shayekh/physics_gre huggingface dataset:
A particle of mass $m$ is moving along the $x$-axis with speed $v$ when it collides with a particle of mass $2 m$ initially at rest. After the collision, the first particle has come to rest, and the second particle has split into two equal-mass pieces that move at equal angles $\theta>0$ with the $x$-axis, as shown in the figure above. Which of the following statements correctly describes the speeds of the two pieces?

A. Each piece moves with speed $v$.

B. One of the pieces moves with speed $v$, the other moves with speed less than $v$.

C. Each piece moves with speed $v / 2$.

D. One of the pieces moves with speed $v / 2$, the other moves with speed greater than $v / 2$.

E. Each piece moves with speed greater than $v / 2$.

SCRATCHWORK


Answer scores (1 = correct):
A: 0
B: 0
C: 0
D: 0
E: 1
A balloon is to be filled with helium and used to suspend a mass of 300 kilograms in air. If the mass of the balloon is neglected, which of the following gives the approximate volume of helium required? (The density of air is 1.29 kilograms per cubic meter and the density of helium is 0.18 kilogram per cubic meter.)

A. $50 \mathrm{~m}^{3}$

B. $95 \mathrm{~m}^{3}$

C. $135 \mathrm{~m}^{3}$

D. $270 \mathrm{~m}^{3}$

E. $540 \mathrm{~m}^{3}$



Answer scores (1 = correct):
A: 0
B: 0
C: 0
D: 1
E: 0
A stream of water of density $\rho$, cross-sectional area $A$, and speed $v$ strikes a wall that is perpendicular to the direction of the stream, as shown in the figure above. The water then flows sideways across the wall. The force exerted by the stream on the wall is

A. $\rho v^{2} A$

B. $\rho v A / 2$

C. $\rho g h A$

D. $v^{2} A / \rho$

E. $v^{2} A / 2 \rho$

Answer scores (1 = correct):
A: 1
B: 0
C: 0
D: 0
E: 0
A proton moves in the $+z$-direction after being accelerated from rest through a potential difference $V$. The proton then passes through a region with a uniform electric field $E$ in the $+x$-direction and a uniform magnetic field $B$ in the $+y$-direction, but the proton's trajectory is not affected. If the experiment were repeated using a potential difference of $2 \mathrm{~V}$, the proton would then be

A. deflected in the $+x$-direction

B. deflected in the $-x$-direction

C. deflected in the $+y$-direction

D. deflected in the $-y$-direction

E. undeflected


Answer scores (1 = correct):
A: 0
B: 1
C: 0
D: 0
E: 0
For an inductor and capacitor connected in series, the equation describing the motion of charge is

$$
L \frac{d^{2} Q}{d t^{2}}+\frac{1}{C} Q=0
$$

where $L$ is the inductance, $C$ is the capacitance, and $Q$ is the charge. An analogous equation can be written for a simple harmonic oscillator with position $x$, mass $m$, and spring constant $k$. Which of the following correctly lists the mechanical analogs of $L, C$, and $Q$ ?

\begin{tabular}{cccc} 
& $\underline{L}$ & $\underline{C}$ & $\underline{Q}$ \\
A. & $m$ & $k$ & $x$ \\
B. & $m$ & $1 / k$ & $x$ \\
C. & $k$ & $x$ & $m$ \\
D. & $1 / k$ & $1 / m$ & $x$ \\
E. & $x$ & $1 / k$ & $1 / m$
\end{tabular}

SCRATCHWORK



Answer scores (1 = correct):
A: 0
B: 1
C: 0
D: 0
E: 0
An infinite, uniformly charged sheet with surfacecharge density $\sigma$ cuts through a spherical Gaussian surface of radius $R$ at a distance $x$ from its center, as shown in the figure above. The electric flux $\Phi$ through the Gaussian surface is

A. $\frac{\pi R^{2} \sigma}{\epsilon_{0}}$

B. $\frac{2 \pi R^{2} \sigma}{\epsilon_{0}}$

C. $\frac{\pi(R-x)^{2} \sigma}{\epsilon_{0}}$

D. $\frac{\pi\left(R^{2}-x^{2}\right) \sigma}{\epsilon_{0}}$

E. $\frac{2 \pi\left(R^{2}-x^{2}\right) \sigma}{\epsilon_{0}}$



Answer scores (1 = correct):
A: 0
B: 0
C: 0
D: 1
E: 0
An electromagnetic plane wave, propagating in vacuum, has an electric field given by $E=E_{0} \cos (k x-\omega t)$ and is normally incident on a perfect conductor at $x=0$, as shown in the figure above. Immediately to the left of the conductor, the total electric field $E$ and the total magnetic field $B$ are given by which of the following?
$\underline{E}$
$\underline{B}$
A. 0
0
B. $2 E_{0} \cos \omega t$
0
C. 0 $\left(2 E_{0} / c\right) \cos \omega t$
D. $2 E_{0} \cos \omega t$ $\left(2 E_{0} / c\right) \cos \omega t$
E. $2 E_{0} \cos \omega t$
$\left(2 E_{0} / c\right) \sin \omega t$


Answer scores (1 = correct):
A: 0
B: 0
C: 1
D: 0
E: 0
A nonrelativistic particle with a charge twice that of an electron moves through a uniform magnetic field. The field has a strength of $\pi / 4$ tesla and is perpendicular to the velocity of the particle. What is the particle's mass if it has a cyclotron frequency of 1,600 hertz?

A. $2.5 \times 10^{-23} \mathrm{~kg}$

B. $1.2 \times 10^{-22} \mathrm{~kg}$

C. $3.3 \times 10^{-22} \mathrm{~kg}$

D. $5.0 \times 10^{-21} \mathrm{~kg}$

E. $7.5 \times 10^{-21} \mathrm{~kg}$

SCRATCHWORK



Answer scores (1 = correct):
A: 1
B: 0
C: 0
D: 0
E: 0
The distribution of relative intensity $I(\lambda)$ of blackbody radiation from a solid object versus the wavelength $\lambda$ is shown in the figure above. If the Wien displacement law constant is $2.9 \times 10^{-3} \mathrm{~m} \cdot \mathrm{K}$, what is the approximate temperature of the object?

A. $10 \mathrm{~K}$

B. $50 \mathrm{~K}$

C. $250 \mathrm{~K}$

D. $1,500 \mathrm{~K}$

E. $6,250 \mathrm{~K}$


Answer scores (1 = correct):
A: 0
B: 0
C: 0
D: 1
E: 0
Electromagnetic radiation provides a means to probe aspects of the physical universe. Which of the following statements regarding radiation spectra is NOT correct?

A. Lines in the infrared, visible, and ultraviolet regions of the spectrum reveal primarily the nuclear structure of the sample.

B. The wavelengths identified in an absorption spectrum of an element are among those in its emission spectrum.

C. Absorption spectra can be used to determine which elements are present in distant stars.

D. Spectral analysis can be used to identify the composition of galactic dust.

E. Band spectra are due to molecules.

$$
C=3 k N_{A}\left(\frac{h v}{k T}\right)^{2} \frac{\mathrm{e}^{h v / k T}}{\left(\mathrm{e}^{h v / k T}-1\right)^{2}}
$$


Answer scores (1 = correct):
A: 1
B: 0
C: 0
D: 0
E: 0
Einstein's formula for the molar heat capacity $C$ of solids is given above. At high temperatures, $C$ approaches which of the following?

A. 0

B. $3 k N_{A}\left(\frac{h v}{k T}\right)$

C. $3 k N_{A} h v$

D. $3 k N_{A}$

E. $N_{A} h v$


Answer scores (1 = correct):
A: 0
B: 0
C: 0
D: 1
E: 0
A sample of radioactive nuclei of a certain element can decay only by $\gamma$-emission and $\beta$-emission. If the half-life for $\gamma$-emission is 24 minutes and that for $\beta$-emission is 36 minutes, the half-life for the sample is

A. 30 minutes

B. 24 minutes

C. 20.8 minutes

D. 14.4 minutes

E. 6 minutes


Answer scores (1 = correct):
A: 0
B: 0
C: 0
D: 1
E: 0
The ${ }^{238} \mathrm{U}$ nucleus has a binding energy of about 7.6 $\mathrm{MeV}$ per nucleon. If the nucleus were to fission into two equal fragments, each would have a kinetic energy of just over $100 \mathrm{MeV}$. From this, it can be concluded that

A. ${ }^{238} \mathrm{U}$ cannot fission spontaneously

B. ${ }^{238} \mathrm{U}$ has a large neutron excess

C. nuclei near $A=120$ have masses greater than half that of ${ }^{238} \mathrm{U}$

D. nuclei near $A=120$ must be bound by about 6.7 $\mathrm{MeV} /$ nucleon

E. nuclei near $A=120$ must be bound by about 8.5 MeV/nucleon

SCRATCHWORK


Answer scores (1 = correct):
A: 0
B: 0
C: 0
D: 0
E: 1
When ${ }_{4}^{7} \mathrm{Be}$ transforms into ${ }_{3}^{7} \mathrm{Li}$, it does so by

A. emitting an alpha particle only

B. emitting an electron only

C. emitting a neutron only

D. emitting a positron only

E. electron capture by the nucleus with emission of a neutrino


Answer scores (1 = correct):
A: 0
B: 0
C: 0
D: 0
E: 1
Blue light of wavelength 480 nanometers is most strongly reflected off a thin film of oil on a glass slide when viewed near normal incidence. Assuming that the index of refraction of the oil is 1.2 and that of the glass is 1.6 , what is the minimum thickness of the oil film (other than zero)?

A. $150 \mathrm{~nm}$

B. $200 \mathrm{~nm}$

C. $300 \mathrm{~nm}$

D. $400 \mathrm{~nm}$

E. $480 \mathrm{~nm}$


Answer scores (1 = correct):
A: 0
B: 1
C: 0
D: 0
E: 0
Light from a laser falls on a pair of very narrow slits separated by 0.5 micrometer, and bright fringes separated by 1.0 millimeter are observed on a distant screen. If the frequency of the laser light is doubled, what will be the separation of the bright fringes?

A. $0.25 \mathrm{~mm}$

B. $0.5 \mathrm{~mm}$

C. $1.0 \mathrm{~mm}$

D. $2.0 \mathrm{~mm}$

E. $2.5 \mathrm{~mm}$


Answer scores (1 = correct):
A: 0
B: 1
C: 0
D: 0
E: 0
The ultraviolet Lyman alpha line of hydrogen with wavelength 121.5 nanometers is emitted by an astronomical object. An observer on Earth measures the wavelength of the light received from the object to be 607.5 nanometers. The observer can conclude that the object is moving with a radial velocity of

A. $2.4 \times 10^{8} \mathrm{~m} / \mathrm{s}$ toward Earth

B. $2.8 \times 10^{8} \mathrm{~m} / \mathrm{s}$ toward Earth

C. $2.4 \times 10^{8} \mathrm{~m} / \mathrm{s}$ away from Earth

D. $2.8 \times 10^{8} \mathrm{~m} / \mathrm{s}$ away from Earth

E. $12 \times 10^{8} \mathrm{~m} / \mathrm{s}$ away from Earth



Answer scores (1 = correct):
A: 0
B: 0
C: 0
D: 1
E: 0
Two identical blocks are connected by a spring. The combination is suspended, at rest, from a string attached to the ceiling, as shown in the figure above. The string breaks suddenly. Immediately after the string breaks, what is the downward acceleration of the upper block?

A. 0

B. $g / 2$

C. $g$

D. $\sqrt{2} g$

E. $2 g$



Answer scores (1 = correct):
A: 0
B: 0
C: 0
D: 0
E: 1
For the system consisting of the two blocks shown in the figure above, the minimum horizontal force $F$ is applied so that block $B$ does not fall under the influence of gravity. The masses of $A$ and $B$ are 16.0 kilograms and 4.00 kilograms, respectively. The horizontal surface is frictionless and the coefficient of friction between the two blocks is 0.50 . The magnitude of $F$ is most nearly

A. $50 \mathrm{~N}$

B. $100 \mathrm{~N}$

C. $200 \mathrm{~N}$

D. $400 \mathrm{~N}$

E. $1,600 \mathrm{~N}$

SCRATCHWORK


Answer scores (1 = correct):
A: 0
B: 0
C: 0
D: 1
E: 0
The Lagrangian for a mechanical system is

$$
L=a \dot{q}^{2}+b q^{4}
$$

where $q$ is a generalized coordinate and $a$ and $b$ are constants. The equation of motion for this system is

A. $\dot{q}=\sqrt{\frac{b}{a}} q^{2}$

B. $\dot{q}=\frac{2 b}{a} q^{3}$

C. $\ddot{q}=-\frac{2 b}{a} q^{3}$

D. $\ddot{q}=+\frac{2 b}{a} q^{3}$

E. $\ddot{q}=\frac{b}{a} q^{3}$

$$
\left(\begin{array}{l}
a_{x}^{\prime} \\
a_{y}^{\prime} \\
a_{z}^{\prime}
\end{array}\right)=\left[\begin{array}{ccc}
1 / 2 & \sqrt{3} / 2 & 0 \\
-\sqrt{3} / 2 & 1 / 2 & 0 \\
0 & 0 & 1
\end{array}\right]\left(\begin{array}{l}
a_{x} \\
a_{y} \\
a_{z}
\end{array}\right)
$$


Answer scores (1 = correct):
A: 0
B: 0
C: 0
D: 1
E: 0
The matrix shown above transforms the components of a vector in one coordinate frame $S$ to the components of the same vector in a second coordinate frame $S^{\prime}$. This matrix represents a rotation of the reference frame $S$ by

A. $30^{\circ}$ clockwise about the $x$-axis

B. $30^{\circ}$ counterclockwise about the $z$-axis

C. $45^{\circ}$ clockwise about the $z$-axis

D. $60^{\circ}$ clockwise about the $y$-axis

E. $60^{\circ}$ counterclockwise about the $z$-axis


Answer scores (1 = correct):
A: 0
B: 0
C: 0
D: 0
E: 1
The mean kinetic energy of the conduction electrons in metals is ordinarily much higher than $k T$ because

A. electrons have many more degrees of freedom than atoms do

B. the electrons and the lattice are not in thermal equilibrium

C. the electrons form a degenerate Fermi gas

D. electrons in metals are highly relativistic

E. electrons interact strongly with phonons

Answer scores (1 = correct):
A: 0
B: 0
C: 1
D: 0
E: 0
An ensemble of systems is in thermal equilibrium with a reservoir for which $k T=0.025 \mathrm{eV}$.

State $A$ has an energy that is $0.1 \mathrm{eV}$ above that of state $B$. If it is assumed the systems obey Maxwell-Boltzmann statistics and that the degeneracies of the two states are the same, then the ratio of the number of systems in state $A$ to the number in state $B$ is

A. $\mathrm{e}^{+4}$

B. $\mathrm{e}^{+0.25}$

C. 1

D. $\mathrm{e}^{-0.25}$

E. $\mathrm{e}^{-4}$


Answer scores (1 = correct):
A: 0
B: 0
C: 0
D: 0
E: 1
The muon decays with a characteristic lifetime of about $10^{-6}$ second into an electron, a muon neutrino, and an electron antineutrino. The muon is forbidden from decaying into an electron and just a single neutrino by the law of conservation of

A. charge

B. mass

C. energy and momentum

D. baryon number

E. lepton number


Answer scores (1 = correct):
A: 0
B: 0
C: 0
D: 0
E: 1
A particle leaving a cyclotron has a total relativistic energy of $10 \mathrm{GeV}$ and a relativistic momentum of $8 \mathrm{GeV} / c$. What is the rest mass of this particle?

A. $0.25 \mathrm{GeV} / c^{2}$

B. $1.20 \mathrm{GeV} / c^{2}$

C. $2.00 \mathrm{GeV} / c^{2}$

D. $6.00 \mathrm{GeV} / c^{2}$

E. $16.0 \mathrm{GeV} / c^{2}$

SCRATCHWORK


Answer scores (1 = correct):
A: 0
B: 0
C: 0
D: 1
E: 0
A tube of water is traveling at $1 / 2 c$ relative to the lab frame when a beam of light traveling in the same direction as the tube enters it. What is the speed of light in the water relative to the lab frame? (The index of refraction of water is $4 / 3$.)

A. $1 / 2 c$

B. $2 / 3 c$

C. $5 / 6 c$

D. $10 / 11 c$

E. $c$


Answer scores (1 = correct):
A: 0
B: 0
C: 0
D: 1
E: 0
Which of the following is the orbital angular momentum eigenfunction $Y_{\ell}^{m}(\theta, \phi)$ in a state for which the operators $\mathbf{L}^{2}$ and $L_{z}$ have eigenvalues $6 \hbar^{2}$ and $-\hbar$, respectively?

A. $Y_{2}^{1}(\theta, \phi)$

B. $Y_{2}^{-1}(\theta, \phi)$

C. $\frac{1}{\sqrt{2}}\left[Y_{2}^{1}(\theta, \phi)+Y_{2}^{-1}(\theta, \phi)\right]$

D. $Y_{3}^{2}(\theta, \phi)$

E. $Y_{3}^{-1}(\theta, \phi)$


Answer scores (1 = correct):
A: 0
B: 1
C: 0
D: 0
E: 0
Let $|\alpha\rangle$ represent the state of an electron with spin up and $|\beta\rangle$ the state of an electron with spin down. Valid spin eigenfunctions for a triplet state $\left({ }^{3} S\right)$ of a two-electron atom include which of the following?
I. $|\alpha\rangle_{1}|\alpha\rangle_{2}$

II. $\frac{1}{\sqrt{2}}\left(|\alpha\rangle_{1}|\beta\rangle_{2}-|\alpha\rangle_{2}|\beta\rangle_{1}\right)$

III. $\frac{1}{\sqrt{2}}\left(|\alpha\rangle_{1}|\beta\rangle_{2}+|\alpha\rangle_{2}|\beta\rangle_{1}\right)$

A. I only

B. II only

C. III only

D. I and III

E. II and III

Answer scores (1 = correct):
A: 0
B: 0
C: 0
D: 1
E: 0
The state of a spin- $\frac{1}{2}$ particle can be represented using the eigenstates $|\uparrow\rangle$ and $|\downarrow\rangle$ of the $S_{z}$ operator.

$$
\begin{aligned}
& S_{z}|\uparrow\rangle=\frac{1}{2} \hbar|\uparrow\rangle \\
& S_{z}|\downarrow\rangle=-\frac{1}{2} \hbar|\downarrow\rangle
\end{aligned}
$$

Given the Pauli matrix $\sigma_{x}=\left(\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right)$, which of the following is an eigenstate of $S_{x}$ with eigenvalue $-\frac{1}{2} \hbar$ ?

A. $|\downarrow\rangle$

B. $\frac{1}{\sqrt{2}}(|\uparrow\rangle+|\downarrow\rangle)$

C. $\frac{1}{\sqrt{2}}(|\uparrow\rangle-|\downarrow\rangle)$

D. $\frac{1}{\sqrt{2}}(|\uparrow\rangle+i|\downarrow\rangle)$

E. $\frac{1}{\sqrt{2}}(|\uparrow\rangle-i|\downarrow\rangle)$



Answer scores (1 = correct):
A: 0
B: 0
C: 1
D: 0
E: 0
An energy-level diagram of the $n=1$ and $n=2$ levels of atomic hydrogen (including the effects of spin-orbit coupling and relativity) is shown in the figure above. Three transitions are labeled $A, B$, and $C$. Which of the transitions will be possible electric-dipole transitions?

A. $B$ only

B. $C$ only

C. $A$ and $C$ only

D. $B$ and $C$ only

E. $A, B$, and $C$

SCRATCHWORK


Answer scores (1 = correct):
A: 0
B: 0
C: 0
D: 1
E: 0
One end of a Nichrome wire of length $2 L$ and cross-sectional area $A$ is attached to an end of another Nichrome wire of length $L$ and crosssectional area $2 A$. If the free end of the longer wire is at an electric potential of 8.0 volts, and the free end of the shorter wire is at an electric potential of 1.0 volt, the potential at the junction of the two wires is most nearly equal to

A. $2.4 \mathrm{~V}$

B. $3.3 \mathrm{~V}$

C. $4.5 \mathrm{~V}$

D. $5.7 \mathrm{~V}$

E. $6.6 \mathrm{~V}$



Answer scores (1 = correct):
A: 1
B: 0
C: 0
D: 0
E: 0
A coil of 15 turns, each of radius 1 centimeter, is rotating at a constant angular velocity $\omega=300$ radians per second in a uniform magnetic field of 0.5 tesla, as shown in the figure above. Assume at time $t=0$ that the normal $\hat{\mathbf{n}}$ to the coil plane is along the $y$-direction and that the selfinductance of the coil can be neglected. If the coil resistance is $9 \mathrm{ohms}$, what will be the magnitude of the induced current in milliamperes?

A. $225 \pi \sin \omega t$

B. $250 \pi \sin \omega t$

C. $0.08 \pi \cos \omega t$

D. $1.7 \pi \cos \omega t$

E. $25 \pi \cos \omega t$

SCRATCHWORK


Answer scores (1 = correct):
A: 0
B: 0
C: 0
D: 0
E: 1
Two spherical, nonconducting, and very thin shells of uniformly distributed positive charge $Q$ and radius $d$ are located a distance $10 d$ from each other. A positive point charge $q$ is placed inside one of the shells at a distance $d / 2$ from the center, on the line connecting the centers of the two shells, as shown in the figure above. What is the net force on the charge $q$ ?

A. $\frac{q Q}{361 \pi \epsilon_{0} d^{2}}$ to the left

B. $\frac{q Q}{361 \pi \epsilon_{0} d^{2}}$ to the right

C. $\frac{q Q}{441 \pi \epsilon_{0} d^{2}}$ to the left

D. $\frac{q Q}{441 \pi \epsilon_{0} d^{2}}$ to the right

E. $\frac{360 q Q}{361 \pi \epsilon_{0} d^{2}}$ to the left

SCRATCHWORK



Answer scores (1 = correct):
A: 1
B: 0
C: 0
D: 0
E: 0
A segment of wire is bent into an arc of radius $R$ and subtended angle $\theta$, as shown in the figure above. Point $P$ is at the center of the circular segment. The wire carries current $I$. What is the magnitude of the magnetic field at $P$ ?

A. 0

B. $\frac{\mu_{0} I \theta}{(2 \pi)^{2} R}$

C. $\frac{\mu_{0} I \theta}{4 \pi R}$

D. $\frac{\mu_{0} I \theta}{4 \pi R^{2}}$

E. $\frac{\mu_{0} I}{2 \theta R^{2}}$



Answer scores (1 = correct):
A: 0
B: 0
C: 1
D: 0
E: 0
A child is standing on the edge of a merry-goround that has the shape of a solid disk, as shown in the figure above. The mass of the child is 40 kilograms. The merry-go-round has a mass of 200 kilograms and a radius of 2.5 meters, and it is rotating with an angular velocity of $\omega=2.0$ radians per second. The child then walks slowly toward the center of the-merry-goround. What will be the final angular velocity of the merry-go-round when the child reaches the center? (The size of the child can be neglected.)

A. $2.0 \mathrm{rad} / \mathrm{s}$

B. $2.2 \mathrm{rad} / \mathrm{s}$

C. $2.4 \mathrm{rad} / \mathrm{s}$

D. $2.6 \mathrm{rad} / \mathrm{s}$

E. $2.8 \mathrm{rad} / \mathrm{s}$

SCRATCHWORK



Answer scores (1 = correct):
A: 0
B: 0
C: 0
D: 0
E: 1
Two identical springs with spring constant $k$ are connected to identical masses of mass $M$, as shown in the figures above. The ratio of the period for the springs connected in parallel (Figure 1) to the period for the springs connected in series (Figure 2) is

A. $\frac{1}{2}$

B. $\frac{1}{\sqrt{2}}$

C. 1

D. $\sqrt{2}$

E. 2

SCRATCHWORK



Answer scores (1 = correct):
A: 1
B: 0
C: 0
D: 0
E: 0
The cylinder shown above, with mass $M$ and radius $R$, has a radially dependent density. The cylinder starts from rest and rolls without slipping down an inclined plane of height $H$. At the bottom of the plane its translational speed is $(8 g H / 7)^{1 / 2}$. Which of the following is the rotational inertia of the cylinder?

A. $\frac{1}{2} M R^{2}$

B. $\frac{3}{4} M R^{2}$

C. $\frac{7}{8} M R^{2}$

D. $M R^{2}$

E. $\frac{7}{4} M R^{2}$



Answer scores (1 = correct):
A: 0
B: 1
C: 0
D: 0
E: 0
Two small equal masses $m$ are connected by an ideal massless spring that has equilibrium length $\ell_{0}$ and force constant $k$, as shown in the figure above. The system is free to move without friction in the plane of the page. If $p_{1}$ and $p_{2}$ represent the magnitudes of the momenta of the two masses, a Hamiltonian for this system is

A. $\frac{1}{2}\left\{\frac{p_{1}^{2}}{m}+\frac{p_{2}^{2}}{m}-2 k\left(\ell-\ell_{0}\right)\right\}$

B. $\frac{1}{2}\left\{\frac{p_{1}^{2}}{m}+\frac{p_{2}^{2}}{m}+2 k\left(\ell-\ell_{0}\right)^{2}\right\}$

C. $\frac{1}{2}\left\{\frac{p_{1}^{2}}{m}+\frac{p_{2}^{2}}{m}-k\left(\ell-\ell_{0}\right)\right\}$

D. $\frac{1}{2}\left\{\frac{p_{1}^{2}}{m}+\frac{p_{2}^{2}}{m}-k\left(\ell-\ell_{0}\right)^{2}\right\}$

E. $\frac{1}{2}\left\{\frac{p_{1}^{2}}{m}+\frac{p_{2}^{2}}{m}+k\left(\ell-\ell_{0}\right)^{2}\right\}$

SCRATCHWORK


Answer scores (1 = correct):
A: 0
B: 0
C: 0
D: 0
E: 1
The solution to the Schrödinger equation for the ground state of hydrogen is

$$
\psi_{0}=\frac{1}{\sqrt{\pi a_{0}^{3}}} e^{-r / a_{0}}
$$

where $a_{0}$ is the Bohr radius and $r$ is the distance from the origin. Which of the following is the most probable value for $r$ ?

A. 0

B. $a_{0} / 2$

C. $a_{0}$

D. $2 a_{0}$

E. $\infty$


Answer scores (1 = correct):
A: 0
B: 0
C: 1
D: 0
E: 0
The raising and lowering operators for the quantum harmonic oscillator satisfy

$a^{\dagger}|n\rangle=\sqrt{n+1}|n+1\rangle, a|n\rangle=\sqrt{n}|n-1\rangle$

for energy eigenstates $|n\rangle$ with energy $E_{n}$.

Which of the following gives the first-order shift in the $n=2$ energy level due to the perturbation

$$
\Delta H=V\left(a+a^{\dagger}\right)^{2}
$$

where $V$ is a constant?

A. 0

B. $V$

C. $\sqrt{2} V$

D. $2 \sqrt{2} V$

E. $5 \mathrm{~V}$



Answer scores (1 = correct):
A: 0
B: 0
C: 0
D: 0
E: 1
An infinite slab of insulating material with dielectric constant $K$ and permittivity $\epsilon=K \epsilon_{0}$ is placed in a uniform electric field of magnitude $E_{0}$. The field is perpendicular to the surface of the material, as shown in the figure above. The magnitude of the electric field inside the material is

A. $\frac{E_{0}}{K}$

B. $\frac{E_{0}}{K \epsilon_{0}}$

C. $E_{0}$

D. $K \epsilon_{0} E_{0}$

E. $K E_{0}$


Answer scores (1 = correct):
A: 1
B: 0
C: 0
D: 0
E: 0
A uniformly charged sphere of total charge $Q$ expands and contracts between radii $R_{1}$ and $R_{2}$ at a frequency $f$. The total power radiated by the sphere is

A. proportional to $Q$

B. proportional to $f^{2}$

C. proportional to $f^{4}$

D. proportional to $\left(R_{2} / R_{1}\right)$

E. zero

SCRATCHWORK



Answer scores (1 = correct):
A: 0
B: 0
C: 0
D: 0
E: 1
A beam of light has a small wavelength spread $\delta \lambda$ about a central wavelength $\lambda$. The beam travels in vacuum until it enters a glass plate at an angle $\theta$ relative to the normal to the plate, as shown in the figure above. The index of refraction of the glass is given by $n(\lambda)$. The angular spread $\delta \theta^{\prime}$ of the refracted beam is given by

A. $\delta \theta^{\prime}=\left|\frac{1}{n} \delta \lambda\right|$

B. $\delta \theta^{\prime}=\left|\frac{d n(\lambda)}{d \lambda} \delta \lambda\right|$

C. $\delta \theta^{\prime}=\left|\frac{1}{\lambda} \frac{d \lambda}{d n} \delta \lambda\right|$

D. $\delta \theta^{\prime}=\left|\frac{\sin \theta}{\sin \theta^{\prime}} \frac{\delta \lambda}{\lambda}\right|$

E. $\delta \theta^{\prime}=\left|\frac{\tan \theta^{\prime}}{n} \frac{d n(\lambda)}{d \lambda} \delta \lambda\right|$

Answer scores (1 = correct):
A: 0
B: 0
C: 0
D: 0
E: 1
Suppose that a system in quantum state $i$ has energy $E_{i}$. In thermal equilibrium, the expression

$$
\frac{\sum_{i} E_{i} e^{-E_{i} / k T}}{\sum_{i} e^{-E_{i} / k T}}
$$

represents which of the following?

A. The average energy of the system

B. The partition function

C. Unity

D. The probability to find the system with energy $E_{i}$

E. The entropy of the system


Answer scores (1 = correct):
A: 1
B: 0
C: 0
D: 0
E: 0