Physics Problems & Solutions: #98

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Thermal Physics - Thermal Equilibrium

Suppose that a system in a quantum state i has energy E_i. In thermal equilibrium, the expression

represents which of the following?

A. The average energy of the system
B. The partition function
C. Unity
D. The probability to find system with energy E_i
E. The entropy of the system

(GR0177 #98)

Solution:

The units of the given expression are the units of E_i which as given is energy.

The average energy is given by

<E> = ^{∑_i E_ie^−E_i/kT}/ _Z

where

Z = ∑_i e^−E_i/kT

is the partition function.

Answer: A

Electromagnetism - Electric Potential

The long thin cylindrical glass rod shown above has length l and is insulated from its surroundings. The rod has an excess charge Q uniformly distributed along its length. Assume the electric potential to be zero at infinite distances from the rod. If k is the constant in Coulomb’s law, the electric potential at a point P along the axis of the rod and a distance l from one end is kQ/l multiplied by

A. 4/9
B. 1/2
C. 2/3
D. ln 2
E. 1

(GR8677 #98)

Solution:

Line Charge Density: $\lambda=\frac{q_{\text{enclosed}}}{l}$
$q_{\text{enc}} = Q$ , and for uniform line charge: $dQ=\lambda dl$

Electric Potential: $V=\frac{kQ}{r}$

$\\\rightarrow dV=\frac{kdQ}{l}=k\lambda \frac{dl}{l} \\\rightarrow V=k\lambda \int_{l}^{2l}\frac{dl}{l}=k\lambda \ln2=\frac{kQ}{l}\ln2$

Answer: D

Quantum Mechanics - Hermitian Matrix

The matrix $A= \left| \begin{array}{ccc} 0 & 1 & 0\\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{array} \right|$ has 3 eigenvalues $\lambda_i$ defined by $Av_i=\lambda_i v_i$ . Which of the following statements is NOT true?

A. $\lambda_1 + \lambda_2 + \lambda_3 = 0$
B. $\lambda_1, \lambda_2,$ and $\lambda_3$ are all the real numbers
C. $\lambda_1\lambda_2 = + 1$ for some pair roots
D. $\lambda_1\lambda_2 + \lambda_2\lambda_3 + \lambda_3\lambda_1= 0$
E. $\lambda_i^3 =+1, i =1,2,3$

(GR9277 #98)

Solution:

Every real symmetric matrix is Hermitian, and therefore all its eigenvalues are real.

A symmetric matrix is a square matrix that is equal to its transpose.
$A=A^T \rightarrow a_{ij} = a_{ji}$

Matrix $A = \left| \begin{array}{ccc} 0 & 1 & 0\\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{array} \right|$

is not symmetric, since $a_{ij} \neq a_{ji}$ . Therefore, its eigenvalues are NOT real.

Answer: B

Quantum Mechanics - Harmonic Oscillator

A particle of mass m is acted on by a harmonic force with potential energy function V(x) = mω²x²/2 (a one dimensional simple harmonic oscillator). If there is a wall at x = 0 so that V = ∞ for x < 0, then the energy levels are equals to

A. 0, ħω, 2ħω, ...
B. 0, ¹⁄₂ħω, ħω, ...
C. ¹⁄₂ħω, ³⁄₂ħω, ⁵⁄₂ħω, ...
D. ³⁄₂ħω, ⁷⁄₂ħω, ¹¹⁄₂ħω, ...
E. 0, ³⁄₂ħω, ⁵⁄₂ħω, ...

(GR9677 #98)

Solution

The probability distributions for the quantum states of the oscillator without the barrier.

image: hyperphysics

Infinite barrier at the origin means a node at origin, or the wave function goes to zero at x = 0.

By symmetry, the ground state will disappear, as well all the even states. Odd values remain.

Answer: D

Nuclear & Particle Physics - Muon Decay

Which of the following is the principal decay mode of the positive muon μ⁺?

A. μ⁺ → e⁺ + ν_e
B. μ⁺ → p + ν_μ
C. μ⁺ → n + e⁺ + ν_e
D. μ⁺ → e⁺ + ν_e+ v̄_μ
E. μ⁺ → π⁺+ v̄_e+ ν_μ

(GR0877 #98)

Solution:

(A) FALSE
μ⁺ → e⁺ + ν_eviolates lepton number: 1 → 1 + 1

(B), (C) FALSE
Muon is elementary particle (lepton), p and n are composite particle (bosons). Elementary particle cannot decay into composite particle, (B), (C) FALSE

(D) TRUE

(E) FALSE
π is pion, which a boson, not a lepton.

Answer: D