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# k i w y

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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?

### 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?

### Problem 3

For one dimentional harmonic oscillator

### 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
or
\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?

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

### 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

### 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 q_i}, p'_i = -\frac{\partial H}{\partial p_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 p_i}, p'_i = -\frac{\partial H}{\partial q_i}

### Problem 9

Consider a single electron atom with orbinal angular momentum L =

### 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

### 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?

### Problem 12

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

### 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?

### 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

### 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?

### Problem 16

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

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

D

### Problem 17

The energy eigenstates

### 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.