5. (a) The Hamiltonian of a two-level quantum system is 1 H = 1 Find the possible initial states in which the probabilit

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5 A The Hamiltonian Of A Two Level Quantum System Is 1 H 1 Find The Possible Initial States In Which The Probabilit 1 (140.71 KiB) Viewed 37 times

5. (a) The Hamiltonian of a two-level quantum system is 1 H = 1 Find the possible initial states in which the probability of the system being in those quantum states do not change with time. Suppose the system is found in one of such initial state of H at t = 0, what will happen to that system if the system is perturbed by a Hamiltonian H' = -ħwo3 ? Will it still remain in the same state? C /p(x) -ħw (b) Prove that the variational principle can only predict the upper bound of the ground state energy of a Hamiltonian. (c) Let us consider the time independent Schroedinger equation in one dimension : h? du 2m dx² " where the eigenfunction corresponding to eigenvalue E is given by v(x) choosing appropriate approximation show that A(x): V(x) = 1 + V(x) = Ev, 1 and p(x) === = + =/ √ √p(x) dx, ħ Vo, 0, ∞, Express your answer in terms of Vo and E= where p(x) is classical linear momentum at x. Does this approximation valid for classical turning points? Explain your answer. (d) Use the WKB approximation to find the allowed energies (En) of an infinite square well with a "self", of height Vo extending half-way across: = if 0 < x < 1/1, if / < x <a, otherwise. (nπh)² 2ma2. A(x)ei(). Now