Question

xCThe State Of Sone System CanEX+ CBirzeit University [PS] https://ritaj.birzeitedu/bzu msgs/attach/887815/Extra problems ch3.pdf 212 If at time t = 0, the operator Q was measured, and the largest eigenvalue was obtained, write the wavefunction at any later time t. (c) If H was measured at any later time t, what might we get and with what probability 5. Consider a three-level system where the Hamiltonian and observable A are given by the matrix -1 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 s of the first measurement is its largest possible value, determine the expectation value < (t Aw)) > for the second measurement. 6. Consider a two level system, where the matrix representation of the position operator and its eigenvectors are given by 21 and that the Hamiltonian of the system is given by: vate Wind How satisfied are 'you with Magnifier? O Type here to search 110 AM 14/2019 3

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.

Answer 1

AX = lambda X Consider the above eigenvalue equation:Vertical-bar -lambda mu 0 mu -lambda mu 0 mu - lambda Vertical-bar= 0 + lambda (lambda^2 - mu^2) + mu (-mu lambda) = 0 lambda (lambda^2 - mu^2) - mu^2 lambda = 0 lambda (lambda^2 - 2 mu^2) = 0 lambda = 0, squareroot 2mu, - squareroot 2 mu The measurement of A can give 0, plusminus squareroot 2 mu In other words do H^^and A^^commute or not. AH = mu hwithstroke omega [0 1 0 1 0 1 0 1 0] [1 0 0 0 1 0 0 0 1] HA = mu hwithstroke omega [1 0 0 0 1 0 0 01] [0 1 0 1 0 1 0 1 0] \r\n No need to go any further Notice that H^^is proportional to identity matrix and AH - HA = 0 Lets find the eigen vector of A (which by the way would be linear combination of H^^s eigen vectors). Corresponding to +