Quantum Physics

5) If  is a Hermitian operator, then *  dx = f(¢) dx for wave functions and (optional proof available in Moodle in section 2). of a Hermitian operator  are orthogonal Using this relation, show that the eigenfunctions provided that they have different eigenvalues. Explain the significance of this result.

6. Now, turn the damping up to "lots" and move the wrench up and down. In a couple of sentences describe what you see happening to the wave as it travels down.

3. Click on the Rulers button on the bottom box. Then set the amplitude and frequency to the numbers below. After the numbers are set, play the wave and pause it after 5 seconds. Measure the distance between the top of one wave to the next wave. Record that as the Wavelength.Amplitude

1. In the simulation, switch to "Oscillate". Play with the Amplitude in the bottom box. What do you see happen to the wave on the string when you increase the amplitude?

2. Play with the Frequency in the bottom box. What do you see happen when you increase the frequency?

A spin-1/2 particle's state space has a basis |+), |-). On this basis the matrix representations of the spin operators are Su The particle is in a uniform magnetic field in the +x-direction, so the Hamiltonian for the particle is H = wS, where w = -7B. At t = 0 the wavefunction of the particle is ST = ħ 01 2 10 0 [[97] 2 i 0 1 ✓10 = ST 1 0 - [84] ħ 2 0 -1 [3]+) +-)]. 1. At t = 0, S₂, is measured. What are the possible results of this measurement, and what is the probability of each being obtained? = 2. Instead of measuring S₂, at t 0, it is measured at some later time t. What are the possible results of this measurement, and what is the probability of each being, obtained? Is S₂, a constant of the motion?

2. (30 points) A quantum mechanical particle is in an eigenstate ) of β with eigenvalue 2ħ²: At a particular moment, the particle also is in an eigenstate of the x component of the angular momentum, Î, with eigenvalue 0. In other words, . Express this eigenstate of I, as a normalized superposition of the familiar eigenstates, |lm), of β and Îz, where ο is the z component of the angular momentum.

a. Using the simple energy band diagram, draw the conduction band and valence band for silicon and label Ec, Ev and the bandgap value. b. In a separate band diagram show the process of the generation of an electron hole pair. c. In a separate band diagram show the process of the recombination of an electron holepair.

1. Discuss why simultaneous eigenstates of L² and L₂ are useful for solving rotationally invariant quantum mechanical problems. 2. What are the commutation relationships between a) L, and X, Y, or Z? b) L, and Px, Py, or P₂? c) L₂ and L, or Ly?

3) Consider a free particle wave packet for which the wave number distribution A(K) = { C for - Ak ≤ k ≤ Ak 0 elsewhere with constant C > 0. a) Determine the constant C so that A² (k) is normalized. b) Sketch A(k). c) Determine the associated spatial probability density (x, 0)|² at time t = 0. Determine the value of y(x,0)|² at x = 0. Use these results to sketch |(x,0)|². d) Approximating the width of A²(k) as Ak and the width Ax as the distance from the origin to the first minimum of y(x, 0)|², determine the product Ax Ap at time t = 0 and comment on your result. e) Write down an explicit integral expression for (x, t). You do not need to solve this integral!