## Quantum Mechanics

### Problem 1

The partition function Z in statistical mechanics can be written as_{r}is the energy of microstate r. For a single quantum mechanical harmonic oscillator with energies

### Solution 1

Answer

### Problem 2

A simple wave function for the deuteron is given by

for a < r < a + B and for r > a + bWhich of the following expression can be used to normalize the wave function ψ?

### Solution 2

Answer

### Problem 3

For one dimentional harmonic oscillator

### Solution 3

Answer

### 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

### 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

Answer

### 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*}. )

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

### Solution 6

By definition...

- \psi^*_i (x) \psi_j (x) = 0Nonsense
- \psi^*_i (x) \psi_j (x) = 1Nonsense
- \int^{\infty}_{-\infty} \psi^*_i \psi_j (x) dx = 0Orthogonality
- \int^{\infty}_{-\infty} \psi^*_i \psi_j (x) dx = 1Normalization
- \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

where a_{0} is the Bohr radius. What is approximate distance that the electron is from the nucleus?

- 0
- a
_{0}/ 2 - a
_{0}/ √2 - a
_{0} - 2 a
_{0}

### Solution 7

Answer

### Problem 8

Which of the following gives Hamilton's canonical equations of motion? (H is the Hamltonian, q_{1} are the generalized coordinates and p_{1} are the generalized momenta.)

- q_1 = \frac{\partial H}{\partial p_i}, p_i = -\frac{\partial H}{\partial q_i}
- q'_1 = \frac{\partial H}{\partial p_i}, p'_i = -\frac{\partial H}{\partial q_i}
- q'_1 = \frac{\partial H}{\partial q_i}, p'_i = -\frac{\partial H}{\partial p_i}
- q_1 = \frac{\partial H}{\partial q_i}, p_i = \frac{\partial H}{\partial p_i}
- \frac{d}{dt} (\frac{\partial H}{\partial p_i}) - \frac{\partial H}{\partial q_i} = 0

### Solution 8

Answer

### Problem 9

Consider a single electron atom with orbinal angular momentum L =

### Solution 9

Answer

### Problem 10

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

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

### Solution 10

Answer

### 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

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

### Solution 11

Answer

### Problem 12

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

2 | i |

-i | 2 |

### Solution 12

Answer

### Problem 13

Consider the Pauli spin matrices σ_{x}, σ_{y}, and σ_{z} and the identity matrix I given above. The commutator [σ_{x}, σ_{y}] ≡ σ_{x}σ_{y} - σ_{y}σ_{x} is equal to which of the following?

### Solution 13

Answer

### 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 S_{z}= -½ ℏ is

### Solution 14

Answer

### 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

Answer

### Problem 16

Let **Ĵ** be a quantum mechanical angular momentum operator. The commutator [Ĵ_{x}, Ĵ_{y}, Ĵ_{x}] is equivalent to which of the following?

- i ℏ Ĵ
_{z} - i ℏ Ĵ
_{z}Ĵ_{x} - -i ℏ Ĵ
_{x}Ĵ_{z} - 0
- i ℏ Ĵ
_{x}Ĵ_{y}

### Solution 16

C

### Problem 17

The energy eigenstates

### Solution 17

Answer

### 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

Answer

### Problem 20

### Solution 20

Answer

### Problem 21

### Solution 21

Answer

### Problem 22

### Solution 22

Answer

### Problem 23

### Solution 23

Answer

### Problem 24

### Solution 24

Answer

### Problem 25

### Solution 25

Answer