Master Molecular Physics: Comprehensive Exam Insights
School
Colorado School of Mines**We aren't endorsed by this school
Course
PHGN 520
Subject
Chemistry
Date
Dec 10, 2024
Pages
9
Uploaded by DrSnow15939
Molecular Physics Comprehensive ExaminationDate: March 15, 2023Duration: 2 hoursTotal Score: 100 pointsInstructions:- Answer all questions on the provided answer sheet.- Use a non-programmable calculator for calculations.- Graphs and charts should be drawn clearly.- Questions with an asterisk (*) indicate the need for detailed solutions.- Answer as many questions as possible within the allocated time.Question 1 (5 points) – Multiple ChoiceThe energy levels of a particle in a one-dimensional infinite square well are quantized. Which of the following statements is correct?A. The energy levels are discrete and increase linearly with quantum number.B. The energy levels are continuous and increase linearly with quantum number.C. The energy levels are discrete and increase exponentially with quantum number.D. The energy levels are continuous and increase exponentially with quantum number.Question 2 (5 points) – Multiple ChoiceConsider a quantum harmonic oscillator. Which of the following statements is correct?A. The energy levels are evenly spaced and can be expressed as E_n = (n + 1/2) * h * .ωB. The energy levels are evenly spaced and can be expressed as E_n = n * h * .ωC. The energy levels are unevenly spaced and can be expressed as E_n = (n + 1/2) * h * .ωD. The energy levels are unevenly spaced and can be expressed as E_n = n * h * .ωQuestion 3 (5 points) – Open-Ended* Calculate the energy difference between the ground state and the first excited state of a quantum harmonic oscillator. Assume that the angular frequency is given.ωQuestion 4 (5 points) – Open-Ended* Derive the expression for the expectation value of the position operator in a quantum harmonic oscillator.Question 5 (5 points) – CalculationA quantum harmonic oscillator is in the ground state. Calculate the probability of finding theoscillator between x = 0 and x = a, where a is a given value.
Question 6 (5 points) – CalculationConsider a hydrogen atom in its ground state. Calculate the expectation value of the kinetic energy.Question 7 (5 points) – Open-Ended* A quantum system is described by the Hamiltonian H = P^2 / 2m + V(x). Derive the time-independent Schrödinger equation for this system.Question 8 (5 points) – CalculationA particle is in a one-dimensional infinite square well of width L. Calculate the probability offinding the particle in the interval (L/4, L/2).Question 9 (5 points) – Open-Ended* The wave function of a quantum system is given by (x) = A * sin(kx). Derive the ψnormalization constant A.Question 10 (5 points) – Multiple ChoiceWhich of the following statements about the hydrogen atom is correct?A. The electron can occupy any energy level.B. The electron can only occupy certain energy levels.C. The energy levels of the electron are continuous.D. The energy levels of the electron are discrete and can be calculated using the quantum harmonic oscillator formula.Question 11 (5 points) – CalculationA hydrogen atom is in the second excited state (n = 3). Calculate the probability of finding the electron in the interval (r = 2a_0, r = 3a_0), where a_0 is the Bohr radius.Question 12 (5 points) – Open-Ended* Derive the expression for the degeneracy of the nth energy level of a hydrogen atom.Question 13 (5 points) – CalculationA quantum system is described by the Hamiltonian H = -ħ^2 / 2m * d^2/dx^2 + V(x). If the potential V(x) is a constant V_0, calculate the energy eigenvalues.Question 14 (5 points) – Open-Ended* A quantum system is described by the Hamiltonian H = -ħ^2 / 2m * d^2/dx^2 + 1/2 * k * x^2. Derive the time-independent Schrödinger equation for this system.Question 15 (5 points) – Multiple ChoiceWhich of the following statements about the Heisenberg uncertainty principle is correct?
A. p * x ≥ ħ / 2, where p is the uncertainty in momentum and x is the uncertainty in ΔΔΔΔposition.B. p * x ≤ ħ / 2, where p is the uncertainty in momentum and x is the uncertainty in ΔΔΔΔposition.C. E * t ≥ ħ / 2, where E is the uncertainty in energy and t is the uncertainty in time.ΔΔΔΔD. E * t ≤ ħ / 2, where E is the uncertainty in energy and t is the uncertainty in time.ΔΔΔΔQuestion 16 (5 points) – Open-Ended* A quantum system is described by the Hamiltonian H = -ħ^2 / 2m * d^2/dx^2 + 1/2 * k * x^2. Calculate the expectation value of the potential energy.Question 17 (5 points) – Open-Ended* The wave function of a quantum system is given by (x) = A * exp(-x^2 / 2a^2). Derive theψnormalization constant A.Question 18 (5 points) – CalculationConsider a quantum system with Hamiltonian H = P^2 / 2m + V(x). If the potential V(x) is a constant V_0, calculate the probability of finding the particle with energy between E and E + E.ΔQuestion 19 (5 points) – Multiple ChoiceWhich of the following statements about the wavefunction in a quantum system is correct?A. The wavefunction must be continuous.B. The wavefunction must be continuous and differentiable.C. The wavefunction must be continuous and integrable.D. The wavefunction must be continuous, differentiable, and integrable.Question 20 (5 points) – CalculationA quantum system is described by the Hamiltonian H = -ħ^2 / 2m * d^2/dx^2 + 1/2 * k * x^2. Calculate the probability of finding the particle in the interval (x = 0, x = a).Question 21 (5 points) – Open-Ended* Derive the expression for the expectation value of the position operator in a quantum harmonic oscillator.Question 22 (5 points) – CalculationA quantum harmonic oscillator is in the ground state. Calculate the probability of finding theoscillator between x = -a and x = 0, where a is a given value.Question 23 (5 points) – Multiple ChoiceWhich of the following statements about the quantum harmonic oscillator is correct?
A. The energy levels are discrete and increase linearly with quantum number.B. The energy levels are continuous and increase linearly with quantum number.C. The energy levels are discrete and increase exponentially with quantum number.D. The energy levels are continuous and increase exponentially with quantum number.Question 24 (5 points) – Open-Ended* Derive the expression for the degeneracy of the nth energy level of a hydrogen atom.Question 25 (5 points) – CalculationA quantum system is described by the Hamiltonian H = -ħ^2 / 2m * d^2/dx^2 + 1/2 * k * x^2. Calculate the expectation value of the potential energy.Question 26 (5 points) – Open-Ended* Calculate the energy difference between the ground state and the first excited state of a quantum harmonic oscillator. Assume that the angular frequency is given.ωQuestion 27 (5 points) – CalculationA hydrogen atom is in the second excited state (n = 3). Calculate the probability of finding the electron in the interval (r = 2a_0, r = 3a_0), where a_0 is the Bohr radius.Question 28 (5 points) – Multiple ChoiceWhich of the following statements about the wavefunction in a quantum system is correct?A. The wavefunction must be continuous.B. The wavefunction must be continuous and differentiable.C. The wavefunction must be continuous and integrable.D. The wavefunction must be continuous, differentiable, and integrable.Question 29 (5 points) – Open-Ended* Derive the expression for the expectation value of the kinetic energy in a hydrogen atom.Question 30 (5 points) – CalculationA quantum system is described by the Hamiltonian H = -ħ^2 / 2m * d^2/dx^2 + 1/2 * k * x^2. Calculate the probability of finding the particle in the interval (x = 0, x = a).Question 31 (5 points) – Multiple ChoiceWhich of the following statements about the Heisenberg uncertainty principle is correct?A. p * x ≥ ħ / 2, where p is the uncertainty in momentum and x is the uncertainty in ΔΔΔΔposition.B. p * x ≤ ħ / 2, where p is the uncertainty in momentum and x is the uncertainty in ΔΔΔΔposition.
C. E * t ≥ ħ / 2, where E is the uncertainty in energy and t is the uncertainty in time.ΔΔΔΔD. E * t ≤ ħ / 2, where E is the uncertainty in energy and t is the uncertainty in time.ΔΔΔΔQuestion 32 (5 points) – CalculationA hydrogen atom is in the ground state. Calculate the expectation value of the kinetic energy.Question 33 (5 points) – Open-Ended* Derive the time-independent Schrödinger equation for a quantum system described by theHamiltonian H = -ħ^2 / 2m * d^2/dx^2 + 1/2 * k * x^2.Question 34 (5 points) – Open-Ended* Calculate the probability of finding a particle in a one-dimensional infinite square well with width L in the interval (L/4, L/2).Question 35 (5 points) – CalculationConsider a quantum harmonic oscillator. Calculate the energy difference between the ground state and the first excited state. Assume that the angular frequency is given.ωQuestion 36 (5 points) – Multiple ChoiceWhich of the following statements about the hydrogen atom is correct?A. The electron can occupy any energy level.B. The electron can only occupy certain energy levels.C. The energy levels of the electron are continuous.D. The energy levels of the electron are discrete and can be calculated using the quantum harmonic oscillator formula.Question 37 (5 points) – Open-Ended* Derive the expression for the expectation value of the potential energy in a quantum harmonic oscillator.Question 38 (5 points) – CalculationA quantum harmonic oscillator is in the ground state. Calculate the probability of finding theoscillator between x = -a and x = 0, where a is a given value.Question 39 (5 points) – Open-Ended* The wave function of a quantum system is given by (x) = A * exp(-x^2 / 2a^2). Derive theψnormalization constant A.Question 40 (5 points) – Multiple ChoiceWhich of the following statements about the wavefunction in a quantum system is correct?
A. The wavefunction must be continuous.B. The wavefunction must be continuous and differentiable.C. The wavefunction must be continuous and integrable.D. The wavefunction must be continuous, differentiable, and integrable.Question 41 (5 points) – Open-Ended* Derive the expression for the expectation value of the kinetic energy in a hydrogen atom.Question 42 (5 points) – CalculationA quantum system is described by the Hamiltonian H = -ħ^2 / 2m * d^2/dx^2 + 1/2 * k * x^2. Calculate the probability of finding the particle in the interval (x = 0, x = a).Question 43 (5 points) – Open-Ended* A quantum system is described by the Hamiltonian H = -ħ^2 / 2m * d^2/dx^2 + 1/2 * k * x^2. Derive the time-independent Schrödinger equation for this system.Question 44 (5 points) – CalculationConsider a quantum harmonic oscillator. Calculate the energy difference between the ground state and the first excited state. Assume that the angular frequency is given.ωQuestion 45 (5 points) – Multiple ChoiceWhich of the following statements about the quantum harmonic oscillator is correct?A. The energy levels are discrete and increase linearly with quantum number.B. The energy levels are continuous and increase linearly with quantum number.C. The energy levels are discrete and increase exponentially with quantum number.D. The energy levels are continuous and increase exponentially with quantum number.Question 46 (5 points) – Open-Ended* Derive the expression for the degeneracy of the nth energy level of a hydrogen atom.Question 47 (5 points) – CalculationA hydrogen atom is in the second excited state (n = 3). Calculate the probability of finding the electron in the interval (r = 2a_0, r = 3a_0), where a_0 is the Bohr radius.Question 48 (5 points) – Open-Ended* Derive the expression for the expectation value of the potential energy in a quantum harmonic oscillator.Question 49 (5 points) – CalculationA quantum system is described by the Hamiltonian H = -ħ^2 / 2m * d^2/dx^2 + 1/2 * k * x^2. Calculate the expectation value of the potential energy.
Question 50 (5 points) – Multiple ChoiceWhich of the following statements about the Heisenberg uncertainty principle is correct?A. p * x ≥ ħ / 2, where p is the uncertainty in momentum and x is the uncertainty in ΔΔΔΔposition.B. p * x ≤ ħ / 2, where p is the uncertainty in momentum and x is the uncertainty in ΔΔΔΔposition.C. E * t ≥ ħ / 2, where E is the uncertainty in energy and t is the uncertainty in time.ΔΔΔΔD. E * t ≤ ħ / 2, where E is the uncertainty in energy and t is the uncertainty in time.ΔΔΔΔQuestion 51 (5 points) – CalculationA hydrogen atom is in the ground state. Calculate the expectation value of the kinetic energy.Question 52 (5 points) – Open-Ended* Derive the time-independent Schrödinger equation for a quantum system described by theHamiltonian H = -ħ^2 / 2m * d^2/dx^2 + 1/2 * k * x^2.Question 53 (5 points) – Open-Ended* Calculate the probability of finding a particle in a one-dimensional infinite square well with width L in the interval (L/4, L/2).Question 54 (5 points) – CalculationA quantum harmonic oscillator is in the ground state. Calculate the probability of finding theoscillator between x = 0 and x = a, where a is a given value.Question 55 (5 points) – Multiple ChoiceWhich of the following statements about the hydrogen atom is correct?A. The electron can occupy any energy level.B. The electron can only occupy certain energy levels.C. The energy levels of the electron are continuous.D. The energy levels of the electron are discrete and can be calculated using the quantum harmonic oscillator formula.Question 56 (5 points) – Open-Ended* A quantum system is described by the Hamiltonian H = -ħ^2 / 2m * d^2/dx^2 + 1/2 * k * x^2. Derive the time-independent Schrödinger equation for this system.Question 57 (5 points) – CalculationConsider a quantum harmonic oscillator. Calculate the energy difference between the ground state and the first excited state. Assume that the angular frequency is given.ω
Question 58 (5 points) – Open-Ended* Derive the expression for the expectation value of the kinetic energy in a hydrogen atom.Question 59 (5 points) – CalculationA hydrogen atom is in the second excited state (n = 3). Calculate the probability of finding the electron in the interval (r = 2a_0, r = 3a_0), where a_0 is the Bohr radius.Question 60 (5 points) – Multiple ChoiceWhich of the following statements about the wavefunction in a quantum system is correct?A. The wavefunction must be continuous.B. The wavefunction must be continuous and differentiable.C. The wavefunction must be continuous and integrable.D. The wavefunction must be continuous, differentiable, and integrable.Question 61 (5 points) – Open-Ended* The wave function of a quantum system is given by (x) = A * exp(-x^2 / 2a^2). Derive theψnormalization constant A.Question 62 (5 points) – CalculationA quantum system is described by the Hamiltonian H = -ħ^2 / 2m * d^2/dx^2 + 1/2 * k * x^2. Calculate the expectation value of the potential energy.Question 63 (5 points) – Open-Ended* Derive the expression for the expectation value of the kinetic energy in a quantum harmonic oscillator.Question 64 (5 points) – Multiple ChoiceWhich of the following statements about the wavefunction in a quantum system is correct?A. The wavefunction must be continuous.B. The wavefunction must be continuous and differentiable.C. The wavefunction must be continuous and integrable.D. The wavefunction must be continuous, differentiable, and integrable.Question 65 (5 points) – Open-Ended* Derive the expression for the degeneracy of the nth energy level of a hydrogen atom.Question 66 (5 points) – Open-Ended* Derive the time-independent Schrödinger equation for a quantum system described by theHamiltonian H = -ħ^2 / 2m * d^2/dx^2 + 1/2 * k * x^2.Question 67 (5 points) – Multiple Choice
Which of the following statements about the quantum harmonic oscillator is correct?A. The energy levels are discrete and increase linearly with quantum number.B. The energy levels are continuous and increase linearly with quantum number.C. The energy levels are discrete and increase exponentially with quantum number.D. The energy levels are continuous and increase exponentially with quantum number.Question 68 (5 points) – CalculationA quantum harmonic oscillator is in the ground state. Calculate the probability of finding theoscillator between x = -a and x = 0, where a is a given value.Question 69 (5 points) – Open-Ended* Calculate the probability of finding a particle in a one