Quantum Physics

A particle of mass m is moving in a one-dimensional potential. (x) are the orthonormal energy eigenstates (their explicit form is not needed) and the energy eigenvalues are En nE with n= 1, 2, 3, ... and E some real constant. In this problem no integrations are needed; use orthonormality of the Un and/or any results derived in class. At t=0 the system is prepared in the state: (a) Find a by requiring that (x, 0) is correctly normalised. You may assume that a is a real and positive number. [2] (b) At t = 0 a measurement of energy is performed and the most likely energy eigenvalue is found. Which eigenvalue is this? Find the corresponding probability and the wave function after the measurement at t > 0. [3] [3] (c) Calculate the expectation values (E) = (Î) and (E²) = (Ĥ²), and the energy uncertainty AE.

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.

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))

a. Sketch the molecular orbital energy level diagram for the diatomic molecule Ne2+ and deduce its ground state electronic configuration. b. Is Ne likely to have a shorter or longer bond length than He2+? Explain.

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?

Problem 3. Consider a region of silicon which is single crystal silicon except for one arsenic atom that has been implanted as shown in the figure below. Note that an electron has been donated to the lattice and it moves from region 1 to region 2.

Discuss the role of the Born-Oppenheimer approximation in the calculation of molecular potential energy curves such as the one below: