Quantum particle in a magnetic field Consider a particle of mamm m and charge e confined to the xy-plane interacting with a magnetic field B = Bk with vector potential

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.

Problem 1- The band diagram of a Silicon sample with Fermi level is shown below, and assume bandgap energy is constant of temperature: Calculate the density of free carriers (both n and p) (a) at room temperature (300 K). (b) at 400 K.

Problem 2- A sample of silicon is doped uniformly with two different dopants: Donor density of 4x10^¹6 cm-³, and Acceptor density of 5x10^¹6 cm-³. a) Calculate electron and hole concentrations in the sample at room temperature. b) Calculate mobility of majority carrier in the sample at room temperature.

Problem 3-A piece of Metal/Silicon is doped with phosphorous, 1 x 10^¹7 cm ^-³. The metal has a work function of 6 eV. Within depletion approximation at Zero bias: (a) Calculate barrier height (b) Calculate build-in potential (c) Calculate depletion thickness (in unit of micrometer) (d) Calculate maximum electric field

calculate the radioactivity of the initial time for a neutron source has 1mg 252Cf 1. Because the content of 14C within a living organism is the same as the content of the 14C in atmosphere and 14C content begins to change after metabolism once stopped, therefore archaeologists use 14C radioactive decay to determine the age of organism in paleontology. It is possible to determine the age of organism in accordance with existing paleontological amount of 14C. If known 14C decay constant À is 0.00012097/year and 14C content of a paleontology fossil is measured as 5% of the content of the 14C in atmosphere, how many years ago did this creature die?

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

2. (30 points) A quantum mechanical particle is in an eigenstate ) of Î² with eigenvalue 2ħ²: At a particular moment, the particle also is in an eigenstate of the x component of the angular momentum, Î, with eigenvalue 0. In other words, . Express this eigenstate of I, as a normalized superposition of the familiar eigenstates, |lm), of Î² and Îz, where Î¿ is the z component of the angular momentum.

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.

3. The states of a two-state system are represented by the orthogonal kets, |1) and 2). In this basis, the Hamiltonian for the system may be written as (a) (10 points) Are [1) and 2) eigenstates of Ĥ? If so, what are their eigenenergies? If not, then express the eigenstates of Ĥ in terms of |1) and 2). (b) (5 points) Write down the time-dependent Schrödinger equation for this system. (c) (15 points) The system initially is in the state (0)) = (1) + |2)) at time t = 0. At a later time, t, it is in another superposition, (t))

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