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.

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.

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 {. }

Consider a one-dimensional chain of sites available for electrons. Let N be the number of sites in the chain. The probabilities of finding the electron on any of the sites is given by 2, where yj, j = 1, ..., N is a set of complex wave function values fully describing the electron state. The dynamical system of equations describing the time dependence of the wave function V¡(t) reads i \hbar \frac{d \Psi_{j}}{d t}=\left\{\begin{array}{cc} -\tau\left(\Psi_{j+1}+\Psi_{j-1}\right), & 1<j<N \\ -\tau \Psi_{j-1}, & j=N \\ -\tau \Psi_{j+1}, & j=1 \end{array}\right. Here is the parameter describing coupling of nearest neighbouring sites. (b) Consider the cases N = 2, 3, 4 and, for each case, write down the system (1) withexcluded time in the form of the matrix equation \mathrm{H}_{N} \Psi=E \Psi c) For the cases N = 2, 3, 4 divide Eq. (2) by the coupling parameter 7 and replace-E/T = 28 to obtain the dimensionless form of the equation: (f) In the general case, prove the following recurrent relation for polynomials PN(E): P_{N}(\varepsilon)=2 \varepsilon P_{N-1}(\varepsilon)-P_{N-2}(\varepsilon) by expressing the matrix determinant in Eq. (4) in terms of its minors. The recurrent relation (6) is common for both types of Chebyshev polynomials TNand UN. Identify which of the two types corresponds to the polynomial PN(=) bychecking the cases N =V = 2, 3.[3 marks] 1) Using the right the expression for one of these polynomials T_{N}(\cos (\theta))=\cos (n \theta), U_{N}(\cos (\theta)) \sin (\theta)=\sin ((N+1) \theta) solve the equation PÂ(ɛ) = 0 for values of . Recover the allowed electron energies E using the relation E = 2ET. ) Addressing the limit N » 1, identify the upper and the lower limits for allowedenergies in terms of the coupling parameter 7. Express the width of the energyband as the difference between the two limits.[2 marks] Prove that the zero energy state E = 0 is allowed for any odd value of N

A silicon device is doped with 1 x 10¹6 cm³ arsenic atoms. The Fermi level is 0.41 eV below Ei at room temperature (300K). Determine the acceptor concentration.

Particles of mass m are traveling through the potential barrier v(x) =

The two dimnsional Harmonic oscillator position sopace wavefunction Find the momentum space wavefunction corresponding to the same energy En.

The oxygen molecule has a vibrational energy that is listed by spectroscopists as 1580 cm ¹. What they're actually describing is the inverse wavelength of the photons with the corresponding energy. Multiplying by hc, the vibrational energy spacing comes out to ε ≈ 0.2 eV. a. Derive the vibrational partition function of a generic harmonic oscillator. Simplify it as much as possible. b. Evaluate the vibrational partition function at room temperature for oxygen at room temperature. Compare c. Calculate what fraction of oxygen molecules are in the first excited vibrational state at room temperature. d. Taking the vibrational ground state to be energy zero, calculate the average energy of an oxygen molecule, both in eV and in proportion to KT.

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.

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!