112 年 國立中山大學物理學系碩士班《近代物理》

An astronaut traveling with a speed $v = 0.90c$ holds a meterstick in his hand. If he measures its length, he will obtain a value of
(A) 1.0 m.
(B) 2.3 m.
(C) 0.19 m.
(D) 0.43 m.
(E) 0.81 m.

The quantity which does not change in magnitude from that observed in system S when observed in system S' moving away from system S at speed v is
(A) $m\vec{a}$.
(B) $m\vec{v}$.
(C) $(\gamma - 1)mc^2$.
(D) $E^2 - p^2c^2$.
(E) $x^2 + y^2 + z^2$.

An electron ($m = 9.1 \times 10^{-31}$ kg) has a speed of $0.90c$. What is the difference between its relativistic momentum and its non-relativistic momentum (in kg m/s)?
(A) $4.3 \times 10^{-22}$
(B) $3.2 \times 10^{-22}$
(C) $5.4 \times 10^{-22}$
(D) $6.5 \times 10^{-22}$
(E) $2.5 \times 10^{-22}$

How much energy is in a 63-kHz photon of AM radiation (Hint: the Planck constant $h = 6.626 \times 10^{-34}$ J·s)?
(A) $1.0 \times 10^{-38}$ J
(B) $6.6 \times 10^{-30}$ J
(C) $4.2 \times 10^{-29}$ J
(D) $3.1 \times 10^{-30}$ J
(E) $13 \times 10^{-29}$ J

What is the maximum velocity (in km/s) of a photoelectron emitted from a surface whose work function is 5.0 eV when illuminated by a light whose wavelength is 200 nm?
(A) 460
(B) 650
(C) 420
(D) 550
(E) 1 480

A neutron has a mass of $1.67 \times 10^{-27}$ kg. The de Broglie wavelength is $1.4 \times 10^{-10}$ m. How fast is the neutron going (in m/s)?
(A) $3.4 \times 10^3$
(B) $2.8 \times 10^3$
(C) $3.9 \times 10^3$
(D) $2.6 \times 10^3$
(E) $1.7 \times 10^3$

Assume we can determine the position of a particle within an uncertainty of 0.5 nm. What will be the resulting uncertainty in the particle's momentum (in kg. m/s)?
(A) $1.9 \times 10^{-25}$
(B) $4.2 \times 10^{-25}$
(C) $1.1 \times 10^{-25}$
(D) $1.3 \times 10^{-24}$
(E) $6.6 \times 10^{-25}$

The average position, or expectation value, of a particle whose wave function $\psi(x)$ depends only on the value of $x$, is given by $<x> =$
(A) $\int_{-\infty}^{\infty} \sqrt{x} \psi(x) dx$.
(B) $\int_{-\infty}^{\infty} x \psi(x) dx$.
(C) $\int_{-\infty}^{\infty} x |\psi(x)|^2 dx$.
(D) $\int_{-\infty}^{\infty} \psi(x) dx$.
(E) $\int_{-\infty}^{\infty} x |\psi(x)|^2 dx$.

The wave function $\psi(x)$ of a particle confined to $0 \le x \le L$ is given by $\psi(x) = Ax$. $\psi(x) = 0$ for $x < 0$ and $x > L$. When the wave function is normalized, the probability density at coordinate $x$ has the value
(A) $\frac{2}{L^2} x$
(B) $\frac{2}{L} x^2$
(C) $\frac{2}{L^3} x^2$
(D) $\frac{3}{L} x^2$
(E) $\frac{3}{L^3} x^2$

Find the kinetic energy (in terms of Planck's constant) of a baseball (m = 1 kg) confined to a one-dimensional box that is 25 cm wide if the baseball can be treated as a wave in the ground state.
(A) $3 h^2$
(B) $2 h^2$
(C) $h^2$
(D) $4h^2$
(E) $0.5 h^2$

The wave function for a particle confined to a one-dimensional box located between $x = 0$ and $x = L$ is given by $\psi(x) = A \sin (\frac{n\pi x}{L}) + B \cos (\frac{n\pi x}{L})$. The constants $A$ and $B$ are determined to be
(A) $2/\sqrt{L}, 0$.
(B) $1/\sqrt{L}, 1/\sqrt{L}$.
(C) $0, 2/\sqrt{L}$.
(D) $\sqrt{2}/\sqrt{L}, \sqrt{2}/\sqrt{L}$.
(E) $2/\sqrt{L}, 0$.

The wave function for a particle in a one-dimensional box is $\psi(x) = A \sin (\frac{n\pi x}{L})$. Which statement is correct?
(A) This wavefunction gives the probability of finding the particle at $x$.
(B) $|\psi(x)|^2$ gives the probability of finding the particle at $x$.
(C) $|\psi(x)|^2 \Delta x$ gives the probability of finding the particle between $x$ and $x + \Delta x$.
(D) $\int_0^L \psi(x) dx$ gives the probability of finding the particle at a particular value of $x$.
(E) $\int_0^L |\psi(x)|^2 dx$ gives the probability of finding the particle between $x$ and $x + \Delta x$.

A particle in a finite potential well has energy $E$, as shown below. The wave function in region I where $x < 0$ has the form
(A) $a e^{-kx}$
(B) $b e^{kx}$
(C) $A \sin kx$
(D) $B \cos kx$
(E) $A \sin kx + B \cos kx$

When a particle approaching a potential energy barrier has a total energy that is greater than the potential energy barrier, what is the probability that the particle will be reflected?
(A) $P < 0$.
(B) $P = 0$.
(C) $P = 1$.
(D) $P > 0$.
(E) $P = \infty$.

The ground state energy of a harmonic oscillator is
(A) $E = \hbar \omega$.
(B) $E = \frac{1}{2} \hbar \omega$.
(C) $E = (2/3) \hbar \omega$.
(D) $E = 0$.
(E) $E = \frac{1}{4} \hbar \omega$.

An electron in a hydrogen atom makes a transition from the $n = 3$ to the $n = 1$ energy state. Determine the wavelength of the emitted photon (in nm). (R = $1.097 \times 10^7$ m$^{-1}$)
(A) 1006
(B) 209
(C) 306
(D) 103
(E) 821

Suppose Bohr had chosen the potential energy of the electron in the hydrogen atom to be $V = 0$ when the electron is in the orbit with $n = 1$. He could do this by
(A) choosing $n = 1$ for the orbit where the kinetic energy of the electron is zero.
(B) adding a constant 13.6 eV to the potential energy for all values of $n$.
(C) adding a constant 27.2 eV to the potential energy for all values of $n$.
(D) subtracting a constant 13.6 eV from the potential energy for all values of $n$.
(E) subtracting a constant 27.2 eV from the potential energy for all values of $n$.

The radial portion of the de Broglie wavefunction for an electron in the ground state of the hydrogen atom is
$R_{1s}(r) = \frac{1}{\sqrt{\pi a_0^3}} e^{-r/a_0}$, where $a_0$ is the Bohr radius. The probability of finding the electron is
(A) $\frac{1}{\sqrt{\pi a_0^3}} \int_{-\infty}^{\infty} (\frac{1}{\sqrt{\pi a_0^3}} e^{-r/a_0}) 4\pi r^2 dr$.
(B) $\int_{-\infty}^{\infty} (\frac{1}{\sqrt{\pi a_0^3}} e^{-r/a_0}) 4\pi r^2 dr$.
(C) $\frac{1}{\sqrt{\pi a_0^3}} \int_{-\infty}^{\infty} (\frac{1}{\sqrt{\pi a_0^3}} e^{-r/a_0}) dr$.
(D) $\frac{1}{\pi a_0^3} \int_{-\infty}^{\infty} (e^{-r/a_0}) dr$.
(E) $\frac{d}{dr} \left(\frac{1}{\sqrt{\pi a_0^3}} e^{-r/a_0}\right) \left(\frac{1}{\sqrt{\pi a_0^3}} e^{-r/a_0}\right)^{1/2}$.

Forbidden transitions and selection rules suggest that
(A) a photon has linear momentum.
(B) a photon has energy.
(C) a photon has angular momentum.
(D) a photon has parity.
(E) a photon has mass.

Zeke says that the magnitude of the orbital angular momentum in the hydrogen atom has the value $L = \ell \hbar$. Ruth says that the maximum magnitude of the projection of the angular momentum along the direction of a constant magnetic field vector $\vec{B}$ is $\sqrt{\ell(\ell+1)}\hbar$. Which one, if either, is correct, and why?
(A) Ruth, because the maximum value of $L$ is $\sqrt{\ell(\ell+1)}\hbar$.
(B) Ruth, because the orbital angular momentum always lines up with a magnetic field so that $\vec{L}$ has its maximum value along the field.
(C) Zeke, because the maximum magnitude of $L$ is $L = \ell\hbar$.
(D) Zeke, because the orbital angular momentum always lines up with a magnetic field so that $L$ has its maximum value along the field.
(E) Neither, because they have interchanged the maximum magnitude of $\vec{L}$, $\sqrt{\ell(\ell+1)}\hbar$, and $\ell\hbar$, its maximum projection along a magnetic field direction.