Consider a three-level system where the Hamiltonian and observable A are given by the matrix Aˆ = µ 0 1 0 1 0 1 0 1 0 Hˆ = ¯hω 1 0 0 0 1 0 0 0 1 (a) What are the possible values obtained in a measurement of A (b) Does a state exist in which both the results of a measurement of energy E and observable A can be predicted with certainty? Why or why not? (c) Two measurements of A are carried out, separated in time by t. If the result of the first measurement is its largest possible value, determine the expectation value < ψ(t)|A|ψ)t) > for the second measurement.
Consider a three-level system where the Hamiltonian and observable A are given by the matrix Aˆ...
Problem 8.3 - A New Two-State System Consider a new two-level system with a Hamiltonian given by i = Ti 1461 – 12) (2) (3) Also consider an observable represented by the operator Ŝ = * 11/21 - *12/11: It should (hopefully) be clear that 1) and 2) are eigenkets of the Hamiltonian. Let $1) be an eigenket of S corresponding to the smaller eigenvalue of S and let S2) be an eigenket of S corresponding to the larger eigenvalue....
3) Consider a system whose Hamiltonian H and an operator A are given by the matrices 71 H = 60 -1 10 -1 1 0 0 0 -1) A = a 10 4 4 0 10 1 o) 1 0 where εo has the dimensions of energy. a) What are the possible values for the measurement of the energy? (3 marks) b) Suppose that the energy is measured, giving E = - Eo. What values are obtained if we subsequently...
(introduction to quantum mechanics) , the Hamiltonian matrix is H- 3. In the basis |1) - (a) Find the eigenvalues En and eigenfunctions Ion) of H. (b) The system is in state 2) initially (t 0). Find the state of the system at t in the basis n). (c) Calculate the expectation value of H. Briefly explain your result. Does it depend on time? Why? , the Hamiltonian matrix is H- 3. In the basis |1) - (a) Find the...
A measurement of a spin 1/2 observable described by the operator Sˆ = (Sˆx + Sˆy) is made and the system is found in a state corresponding to the largest eigenvalue of Sˆ. Find the probability that at a later time, a measurement of Sˆx will yield the value +ħ/2.
The Hamiltonian of a system in the basis In > is given by H = hw(" >< 0,1 + il" >< 421-142 >< 0,1 -21°3 >< $3D Here w is a constant. Write the Hamiltonian in the form of a matrix and obtain its eigenvalues and eigenfunctions. Express the eigenfunctions in terms of the basis In > and in its eigenvalues as En = hwe If the system is initially in the state | (0) >= 10 > a. What...
Let 0 E1 0 A 0 E/ Be the matrix representation of the Hamiltonian for a three-state system with basis |1), |2), |3) If the state of the system at time t-0 is lp(0)) 13), what are lp(t)) and c) d) e) If the state of the system at time t-0 is lp(0)) 2), what are lp(t)) and @ct-0)|ψ(t))? In terms of time evolution, explain the difference between them? Why are they different?
3. Consider a rigid rotor whose Hamiltonian is given by H L2(21) where L is the angular momentum operator and I is the moment of inertia of the rotator. Its rotation is described by a wave function: (0, N{Yo0(0,6)(1 3i) Y1-1(0,6) 2 Y21(0.0) Y20(0.) Find the normalization constant, N. (i) Find the probability to occupy state Yo0- (ii Find the expectation value of L2 of this state (iii Find the expectation value of L2 of this state (iv) Find (L2L2/21...
Please solve the problem as soon as possible. Problem 1: Consider a two level system with Hamiltonian: Using the first order time-dependent perturbation theory, obtain the probability coefficients cn (t) if the perturbation is applied at t >0 and the system is originally in the ground state. Hint: When solving the problem, first you may need to find the energies and wave functions of the unperturbed Hamiltonian A0. Problem 1: Consider a two level system with Hamiltonian: Using the first...
1. Consider the system described by: *(t) - 6 m (0) + veu(t): y(t) = 01 (1) 60 = {1, 1421 a) Find the state transition matrix and the impulse response matrix of the system. b) Determine whether the system is (i) completely state controllable, (ii) differentially control- lable, (iii) instantaneously controllable, (iv) stabilizable at time to = 0. c) Repeat part (b) for to = 1. d) Determine whether the system is (i) observable, (ii) differentially observable, (iii) instanta-...
1. Given a state y(r) expanded on the eigenstates of the Hamiltonian for the electron, H, in a hydrogen atom: where the subscript of E is n, the principal quantum number. The other two numbers are the 1 and m values, find the expectation values of H (you may use the eigenvalue equation to evaluate for H), L-(total angular momentum operator square), Lz (the z-component of the angular momentum operator) and P (parity operator). Draw schematic pictures of 1 and...