This phys 234 final is hot bullshit

Pls help fuck this exam the teacher didn’t teach it properly at all.

You know what, lets do a gamer moment, here is the exam: A particle of mass m is moving in a one-dimensional harmonic oscillator potential, V (x) = 1 2mω2x 2 . The particle starts out in the following state: |φ(t = 0)i = 2|0i + |1i, where |ni denotes the n th number state (n = 0, 1, 2, ...). (a) If you measure the energies of the particle, what values will you get and with what probabilities? (2 points) (b) Is hEi time dependent and why? (2 points) (c) Calculate ha † i at time t. Here, a † is the raising operator. (2 points) (d) Is a † an observable? Explain your answer. (1.5 points) (e) Is it possible to experimentally measure ha † i for the particle? Explain. (1.5 points) (f) Compute the average value of momentum of the particle at time t. (3 points) (g) Show that the uncertaintly principle in ∆x∆p holds for this state. (3 points) [You can use the textbook to find the Harmonic oscillator energy states as well as definitions of the ladder operators].

An atom could be in one of three energy states |E1i, |E2i, and |E3i, with En denoting the energy of the state |Eni. An experiment is set up in such a way that the atom in the energy state |Eni scatters 5n photons when illuminated by a laser. The atom is prepared in the state, |ψi = |E1i + 2|E2i + e iπ/6 |E3i. (a) Assuming a perfect detector capable of capturing all the photons emitted by the atom, what are the possible values and probabilities of photon counts on the detector when the state of the atom is measured by the laser beam? (3 points) (b) What is the dimensionality of the Hilbert space? (1 point) (c) If the experiment in repeated many times, each time starting with the same state |ψi, what are the average and the standard deviation of number of photons received on the detector? (3 points)

Consider the same system and the same state |ψi as in the previous problem. (a) Write down the state |ψi as a matrix in the energy basis. (2 points) (b) Write a projector operator (in Dirac notation in terms of energy kets and bras) that projects an arbitrary quantum state of the atom onto a state that is orthonormal to |ψi. Is this operator unique? Explain. (3 points) (c) Write the same projector operator (as in the previous part) as a matrix in the energy basis. (2 points)

An electron is confined in a one dimensional infinite wall, 0 < x < L. The electron is prepared in the first excited energy state above the ground state. It then jumps to the ground state, and in the process emits a photon of wavelength 396 nm. What is the probability that the electron is observed in an interval 0 < x < 0.3 nm in the ground state? You can treat the electron as a point particle of mass 9.11 × 10−31 kg. The energy of a photon of wavelength λ is hc/λ, with c = 3 × 108 m/s being the speed of light in vacuum and h = 6.63 × 10−34m2kg/s the Planck’s constant. You are permitted to use the energy eigenstates and eigenvalues from the lecture notes/textbook and a computer program to calculate integrals. (7 points)

A spin-1/2 particle is prepared in a state |ψi such that the probabilities of measuring its spin angular momentum along various basis vectors are as follows: P(|+iz) = 0.5, P(|−iz) = 0.5, P(|+ix) = 0.75, P(|−ix) = 0.25, P(|+iy) = 0.933, P(|−iy) = 0.067. (a) What is the uncertainty (standard deviation) of Sx in this state? (5 points) (b) Draw the approximate location of the state on a Bloch sphere. (2 points)

Find the energy and the corresponding wavefunctions of the three lowest energy eigenstates of a particle of mass m in the following potential energy well, V (x) = 1 2 mω2x 2 , −∞ < x < 0 = ∞, 0 ≤ x < ∞. Approximately sketch the wavefunctions. [You are allowed to use the form of Harmonic oscillator wavefunctions from the textbook, if you need them.] (7 points)

Pls send help fuck a kaji rajbul islam