For your reference, the difficulty level is provided to each problem: Easy = ♢,
Standard = ♢♢, Difficult = ♢♢♢. Submit your exam sheet to the instructor when you
are ...
Quantum Chemistry I (Winter 2013) Final Exam Instructions about the final exam You have 90 minutes. There are five problems to solve, each of which weighs 20 points. In addition to them, there is an additional bonus problem which also weighs 20 points. Yet, your grade is still based on the 100-point scale (lucky for you). The problems are based on the materials covered in the lectures. Use English to answer the problems. No points will be given to you if you use Japanese. You are allowed to use your own single A4-sized cheat sheet. You are NOT allowed to use your smart phone, textbook, lecture notes, notebooks, dictionary, and calculator. For your reference, the difficulty level is provided to each problem: Easy = ♦, Standard = ♦♦, Difficult = ♦♦♦. Submit your exam sheet to the instructor when you are done while you can take this exam sheet with you. You can leave early if you are done early. The solutions to the problems will be uploaded on the course webpage. Good luck! Problem 1: Harmonic oscillator (20 points) ♦ A particle of mass in the 1D harmonic oscillator potential starts out in the state Ψ ,0
1
2
,
for some constant . The wave functions for the ground state, first excited state, and second exited state are given by /2 , / / √2 exp /2 , and / / 2 1 exp /2 /√2, respectively, where / / exp / and √ . Note that this problem is almost exactly the same as Problem 2.41 in Homework No. 2. (a) Find the normalization constant . Hint: Decompose the original state into a linear combination of stationary states. (5 points) (b) What is the probability of finding the particle in the ground state, first excited state, and second excited state, respectively? (5 points) (c) What is the expectation value of the energy? (5 points) (d) At some later time , the wave function is given by Ψ ,
1
2
,
for some constant . What is the smallest possible value of ? (5 points) Problem 2: Hydrogen atom (20 points) ♦ A hydrogen atom starts out in the following linear combination of the stationary states 2, 1, 1 : . Ψ ,0 Use the following wave functions for the stationary states: / sin
, ,
2, 1, 1 and
, ,
√
/
and
√
√
sin .
Note that this problem is exactly the same as Problem 4.15 in Homework No. 3. (a) Construct Ψ , . Simplify it as much as you can. (10 points) 〉. Express your answer in terms of the Bohr energy. 〈 /4 (b) Find the expectation value of the potential energy, 〈 〉 (10 points) Problem 3: Pauli exclusion principle (20 points) ♦♦ Suppose we have two noninteracting particles of mass and ). The one-particle states are given by
in the infinite square well (
sin
0 for 0
and
∞ for
0
,
where and /2 for convenience. Use and for the positions of particles 1 and 2 and and for the quantum numbers of particles 1 and 2, respectively. Note that this problem is almost identical to Example 5.1 in Griffith. (a) If the particles are distinguishable, find the composite wave function and energy for the ground state. (2 points) (b) If the particles are distinguishable, find the composite wave function and energy for the first excited state. (2 points) (c) If the particles are identical bosons with the same spin, find the composite wave function and energy for the ground state. (4 points) (d) If the particles are identical bosons with the same spin, find the composite wave function and energy for the first excited state. (4 points)
(e) If the particles are identical fermions with the same spin, find the composite wave function and energy for the ground state. (4 points) (f) If the particles are identical fermions with the same spin, find the composite wave function and energy for the first excited state. (4 points) Problem 4: Perturbation theory (20 points) ♦♦ An electric field is applied to a particle of mass the particle is given by
and charge
in the 1D harmonic oscillator potential. The Hamiltonian for
. Do not be confused between the electric field and energy while we use the same notation. as a perturbation to the harmonic oscillator and using the perturbation (a) Estimate the energy levels by treating | , theory up to the second order. You may find the following equations useful: /2 | ′| , ∑ | | | | / 1| . (10 points) 1, | 1, √ √ | (b) Find the exact energy levels and compare them with the estimated energy levels you find in (a). Hint: Use a well-known mathematical trick on the quadratic part of the Hamiltonian. (10 points) Problem 5: Tritium decay (20 points) ♦ Consider a tritium atom (3H) also known as hydrogen-3 that consists of one proton, two neutrons, and one electron. Suddenly, the atom undergoes a beta decay, that is, one of the neutrons decays to a proton by emitting an electron and neutrino, such that the atom becomes a helium-3 ion (3He+) that consists of two protons, one neutron, and one electron. Use the following ground state wave function for a hydrogenic atom if necessary: Ψ ,0
/
√
/
.
3
(a) What is the ground state wave function of the tritium atom ( H) before the beta decay? (5 points) (b) What is the ground state wave function of the helium-3 ion (3He+) after the beta decay? (5 points) (c) What is the probability that the electron remains in the ground state after the beta decay? (10 points) Bonus Problem: Hyperfine splitting in positronium (20 points) ♦♦♦ Consider the positoronium atom that consists of one electron and one positron (the anti-particle of the electron with the same mass and gyromagnetic ratio, but opposite charge as the electron). The energy of a hydrogenic atom is given by , / and / , where the reduced mass of the system and the fine structure constant are given by respectively. The radii of orbits are given by , where / . The ratio of the mass of the proton to that of the / 1836, but use the approximate value of the ratio ( / ≅ 2000) in your calculations for electron is given by simplicity. (a) What is the binding energy of the ground state of positronium as compared to that of hydrogen. (4 points) (b) What is the ratio of the radii of the ground state orbit of positronium to that of hydrogen? (4 points) (c) The hyperfine coupling for the hydrogenic atom is given by | 0 | ∙ , 0 | is the wave function at the origin for the orbital with principal quantum number (which is a function of where | only). Write down the expression for the singlet-triplet energy splitting in terms of , , , , and | 0 | . (4 points) (d) What is the ratio of the hyperfine splitting in any state with 0 between positronium and hydrogen in terms of the ratios of gyromagnetic factors / and masses 0 | ) depends on the masses of / . Hint: Do not forget that (hence | the orbiting particles. In the ground state or any other state for which 0, the wave function is spherically symmetrical. The ground state wave function for the hydrogenic atom is provided in Problem 5. (4 points) (e) If the hydrogen hyperfine splitting corresponds to a wavelength of 21 cm, what is the corresponding wavelength for the 2.8. (4 points) hyperfine splitting in positronium? Use /
