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Quantum Mechanics

Problem 1

The partition function Z in statistical mechanics can be written as
Z = \sum\limits_{r} e^{-E_r / k T}
where the index r ranges over all possible microstates of a system and Er is the energy of microstate r. For a single quantum mechanical harmonic oscillator with energies
E_n = (n + \frac{1}{2}) \hbar \omega, n = 0,1,2,...
What is the partition function Z?

Solution 1


Problem 2

A simple wave function for the deuteron is given by

for a < r < a + B and
for r > a + b

Which of the following expression can be used to normalize the wave function ψ?

Solution 2


Problem 3

For one dimentional harmonic oscillator

Solution 3


Problem 4

The energy from electromagnetic waves in equilibrium in a cavity is used to melt ice. If the Kelvin temperature of the cavity is increased by a factor of 2, the mass of ice that can be melted in a fixed amount of time is increased by a factor of what?

Solution 4

Energy from electromagnetic wave implies radiation energy. Therefore, rate of energy supplied to ice is

\frac{dQ}{dt} = P(rad) = \sigma T^4
\frac{d(mL)}{dt} \propto T^4
since Q = mL, m is the mass of ice melted and L is latent heat, or
\frac{dm}{dt} \propto T^4
When, the temperature is increased by factor 2, dm/dt is increased by factor 24 i.e. 16. So, the mass of ice that can be melted increased by 16 times

Problem 5

A quantum mechanical harmonic oscillator has an angular frequency ω. The Schrodinger equation predicts that the ground state energy of the oscillator will be how many hbar omega?

Solution 5


Problem 6

Consider a set of wave functions ψ i (x) . Which of the following conditions guarantees that the functions are normalized and mutually orthogonal? (The indices i and j take on the values in the set {1, 2, . . . , n}. )

  1. \psi^*_i (x) \psi_j (x) = 0
  2. \psi^*_i (x) \psi_j (x) = 1
  3. \int^{\infty}_{-\infty} \psi^*_i \psi_j (x) dx = 0
  4. \int^{\infty}_{-\infty} \psi^*_i \psi_j (x) dx = 1
  5. \int^{\infty}_{-\infty} \psi^*_i \psi_j (x) dx = \delta_{ij}

Solution 6

By definition...

  1. \psi^*_i (x) \psi_j (x) = 0
  2. \psi^*_i (x) \psi_j (x) = 1
  3. \int^{\infty}_{-\infty} \psi^*_i \psi_j (x) dx = 0
  4. \int^{\infty}_{-\infty} \psi^*_i \psi_j (x) dx = 1
  5. \int^{\infty}_{-\infty} \psi^*_i \psi_j (x) dx = \delta_{ij}
    Normalized and Mutually Orthogonal (aka orthonormal)

So the answer is E

Problem 7

The normalized ground state wave function of hydrogen is

\psi_{100} = \frac{2}{(4 \pi)^{1/2} a_0^{3/2}} e^{-r/a_0}

where a0 is the Bohr radius. What is approximate distance that the electron is from the nucleus?

  1. 0
  2. a0 / 2
  3. a0 / √2
  4. a0
  5. 2 a0

Solution 7


Problem 8

Which of the following gives Hamilton's canonical equations of motion? (H is the Hamltonian, q1 are the generalized coordinates and p1 are the generalized momenta.)

  1. \frac{d}{dt} (\frac{\partial H}{\partial p_i}) - \frac{\partial H}{\partial q_i} = 0
  2. q'_1 = \frac{\partial H}{\partial p_i}, p'_i = -\frac{\partial H}{\partial q_i}
  3. q_1 = \frac{\partial H}{\partial q_i}, p_i = \frac{\partial H}{\partial p_i}
  4. q_1 = \frac{\partial H}{\partial p_i}, p_i = -\frac{\partial H}{\partial q_i}
  5. q'_1 = \frac{\partial H}{\partial q_i}, p'_i = -\frac{\partial H}{\partial p_i}

Solution 8


Problem 9

Consider a single electron atom with orbinal angular momentum L =

Solution 9


Problem 10

Characteristics of the quantum harmonic oscillator include which of the following?

  1. A spectrum of evenly spaced energy states
  2. A potential energy function that is linear in the position coordinate
  3. A ground state that is characterized by zero kinetic energy
  4. A nonzero probability of finding the oscillator outside the classical turning points

Solution 10


Problem 11

A particle is in an infinite square well potential with walls at x = 0 and x = L . If the particle is in the state

\psi(x) = A \mathrm{sin}(\frac{3 \pi x}{L})

where A is a constant, what is the probability that the particle is between x = ⅓L and x = ⅔L?

Solution 11


Problem 12

Which of the following are the eigenvalues of the Hermitian matrix?

2 i
-i 2

Solution 12


Problem 13

Consider the Pauli spin matrices σx, σy, and σz and the identity matrix I given above. The commutator [σx, σy] ≡ σxσy - σyσx is equal to which of the following?

Solution 13


Problem 14

A spin-½ particle is in a state described by the spinor

where A is a normalization constant. The probability of finding the particle with spin projection Sz = -½ ℏ is

Solution 14


Problem 15


An electron with total energy E in the region x < 0 is moving in the +x-direction. It encounters a step potential at x = 0. The wave function for x ≤ 0 is given by

and the wave function for x > 0 is given by

Which of the following gives the reflection coefficient for the system?

Solution 15


Problem 16

Let be a quantum mechanical angular momentum operator. The commutator [Ĵx, Ĵy, Ĵx] is equivalent to which of the following?

  1. i ℏ Ĵxy
  2. 0
  3. i ℏ Ĵz
  4. -i ℏ Ĵxz
  5. i ℏ Ĵzx

Solution 16


Problem 17

The energy eigenstates

Solution 17


Problem 18

Let | α > represent the state of an electron with spin up, and | β > the state of an electron with spin down. Valid spin eigenfuctions for a triplet state

Solution 18

David J. Griffiths vanity alert---the QM problems thus far are all straight out of his textbook, An Introduction to Quantum Mechanics.

Problem 19


Solution 19


Problem 20


Solution 20


Problem 21


Solution 21


Problem 22


Solution 22


Problem 23


Solution 23


Problem 24


Solution 24


Problem 25


Solution 25