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.

What is the ground state electronic configuration and bond order of: a. F₂- b. N₂- c. 02^2-

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

Determine the eigenvalues and the corresponding eigenfunctions of the operator F= aP+BX, where a and b and real consultants, p is the

a) (10 points for 5383/10 points for 7383) If the messages are of variable length,comprised of strings of the symbols, and are all equally likely, compute the density matrix of the alphabet, pë, using the computational basis in your intermediate calculations. Show all intermediate calculations and clearly state all assumptions. d) (15 points for 5383/10 points for 7383) Assume the symbol |) is measured with the Observable A as given below. What is/are the possible measurement outcome(s)? And what is/are the possible “collapsed” state(s) associated with the outcome(s)? And, what are the probabilities that the measured state(s) collapse for each possible collapsed state(s)? \mathbf{A}=\left[\begin{array}{cc} 0 & -i \\ i & 0 \end{array}\right]

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.

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

\text { c) If we only consider the coulomb potential } V(r)=\frac{1}{4 \pi \epsilon_{0} r} \text { in our Hamiltonian for hydrogen, the energy } only depends on the principle quantum number n. d) When measuring the absorption spectra from 1s to 2p of hydrogen we observe 3 lines due to spin-orbit coupling. The number of nodes in a hydrogen atom wavefunction is set by the principle quantum number.

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?

Show that Bohr's postulate which states that the angular momentum of an electron in the nthorbit (pe nh) is equivalent to an integer number of de Broglie waves fitting within the circumference of a Bohr circular orbit.=