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.

Provide a physical explanation for the shape of the wavefunction for a particle in box with changes in particle mass and box width. Explain for the energy of the particle in box with changes in particle mass and box width. What causes the particle to have discrete (quantized energies)? Hint: The free particle Schrodinger equation is the same as the particle in a box wavefunction, but its energies are not quantized. You have non-interacting (ideal gas) molecules of mass m in a vessel of width L. You compress the vessel adiabatically (in QM, this means, you compress slowly so that the particles stay in their same quantum states as you compress). Explain how much work was done to compress the gas. Did the temperature change?

3. X-rays of wavelength 0.207 nm are scattered from NaCl. What is the angular separation between first- and second-order diffraction peaks? Assume scattering planes that are parallel to the surface. Note that for NaCl, d = 0.282 nm.

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.

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

The Schrondinger equation for a paticle of mass m in a potential V(x,y) in two dimensions is

At time t = 0 a particle is represented by the wave function \psi(x)=N \times\left\{\begin{array}{ll} e^{x} & x<0 \\ \cos x & 0 \leq x \leq \pi / 2 \\ 0 & x>\pi / 2 \end{array}\right. (a) Find the normalisation constant N assuming it to be real and positive. (b) Sketch the wave function. What is the probability of detecting the particle at x > 0? (c) Compute (p), (p²), and the standard deviation op.

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!