Advanced Quantum Mechanics – I — DU B.Sc. (Hons) Physics Previous Year Question Papers

2. Attempt FIVE questions in all. All questions cany equal marks. 4. Attempt FOUR more questions from the rest of the paper. 5. A few useful formulae are given at the end. Prove that all eigenvalues of a unitary operator lie on the unit circle on the complex plane, i.e. in is the eigenvalue of a unitary operator, then |d|2 = 1. Consider a state |Y) = ± |®i) + ± |®2) + a |@3) which is given in tens of three orthonormal eigenstates |®i), |a>2) and |a>3) of an operator G such that j5|®7t) = n2 lan). Find the expectation value of B for the state |T). Consider a system whose state is given in terms of an orthonormal set of three vectors: Let an ensemble of 810 identical systems, each one of them in the state |T). If measurements are done on all of them, how many systems will be found in each of the states: |®i), |®2) and |@3) ? d. For a Hamiltonian of a system defined as fi = £+ ¢(x), calculate [£,ff]. Also, calculate the uncertainty relation between f and fi. e. Prove that the commutation relation of two Pauli operators is f. For the case of angular momentum j = 1, find the matrices representing the operators f2 and /z. g. Let p be a density operator. Show that Tr(p) ≤ 1, with equality if and only if p is a pure state. a. Consider a system whose Hamiltonian is given by F= a(1ФХФ21+1Ф2ХФ11, where a is a real number having the dimensions of energy and ($1), |$z) are normalized eigenstates of a Hermitian operator A that has no degenerate eigenvalues. i. Prove that a 2H? is a projection operator and A is not. 4 marks i. Show that |Ф1) and |$2) are not eigenstates of A 3 marks iii. Calculate the commutators [E, |Ф,ХФ, |] and [A, |$2)Ф2|]. Then, find the relation that may exist between them. 3 marks iv. Find the normalized eigenstates of H and their corresponding energy eigenvalues. 12 and 3) are orthonormal, show that the states |4) = «($1) + 31|$2)- 1Ф3) and |X\ = (Ф1— ‹Ф½ + 5Ф3) satisfy the Schwarz inequality. 4 marks Consider a system whose initial state |Y (0)) and Hamiltonian F are given by If a measurement of energy is carried out, what values would we obtain and with what probabilities? 3 marks ii. Find the state of the system at a later time t, I*(t)). 3 marks iii. Find the total energy of the system at time t = 0 and at any later time t. i. Write down the Hamiltonian for a particle of mass m in a one-dimensional harmonic oscillator potential in terms of momentum Px and position i operators. il. If one defines new operators Show that the Hamiltonian can be expressed as 3 marks ili. Derive the commutation relation [5, 5t]. 3 marks iv. Using your result from (iii), show that the Hamiltonian is A = 7w (bbt + 4) 2 marks a. If  and B are Hermitian operators, show that (ÂB + BA) is Hermitian and (ÂB - BA) is not. 4 marks b. Consider a particle subjected to a one-dimensional harmonic oscillator potential.