Quantum Physics

Matrix quantum mechanics: The Hamiltonian of a two-state system is given by H = (a) Write down the energy eigenvalues, and the energy eigenvectors U(1) and U(2) of H. () [2] (c) Assume the state of the system at t = 0 is given by C(0) = V. Find the expectation values (H) and (H²). [3] (b) Show that V = is a normalised eigenvector of A. What is the corresponding eigenvalue? [1] (d) Assuming the system is in the state C(0) = V at t = 0 find the state C(t) at later times t > 0 using the expansion theorem. Hint: write V as a linear combination of U(1) and U(2) and put "wiggle factors". (e) Calculate the expectation values (A) and (A²) using the time dependent state C(t). Hence, find the un- certainty AA. For which values of time t does AA vanish? Hint: recall the expectation values of operators in matrix quantum mechanics are defined as (O) = cioc where O is the matrix form of the operator and C is the state; either C(0) or C(t) in the examples above.

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.

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?

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 {. }

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.

1. The distance between adjacent atomic planes in calcite (CaCO3) is 0.300 nm. Find the smallest angle (in degrees) of Bragg scattering for 0.030 nm x-rays?

2. X-rays scattered from a crystal have a first-order diffraction peak at 0 = 12.5°. At what angle will the second- and third-order peaks appear?

Normalize the wave function Areαr/afrom r = 0 to ∞o where a and A are constants. Use a table of integrals.

Consider a proton confined within typical nuclear dimensions of 5 × 10-¹5 m. Estimate the minimum kinetic energy of the proton. Repeat this calculation for an electron confined within typical nuclear dimensions. Comment briefly on the physical significance of your results, given that the nuclear binding energy for a proton is typically in the range 1 -10 MeV.

a) Show that the particle in a box wavefunctions w = Nsin(ntx/L) with 0 < x <L are orthogonal to each other. b) Write a superposition wavefunction for the particle in a box. Why do we write superposition wavefunctions and not just eigenstate wavefunctions? \text { Show that ground state wavefunction of the } \mathrm{H} \text { atom, } \psi=N \exp \left(-r / 2 a_{0}\right) \text {, is orthogonal to the } 2 \mathrm{~s} \text { orbital } \psi=N\left(2-r / a_{0}\right) \exp \left(-r / 2 a_{0}\right) \text { and the } 2 \mathrm{p}_{\mathrm{x}} \text { orbital } \psi=N x \cdot \exp \left(-r / 2 a_{0}\right) Explain why orthogonality is important in creating superposition wavefunctions.