Master Quantum Mechanics: Midterm Exam Questions and Concepts
School
Averett Unversity**We aren't endorsed by this school
Course
PHYSICS MISC
Subject
Chemistry
Date
Dec 12, 2024
Pages
3
Uploaded by JudgeCamelMaster1266
### Midterm Exam in Quantum Mechanics**Exam Time:** 2 hours **Total Score:** 100 points **Instructions:**1. Answer all questions. 2. Show all calculations and derivations. 3. Questions are worth varying points; read the question carefully for the point value. **Exam Content:**---**Question 1 (10 points)** A particle is in a one-dimensional box of length L. The ground state wavefunction is given by: \[ \psi_1(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{\pi x}{L}\right) \] Determine the probability of finding the particle in the right half of the box.---**Question 2 (12 points)** Given the normalized wavefunction for a quantum state: \[ \psi(x) = \frac{1}{\sqrt{2\pi}} \left( \frac{1}{2} e^{-i\pi x} + \frac{1}{2} e^{i\pi x} \right) \] Calculate the average momentum of the system and sketch the wavefunction.---**Question 3 (15 points)** An electron is confined to a circular region of radius R in a hydrogen-like atom. Calculate theenergy levels using the Schrödinger equation and Bohr radius model. Compare your results with the hydrogen atom.---**Question 4 (8 points)** Choose the correct statement regarding the uncertainty principle:A) xp ≥ h/4Δ ΔπB) xp ≤ h/2 Δ ΔC) xp ≥ h Δ Δ
D) xp = h Δ Δ---**Question 5 (10 points)** In a two-state quantum system, the Hamiltonian is given by: \[ H = \frac{\hbar^2}{2m} \left[ \begin{array}{cc} 0 & V \\ V & 0 \end{array} \right] \] Determine the eigenvalues and eigenstates of the Hamiltonian.---**Question 6 (8 points)** Sketch the potential energy surface for a particle undergoing a quantum tunneling event. Describe the role of the energy barrier and the transmission coefficient.---**Question 7 (20 points)** For the harmonic oscillator, derive the energy eigenvalues and wavefunctions from the Schrödinger equation. Then, calculate the expectation value of the position operator and sketch the ground state wavefunction.---**Question 8 (10 points)** Using the Born rule, calculate the probability of measuring the energy of a quantum state to be in the range E to E + E, where E is the energy eigenvalue.Δ---**Question 9 (10 points)** Explain the concept of a coherent state in quantum mechanics. What is its significance, and how is it related to the classical limit?---**Question 10 (10 points)** Determine the commutator [x, p] for the quantum harmonic oscillator and use it to verify Heisenberg's uncertainty principle.---**Question 11 (12 points)**
Derive the time evolution of the wavefunction for a free particle using the Schrödinger equation. Discuss the implications of this solution for the propagation of particles.---**Question 12 (15 points)** Calculate the ground state energy and wavefunction of a particle in a three-dimensional box using the Schrödinger equation. Sketch the probability density in the ground state.---**Question 13 (10 points)** State and prove the Ehrenfest theorem. Use it to explain how classical mechanics emerges from quantum mechanics in the limit of large quantum numbers.---**Question 14 (15 points)** A quantum system has the following time-independent wavefunction: \[ \psi(x,t) = e^{-i\omega t} \sqrt{\frac{1}{\pi \alpha^2}} e^{-\frac{x^2}{2\alpha^2}} \] Identify the system and discuss its properties.---**Question 15 (15 points)** Consider the hydrogen atom with a wavefunction: \[ \psi(r,\theta,\phi) = R(r)Y_{lm}(\theta,\phi) \] Derive the energy eigenvalues and explain the quantum numbers associated with the wavefunction.