a) A particle is placed in the well with an energy E< Uo. Sketch the first three energy levels, and make sure to label each one. V(x) U. +2(x) L X X

7c.2. Evaluate for the total energy & at the following k-directions and evaluate at the BZ boundary. a. (1,0,0) b. (1,1,0) c. (1,1,1)

The kinetic energy operator in 3-D is: a. Î= – Invalid element O b. O C. O d. T= ħ2² 2m - ·D2 ħ 2m ħ 2m V V

6. An electron in hydrogen atom is in initial state Þ(r, 0) = A(2410o + iÞ210 + 421–1 – 2ib211) where wnim are the eigenfunctions of the hydrogen atom a. Determine the constant A b. What is the probability of finding the electron in the first excited state? hw = - n2 c. Write the state Þ(r, t) at time t, using energy eigenvalues as En d. Find the expectation value of L in the state Þ(r,t e. Find the expectation values of Lx and Ly in the state (r, t f. If measurement of Lz led to the value –ħ what will be results of measurement of energy and the square of total orbital momentum immediately afterwards and what are their probabilities?

Show that the equation in the picture is a fundamental equation by answering the following: a. Show that it is a homogenous first order function. b. Show that it has an exact differentiaI. c. Find the 3 equations-of-state. d. Find P=P(T, V, N)

A particle of mass m is in a harmonic oscillator potential with angular frequency o. It is in the state u(x) = (3u{n=1}(x) + 4u{n=3}(x))/5). a. Give the result of a single measurement of the energy. b. Give the expectation value of the energy. c. Give the expectation value of the momentum. d. Give the result of a single measurement of the momentum. Hint: Average energy is shared equally between the kinetic and potential energy for a simple harmonic oscillator.

The probability of finding a particle moves in the left quarter of a one-dimensional box of length a for energy level n =2 is: A. 15% В. 20% C. 25% D. 35% E. 50% A C B C c C D C E C

01: Choose the correct answer: a) The zero point energy of the 1-dimensional harmonic oscillator is: 1) hw, 2) zero 3)-hw.. b) The wave function has definite parity when the potential is: 1) symmetric 2) infinite 3) antisymmetric. c) In ô-function potential the bound states are founds when the energy of the particle (E) is: 1) E 0 3) E-0 3) L/2n. d) The wavelength associated with a particle in 1-D box of length L is: 1) 2L/n 2) L/n ay e) At boundary condition is always continuous except where: 1) V(x) = 2) V(x)=0 2) n=0 3) V(x)=constant. 3) n=00 1) n=1 f) The zero point energy of the 1-dimensional box occurred with: g) In classically forbidden regions, the wave function behavior is: 1) oscillatory 2) decay or grow exponentially.

In how many different configurations can four particles be distributed over a set of evenly spaced energy levels (AE=ɛ) such that the total energy is 4ɛ? Select one: а. 8 b. 11 С. 5 d. 3

P1) a) Using Numerov method (in an excel sheet), solve Schrodinger's equation for a particle in an infinite box; 2m dx? maž4 = Ep. where 0 at x = 0 and at x = 5 units. b) find the reduced energies E, for the first four levels. c) draw the reduced wavefunction 4, for the fourth level. 00 V(x) ma? - 8