Mastering Advanced Quantum Electrodynamics: Exam Guide
School
Albertus Magnus College**We aren't endorsed by this school
Course
SCIENCE 410
Subject
Chemistry
Date
Dec 12, 2024
Pages
4
Uploaded by AdmiralFog15907
**Name of Exam:** Advanced Quantum Electrodynamics Assessment**Exam Time:** 3 hours**Total Score:** 100 points**Instructions:**1. Answer all questions clearly and concisely.2. Show all calculations for calculation questions.3. For graphing and drawing problems, use the space provided or attach additional pages if necessary.4. Indicate your answer clearly for multiple-choice questions.5. Answers should demonstrate understanding of both the theory and its application.6. Submit your test paper with all the questions answered within the allocated time.---**Question 1 (5 points)**Calculate the probability amplitude for an electron to transition from an excited state to the ground state in a hydrogen atom, given the following quantum numbers:- Initial state: \(n_i = 3\)- Final state: \(n_f = 1\)- Transition frequency: \(\nu = 2.47 \times 10^{15}\) Hz**Question 2 (7 points)**Derive the wave function of a one-dimensional quantum particle confined within an infinitely deep potential well of width \(L\). Explain how this wave function can be used to calculate the energy eigenvalues and eigenstates.**Question 3 (6 points)**Given the following wave function: \(\psi(x) = Ce^{ikx}\), where \(C\) and \(k\) are constants, and \(x\) represents the position of the particle in one dimension.a. Determine the normalization constant \(C\).b. Calculate the probability density for finding the particle at a specific position \(x\).c. Describe the significance of the term \(e^{ikx}\) in terms of the particle's wave-like properties.
**Question 4 (8 points)**A system is described by the following Hamiltonian: \(H = \frac{p^2}{2m} + \frac{1}{2}m\omega^2x^2\). This system can be described as a harmonic oscillator.a. Derive the energy eigenvalues of this system.b. Write down the normalized wave functions corresponding to these eigenvalues.c. Calculate the expectation value of the position operator \(\langle x \rangle\) for the ground state.**Question 5 (6 points)**A hydrogen atom is in a superposition of its first two energy levels. Calculate the probabilityof measuring the atom in the ground state after a time \(t\).---**Question 6 (5 points)**In the double-slit experiment, a beam of electrons is passed through two slits separated by adistance \(d\). The interference pattern is observed on a screen placed a distance \(D\) away from the slits.a. Determine the distance \(y\) between the central maximum and the first minimum of the interference pattern.b. Explain how the wave nature of electrons is demonstrated by this experiment.**Question 7 (7 points)**Calculate the probability of finding a particle in a state described by the wave function \(\psi(x) = A\sin(kx)\), where \(A\) and \(k\) are constants, and \(x\) is the position of the particle.**Question 8 (6 points)**Given the following quantum state: \(\psi(x) = Be^{-kx}\), where \(B\) and \(k\) are constants, and \(x\) is the position of the particle.a. Determine the normalization constant \(B\).b. Calculate the expectation value of the position operator \(\langle x \rangle\) for this state.**Question 9 (8 points)**
A two-level system is described by the Hamiltonian: \(H = \frac{\hbar^2 k^2}{2m}\sigma_z\), where \(m\) is the mass of the particle, \(\hbar\) is the reduced Planck constant, and \(\sigma_z\) is the Pauli z-matrix.a. Calculate the energy eigenvalues and eigenstates of this system.b. Write down the time-evolution operator for this system.**Question 10 (5 points)**In a system of two interacting spins, the Hamiltonian is given by: \(H = -J\mathbf{S}_1 \cdot\mathbf{S}_2\), where \(J\) is a constant and \(\mathbf{S}_1\) and \(\mathbf{S}_2\) are the spin operators for the two particles.a. Determine the eigenvalues and eigenstates of this Hamiltonian.b. Explain how the interaction between the two spins can be used to model a simple magnetic material.**Question 11 (7 points)**A beam of particles is incident on a potential barrier with a height \(V_0\) and width \(L\). The wave function of the incident beam is given by: \(\psi_i(x) = A\sin(kx)\), where \(A\) and \(k\) are constants, and \(x\) is the position of the particle.a. Calculate the transmission and reflection coefficients for the barrier.b. Discuss the implications of these coefficients for the tunneling phenomenon.**Question 12 (6 points)**A system is described by the following Hamiltonian: \(H = \frac{p^2}{2m} + V(x)\), where \(m\) is the mass of the particle, \(p\) is its momentum operator, and \(V(x)\) is the potential energy function.a. Write down the Schrödinger equation for this system.b. Explain how the potential energy function \(V(x)\) affects the energy eigenvalues and eigenstates.**Question 13 (8 points)**In a system of two interacting spins, the Hamiltonian is given by: \(H = -J\mathbf{S}_1 \cdot\mathbf{S}_2\), where \(J\) is a constant and \(\mathbf{S}_1\) and \(\mathbf{S}_2\) are the spin operators for the two particles.
a. Calculate the energy eigenvalues and eigenstates of this system.b. Write down the time-evolution operator for this system.**Question 14 (5 points)**A beam of particles is incident on a potential barrier with a height \(V_0\) and width \(L\). The wave function of the incident beam is given by: \(\psi_i(x) = A\sin(kx)\), where \(A\) and \(k\) are constants, and \(x\) is the position of the particle.a. Calculate the transmission and reflection coefficients for the barrier.b. Discuss the implications of these coefficients for the tunneling phenomenon.**Question 15 (7 points)**A system is described by the following Hamiltonian: \(H = \frac{p^2}{2m} + V(x)\), where \(m\) is the mass of the particle, \(p\) is its momentum operator, and \(V(x)\) is the potential energy function.a. Write down the Schrödinger equation for this system.b. Explain how the potential energy function \(V(x)\) affects the energy eigenvalues and eigenstates.---**End of Exam**Please ensure all answers are submitted before the allocated time has expired.