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.

Gallium nitride (GaN) is a wide direct bandgap semiconductor material with Eg = 3.425 eV. a. Draw a simple band model showing Ec, Ev, Eg and the wavelength of light that would be emitted for an electron hole pair recombination process. b. The effective density of states in the conduction band is Ne = 1 x 1019 cm3 and the effective density of states in the valence band is NV = 5 x 1018 cm-3. Determine the intrinsic carrier concentration ni at room temperature (300K). c. If silicon donor atoms are implanted into the semiconductor such that the Fermi energyl evel is 0.167 eV below the conduction band, how many atoms of silicon are implanted assuming that all silicon atoms are ionized?

Thermodynamics and quantum mechanics have been used to estimate the fundamental limits for energy transfer in binary switching of an electronic switch. Use the Heisenberg uncertainty principle to determine the transition time to for the minimum switching energy AE where AE is based on the Landauer principle AE = küТln(2). (kb is Boltzman's constant and T is the room temperature or 300K).

2. Qubit Mathematical Models. A qubit, ly), is in a state as given (differently in some cases) in the sub-questions at time to. The qubit evolves in time according to the following three time-dependent Hamiltonian in the following diagram at the times indicated. Please note that this diagram is NOT a "standard” quantum logic circuit as Hamiltonian are given and not the unitary transformation matrices. b) (15 points for 5383/10 points for 7383) Assume that [y(to)) is initialized as given.Determine the value of ly) at time t2, ly(t2)). Show all work and clearly explain each step of your approach to find the result. \left|\Psi\left(t_{0}\right)\right\rangle=\frac{1}{\sqrt{2}}|0\rangle+\frac{1}{\sqrt{2}} e^{t \frac{\pi}{2}}|1\rangle c) (10 points for 5383/10 points for 7383) Determine the overall transfer matrix, U, that describes the time evolution of [y(t3))=U]y(to)). Clearly state your assumptions and-show all steps in your derivation of U. d) (7383 STUDENTS ONLY: 10 points for 7383) Assume that ly(t3)) evolves to state|y(t3))= |1). In this case, what was the initial state, ly(to))? Show all work and clearly-explain each step of your approach to find the result.

b) Briefly describe how the general shape of the wavefunction can be described without actually solving the Schrodinger equation. c) Briefly explain the fundamental relationship between operators and physical observables in quantum mechanics. What is a physical observable and what kinds of operators are associated with physical observables? d) Describe the composition of a localized particle written as a superposition of momentum states(eigenfunctions of momentum) and how this composition changes as the particle becomes more localized. a) Briefly describe the relationships between the probability, probability density and probability amplitude.

6. Normalize the wave function ei(kx-w ) in the region x = 0 to a.

5. Switch to "Manual". Change the speed from “normal" to “slow motion". Move the wrench up and down and watch how the wave travels towards the clamp. In a couple of sentences describe what you see happening to the wave as it travels.

Question 2. A particle of mass m in the presence of an infinite potential well of width astarts 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. ) 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 (x) and (p) at time t = 0. State your answers as multiples of a.

A semiconductor has a band gap of Eg = 1.1 eV. The effective density of states in the conduction band (Ne) is equal to the effect density of states in the valence band (Nv). The doping level is 3.0 x 10¹5 cm3 of donor atoms, and the donor atom energy level is 0.2eV below the conduction band. The Fermi energy level is 0.25 eV below the conduction band edge. a. Draw the simple energy band diagram showing the valence band (Ev), conduction band(Ec), the Fermi energy level (Ef), the intrinsic energy level (Ei) and the band gap. b. Determine the intrinsic carrier concentration n; and the carrier concentration of electrons and holes at 300K.

\text { Consider the } \mathrm{X} \text {-force equilibriu } \mathrm{m}, \sum \mathrm{F}_{\mathrm{x}}=0 \text {; } Cancelling out equal terms in opposite faces \frac{\partial \sigma_{x}}{\partial x} d x d y d z+\frac{\partial \tau_{y x}}{\partial y} d x d y d z+\frac{\partial \tau_{z x}}{\partial z} d x d y d z+X d x d y d z=0 So we get the x- equilibrium equation \frac{\partial \sigma_{x}}{\partial x}+\frac{\partial \tau_{y x}}{\partial y}+\frac{\partial \tau_{z x}}{\partial z}+X=0 \frac{\partial \tau_{y x}}{\partial x}+\frac{\partial \sigma_{y}}{\partial y}+\frac{\partial \tau_{y z}}{\partial z}+Y=0 \frac{\partial \tau_{z x}}{\partial x}+\frac{\partial \tau_{z y}}{\partial y}+\frac{\partial \sigma_{z}}{\partial z}+Z=0 in short, the equilibrium equations in tensor notation Ou, +X, = 0 (i, j = x, y, z) Take moment quilibrium about an axis through the center and parallel to z - axis \tau_{x y} d y d z \frac{d x}{2}+\left(\tau_{x y}+\frac{\partial \tau_{x y}}{\partial x} d x\right) d y d z \frac{d x}{2}-\tau_{y x} d x d z \frac{d y}{2}-\left(\tau_{y x}+\frac{\partial \tau_{y x}}{\partial y} d y\right) d x d z \frac{d y}{2}=0 \tau_{x y} d x d y d z-\tau_{y x} d x d y d z=0 \therefore \tau_{x y}=\tau_{y x} \tau_{y z}=\tau_{z y} \tau_{z x}=\tau_{x z}