the system will remain in the ground state of 0Ĥ . Again this does not mean that that there is no probability of finding the state in the ground state of fĤ . Show that the probability that the system is in the ground state of 0Ĥ at t=T for adiabatic time variations is exactly the same as the probability that it is in the ground state of fĤ for sudden ones. 2. A particle of mass m is in the ground state of a harmonic oscillator with natural frequency 0ω at t=0. At t=0 a perturbation of the form )1('ˆ /2202 1 τω texmH −−= is added on. Thus as ∞→t the system finds itself in a harmonic oscillator with a frequency of 02 ωω = . a. Find the state of the system at long time for the regime 10 >>τω (you may neglect phases). b. Find the state of the system at long time for the regime 10 <<τω (you may neglect phases). c. For the regime 10 >>τω what is the probability that the system at long times is in the ground state of the original harmonic oscillator. d. For the regime 10 <<τω what is the probability that the system at long times is in the ground state of the final harmonic oscillator. 3. A particle of mass m is in the ground state of an infinite spherical well of radius R. The walls of the well are slowly expanded to 2R. How much work does the particle to on the wall during this expansion?