109 年 國立政治大學應用物理研究所《近代物理》

1. [50 points] Short answer questions

(a) If a container has volume $V'$ in its own rest frame, $S'$, what is its volume as measured by an observer in $S$, with respect to which it is moving at high speed $v$?

(b) Particle $X_1$ of mass $m_1$ is moving with speed $v_1 > 0.5c$ and kinetic energy $K_1$. It collides with particle $X_2$ of mass $m_2$ that is initially at rest. The collision produces only a new particle $X_3$ of mass $m_3$ and kinetic energy $K_3$ (that is, $X_1 + X_2 \to X_3$). Is $m_3$ greater than, less than, or equal to the sum of $m_1 + m_2$? Explain your answer.

(c) Let $E$ represent the average energy of spin-1/2 electrons in a certain block of metal at a temperature of $0K$. Now suppose the electrons have spin 1 instead of spin 1/2. Would you expect the average energy of the spin = 1 electrons in an otherwise identical block of metal at $0K$ to be greater than, the same as, or less than the energy $E$? Explain your answer.

(d) A matter wave of energy $E > 0$ and wave number $k$ is incident from the left on a potential well of width $L$ and depth $V_0$. The top of the well is at zero energy and the bottom of the well is at $-V_0$, as shown in the figure below. Write down the spatial part of the wave function in Region 3.

🖼️【此處有附圖，請對照原卷】
E
0
Region 1
Region 2
Region 3
-V0
x = 0
L

(e) Please give an approximate value for each of the items below (2 points each).
(i) Energy (in units of eV) of a photon of visible light.
(ii) Diameter of an atom.
(iii) Diameter of a nucleus.

(f) Is $\psi(x,t) = A \sin(kx - \omega t)$ an acceptable solution to the time-dependent Schrödinger wave equation? Explain.

(g) The resistivity of most metals increases with increasing temperature, whereas the resistivity of a semiconductor decreases with increasing temperature. What explains this difference in behavior?

2. [15 points] Molecular Energy Levels

The vibrational energy levels of a certain diatomic molecule are given by $E_n = \hbar\omega(n + 1/2)$, where $n$ is an integer and $\hbar\omega = 0.124 \text{ eV}$. (Note that $hc = 1240 \text{ eV}\cdot\text{nm}$.)

(a) What is the energy (in units of eV) of the longest-wavelength photon that can be emitted by transitions between vibrational states of this molecule?

(b) What would the wavelength of this photon be?

(c) Considering whatever value you found for part (b), is this wavelength representative of photons in the visible, infrared, or ultraviolet?

3. [15 points] Stern-Gerlach experiment

In a Stern-Gerlach type of experiment, the magnetic field varies with distance in the z direction according to $dB_z/dz = 1.4 \text{ T/mm}$. The silver atoms travel a distance $x = 3.5 \text{ cm}$ through the magnet. The most probable speed of the atoms emerging from the oven is $v = 750 \text{ m/s}$. Find the separation of the two beams as they leave the magnet. The mass of a silver atom is $1.8 \times 10^{-25} \text{ kg}$, and its magnetic moment is about 1 Bohr magneton ($\mu_B \approx 9.27 \times 10^{-24} \text{ J}\cdot\text{T}^{-1}$).

4. [20 points] Particle on a ring

We consider a particle with mass m.

(a) Write down the time-independent Schrödinger equation for this particle confined to move on a circle of radius R.

(b) The wave functions must be single-valued functions on the circle, i.e. the solutions to the Schrödinger equation must satisfy the condition: $\psi(x) = \psi(x + 2\pi R)$. Find the energy eigenvalues.