problem 3b
Problem 3a Assume the states(ln), n = 0,1,2, ) are mutually orthonormal: (nlm)-δυǐn ....
Problem 3a Assume the states(ln), n = 0,1,2, ) are mutually orthonormal: (nlm)-δυǐn . It is known that the operators a, and a. have the following properties: a,In) = vn + 11n + 1),n20 a-10)-0 The system's Hamiltonian is given by H-h Now, assume the system is prepared in a state described by the (unnormalized) superposition: V 1o) +11) a) Normalize this wavefunction. b) Compute the commutator of operators a, and a c) Compute the average energy (expectation value) of the system in this state. Hint: use the recursive relationships, bosonic operators and the representation of Hamiltonian in terms of these operators Problem 3b The conditions are the same as in problem 3a. Now, lets consider the action of the position operator on the states (In)) Here, a is just a constant parameterizing the states. Using this property, compute the expectation value of the position operator in the state Ψ-0; α1+ 11a).