Quantum Physics

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

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?

2. Why a heavy nucleus split into two daughter nuclei fission process will release neutrons? How is the process when two light nuclei combine to one?

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.

1.Two particles are attracted to each other by a force that is described by a central potential, where q is a (dimensionless) constant, μ is the particles' reduced mass, and r is their separation. The particles' wave function can be factored into radial and angular components, where Yem (0, 0) is one of the standard spherical harmonics and Rne(r) is the radial wave function for this system. For what values of the angular momentum quantum number, I can this system have bound-state solutions?

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))

2. (30 points) A quantum mechanical particle is in an eigenstate ) of β 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 β and Îz, where ο is the z component of the angular momentum.