Quantum Physics

4) a) Determine the adjoint operator to â = c + id, where c and d are real numbers and i = √-1. Is â Hermitian? b) By evaluating (E) - (E) with respect to a general normalized wave function, determine whether or not the operator = this Hermitian. Interpret your result.

What is the ground state electronic configuration and bond order of: a. F₂- b. N₂- c. 02^2-

Discuss the role of the Born-Oppenheimer approximation in the calculation of molecular potential energy curves such as the one below:

a) (10 points for 5383/10 points for 7383) If the messages are of variable length,comprised of strings of the symbols, and are all equally likely, compute the density matrix of the alphabet, pë, using the computational basis in your intermediate calculations. Show all intermediate calculations and clearly state all assumptions. d) (15 points for 5383/10 points for 7383) Assume the symbol |) is measured with the Observable A as given below. What is/are the possible measurement outcome(s)? And what is/are the possible “collapsed” state(s) associated with the outcome(s)? And, what are the probabilities that the measured state(s) collapse for each possible collapsed state(s)? \mathbf{A}=\left[\begin{array}{cc} 0 & -i \\ i & 0 \end{array}\right]

Problem 3. Consider a region of silicon which is single crystal silicon except for one arsenic atom that has been implanted as shown in the figure below. Note that an electron has been donated to the lattice and it moves from region 1 to region 2.

The graph below is a plot of intrinsic carrier concentration versus inverse temperature for 3 different semiconductors. Use the equation below combined with data points taken from the graph to determine the energy band gap for Silicon and GaAs. Assume that Nc and Ny do not vary with temperature. Hint: Set up a ratio of the equation using data points from the graph.

Question 4. A particle of mass m incoming from the left hand side is scattered from thedouble delta-function potential barrier V(x)=\frac{\alpha \hbar^{2}}{m}[\delta(x-a)+\delta(x+a)], \quad \alpha>0 . Draw the scattering diagram and show that the transmission coefficient can be written as T=\left|\frac{4}{(1+\beta)^{2}-\gamma^{4}(1-\beta)^{2}}\right|^{2} \text { where } \beta=1+2 i \alpha / k \text { and } \gamma=e^{i k a} \text { with } k=\sqrt{2 m E} / \hbar \text {. }

Show that Bohr's postulate which states that the angular momentum of an electron in the nthorbit (pe nh) is equivalent to an integer number of de Broglie waves fitting within the circumference of a Bohr circular orbit.=

Question 2. A particle of mass m in the presence of an infinite potential well of width a starts out in the wave function \Psi(x, 0)=\left\{\begin{array}{ll} \frac{2}{\sqrt{a}} \cos \frac{\pi x}{2 a} \sin \frac{3 \pi x}{2 a} & 0 \leq x \leq a \\ 0 & \text { otherwise } \end{array}\right. (a) Write the initial wave function as a superposition of stationary states (b) At time t > 0 the energy of the particle has been measured. What are the possible outcomes and what are the corresponding probabilities? Compute (H). (c) Compute (r) and (p) at time t = 0. State your answers as multiples of a.

This problem examines the number of energy states available in the conduction band that are very close to the band edge Ec. a. Using the formula below for the density of energy states per unit volume, perform the integral from the bottom of the conduction band (Ec) to an energy band 1.2kT above the edge and determine the number of states available per cm³. N(E) d E=\frac{\sqrt{2}}{\pi^{2}}\left(\frac{m_{n}{ }^{*}}{\hbar^{2}}\right)^{3 / 2} \sqrt{E} d E b. Compare your result to the effective density of states in the conduction band for silicon at room temperature (300K) given by the formula N_{c}=2\left(\frac{2 \pi m_{n}^{*} k_{B} T}{h^{2}}\right)^{3 / 2} C. Compare your result to the number of silicon atoms per cm³ you calculated HW1 and determine the ratio of the number of energy states/cm³ to the number of silicon atoms/cm³ and comment.