shayekh/physics_gre huggingface dataset:
The state of a quantum mechanical system is described by a wave function $\psi$. Consider two physical observables that have discrete eigenvalues: observable $A$ with eigenvalues $\{\alpha\}$, and observable $B$ with eigenvalues $\{\beta\}$. Under what circumstances can all wave functions be expanded in a set of basis states, each of which is a simultaneous eigenfunction of both $A$ and $B$ ?

A. Only if the values $\{\alpha\}$ and $\{\beta\}$ are nondegenerate

B. Only if $A$ and $B$ commute

C. Only if $A$ commutes with the Hamiltonian of the system

D. Only if $B$ commutes with the Hamiltonian of the system

E. Under all circumstances



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A particle of mass $m$ is confined to an infinitely deep square-well potential:

$$
egin{aligned}
& V(x)=\infty, x \leq 0, x \geq a \
& V(x)=0,0<x<a .
\end{aligned}
$$

The normalized eigenfunctions, labeled by the quantum number $n$, are $\psi_{n}=\sqrt{rac{2}{a}} \sin rac{n \pi x}{a}$.
For any state $n$, the expectation value of the momentum of the particle is

A. 0

B. $\frac{\hbar n \pi}{a}$

C. $\frac{2 h n \pi}{a}$

D. $\frac{\hbar n \pi}{a}(\cos n \pi-1)$

E. $\frac{-i \hbar n \pi}{a}(\cos n \pi-1)$


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A particle of mass $m$ is confined to an infinitely deep square-well potential:

$$
egin{aligned}
& V(x)=\infty, x \leq 0, x \geq a \
& V(x)=0,0<x<a .
\end{aligned}
$$

The normalized eigenfunctions, labeled by the quantum number $n$, are $\psi_{n}=\sqrt{rac{2}{a}} \sin rac{n \pi x}{a}$.
The eigenfunctions satisfy the condition $\int_{0}^{a} \psi_{n}{ }^{*}(x) \psi_{\ell}(x) d x=\delta_{n \ell}, \delta_{n \ell}=1$ if $n=\ell$, otherwise $\delta_{n \ell}=0$. This is a statement that the eigenfunctions are

A. solutions to the Schrödinger equation

B. orthonormal

C. bounded

D. linearly dependent

E. symmetric


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A particle of mass $m$ is confined to an infinitely deep square-well potential:

$$
egin{aligned}
& V(x)=\infty, x \leq 0, x \geq a \
& V(x)=0,0<x<a .
\end{aligned}
$$

The normalized eigenfunctions, labeled by the quantum number $n$, are $\psi_{n}=\sqrt{rac{2}{a}} \sin rac{n \pi x}{a}$.
A measurement of energy $E$ will always satisfy which of the following relationships?

A. $E \leq \frac{\pi^{2} \hbar^{2}}{8 m a^{2}}$

B. $E \geq \frac{\pi^{2} \hbar^{2}}{2 m a^{2}}$

C. $E=\frac{\pi^{2} \hbar^{2}}{8 m a^{2}}$

D. $E=\frac{n^{2} \pi^{2} \hbar^{2}}{8 m a^{2}}$

E. $E=\frac{\pi^{2} \hbar^{2}}{2 m a^{2}}$



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C: 0
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A rectangular loop of wire with dimensions shown above is coplanar with a long wire carrying current $I$. The distance between the wire and the left side of the loop is $r$. The loop is pulled to the right as indicated.
What are the directions of the induced current in the loop and the magnetic forces on the left and the right sides of the loop as the loop is pulled?

\begin{tabular}{lll} 
Induced Current & \begin{tabular}{l} 
Force on \\
Left Side
\end{tabular} & \begin{tabular}{l} 
Force on \\
Right Side
\end{tabular} \\
A. Counterclockwise & To the left & To the right \\
B. Counterclockwise & To the left & To the left \\
C. Counterclockwise & To the right & To the left \\
D. Clockwise & To the right & To the left \\
E. Clockwise & To the left & To the right
\end{tabular}


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A rectangular loop of wire with dimensions shown above is coplanar with a long wire carrying current $I$. The distance between the wire and the left side of the loop is $r$. The loop is pulled to the right as indicated.
What is the magnitude of the net force on the loop when the induced current is $i$ ?

A. $\frac{\mu_{0} i I}{2 \pi} \ln \left(\frac{r+a}{r}\right)$

B. $\frac{\mu_{0} i I}{2 \pi} \ln \left(\frac{r}{r+a}\right)$

C. $\frac{\mu_{0} i I}{.2 \pi} \frac{b}{a}$

D. $\frac{\mu_{0} i I}{2 \pi} \frac{a b}{r(r+a)}$

E. $\frac{\mu_{0} i I}{2 \pi} \frac{r(r+a)}{a b}$

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If $v$ is frequency and $h$ is Planck's constant, the ground state energy of a one-dimensional quantum mechanical harmonic oscillator is

A. 0

B. $\frac{1}{3} h v$

C. $\frac{1}{2} h v$

D. $h v$

E. $\frac{3}{2} h v$



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A uniform and constant magnetic field $B$ is directed perpendicularly into the plane of the page everywhere within a rectangular region as shown above. A wire circuit in the shape of a semicircle is uniformly rotated counterclockwise in the plane of the page about an axis $A$. The axis $A$ is perpendicular to the page at the edge of the field and directed through the center of the straight-line portion of the circuit. Which of the following graphs best approximates the emf $\varepsilon$ induced in the circuit as a function of time $t$ ?

A.



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The ground state configuration of a neutral sodium atom $(Z=11)$ is

A. $1 s^{2} 2 s^{2} 2 p^{5} 3 s^{2}$

B. $1 s^{2} 2 s^{3} 2 p^{6}$

C. $1 s^{2} 2 s^{2} 2 p^{6} 3 s$

D. $1 s^{2} 2 s^{2} 2 p^{6} 3 p$

E. $l s^{2} 2 s^{2} 2 p^{5}$


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C: 1
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The ground state of the helium atom is a spin

A. singlet

B. doublet

C. triplet

D. quartet

E. quintuplet


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C: 0
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An electron in a metal has an effective mass $m^{*}=0.1 m_{e}$. If this metal is placed in a magnetic field of magnitude 1 tesla, the cyclotron resonance frequency, $\omega_{c}$, is most nearly

A. $930 \mathrm{rad} / \mathrm{s}$

B. $8.5 \times 10^{6} \mathrm{rad} / \mathrm{s}$

C. $2.8 \times 10^{11} \mathrm{rad} / \mathrm{s}$

D. $1.8 \times 10^{12} \mathrm{rad} / \mathrm{s}$

E. $7.7 \times 10^{20} \mathrm{rad} / \mathrm{s}$


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A long, straight, and massless rod pivots about one end in a vertical plane. In configuration I, shown above, two small identical masses are attached to the free end; in configuration II, one mass is moved to the center of the rod. What is the ratio of the frequency of small oscillations of configuration II to that of configuration I ?

A. $(6 / 5)^{\frac{1}{2}}$

B. $(3 / 2)^{\frac{1}{2}}$

C. $6 / 5$

D. $3 / 2$

E. $5 / 3$


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C: 0
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A mole of ideal gas initially at temperature $T_{0}$ and volume $V_{0}$ undergoes a reversible isothermal expansion to volume $V_{1}$. If the ratio of specific heats is $c_{p} / c_{v}=\gamma$ and if $R$ is the gas constant, the work done by the gas is

A. zero

B. $R T_{0}\left(V_{1} / V_{0}\right)^{\gamma}$

C. $R T_{0}\left(V_{1} / V_{0}-1\right)$

D. $c_{v} T_{0}\left[1-\left(V_{0} / V_{1}\right)^{\gamma^{-1}}\right]$

E. $R T_{0} \ln \left(V_{1} / V_{0}\right)$


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Which of the following is true if the arrangement of an isolated thermodynamic system is of maximal probability?

A. Spontaneous change to a lower probability occurs.

B. The entropy is a minimum.

C. Boltzmann's constant approaches zero.

D. No spontaneous change occurs.

E. The entropy is zero.


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C: 0
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If an electric field is given in a certain region by $E_{x}=0, E_{y}=0, E_{z}=k z$, where $k$ is a nonzero constant, which of the following is true?

A. There is a time-varying magnetic field.

B. There is charge density in the region.

C. The electric field cannot be constant in time.

D. The electric field is impossible under any circumstances.

E. None of the above



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Two point charges with the same charge $+Q$ are fixed along the $x$-axis and are a distance $2 R$ apart as shown. A small particle with mass $m$ and charge $-q$ is placed at the midpoint between them. What is the angular frequency $\omega$ of small oscillations of this particle along the $y$-direction?

A. $\frac{Q q}{2 \pi \epsilon_{0} m R^{2}}$

B. $\frac{Q q}{4 \pi \varepsilon_{0} m R^{2}}$

C. $\frac{Q q}{2 \pi \epsilon_{0} m R^{3}}$

D. $\left(\frac{Q q}{4 \pi \epsilon_{0} m R^{2}}\right)^{\frac{1}{2}}$

E. $\left(\frac{Q q}{2 \pi \epsilon_{0} m R^{3}}\right)^{\frac{1}{2}}$


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A thin uniform steel chain is 10 meters long with a mass density of 2 kilograms per meter. One end of the chain is attached to a horizontal axle having a radius that is small compared to the length of the chain. If the chain initially hangs vertically, the work required to slowly wind it up on to the axle is closest to

A. $100 \mathrm{~J}$

B. $200 \mathrm{~J}$

C. $1,000 \mathrm{~J}$

D. $2,000 \mathrm{~J}$

E. $10,000 \mathrm{~J}$



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A steady beam of light is normally incident on a piece of polaroid. As the polaroid is rotated around the beam axis, the transmitted intensity varies as $A+B \cos 2 \theta$, where $\theta$ is the angle of rotation, and $A$ and $B$ are constants with $A>B>0$. Which of the following may be correctly concluded about the incident light?

A. The light is completely unpolarized.

B. The light is completely plane polarized.

C. The light is partly plane polarized and partly unpolarized.

D. The light is partly circularly polarized and partly unpolarized.

E. The light is completely circularly polarized.


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The angular separation of the two components of a double star is 8 microradians, and the light from the double star has a wavelength of 5500 angstroms. The smallest diameter of a telescope mirror that will resolve the double star is most nearly

A. $1 \mathrm{~mm}$

B. $1 \mathrm{~cm}$

C. $10 \mathrm{~cm}$

D. $1 \mathrm{~m}$

E. $100 \mathrm{~m}$

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A fast charged particle passes perpendicularly through a thin glass sheet of index of refraction 1.5. The particle emits light in the glass. The minimum speed of the particle is

A. $\frac{1}{3} c$

B. $\frac{4}{9} c$

C. $\frac{5}{9} c$

D. $\frac{2}{3} c$

E. $c$


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A monoenergetic beam consists of unstable particles with total energies 100 times their rest energy. If the particles have rest mass $m$, their momentum is most nearly

A. $m c$

B. $10 \mathrm{mc}$

C. $70 \mathrm{mc}$

D. $100 m c$

E. $10^{4} \mathrm{mc}$



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A system in thermal equilibrium at temperature $T$ consists of a large number $N_{0}$ of subsystems, each of which can exist only in two states of energy $E_{1}$ and $E_{2}$, where $E_{2}-E_{1}=\epsilon>0$. In the expressions that follow, $k$ is the Boltzmann constant.
For a system at temperature $T$, the average number of subsystems in the state of energy $E_{1}$ is given by

A. $\frac{N_{0}}{2}$

B. $\frac{N_{0}}{1+e^{-\epsilon / k T}}$

C. $N_{0} e^{-\epsilon / k T}$

D. $\frac{N_{0}}{1+e^{\epsilon / k T}}$

E. $\frac{N_{0} e^{\epsilon / k T}}{2}$


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A system in thermal equilibrium at temperature $T$ consists of a large number $N_{0}$ of subsystems, each of which can exist only in two states of energy $E_{1}$ and $E_{2}$, where $E_{2}-E_{1}=\epsilon>0$. In the expressions that follow, $k$ is the Boltzmann constant.
The internal energy of this system at any temperature $T$ is given by $E_{1} N_{0}+\frac{N_{0} \epsilon}{1+e^{\epsilon / k T}}$. The heat capacity of the system is given by which of the following expressions?

A. $N_{0} k\left(\frac{\epsilon}{k T}\right)^{2} \frac{e^{\epsilon / k T}}{\left(1+e^{\epsilon / k T}\right)^{2}}$

B. $N_{0} k\left(\frac{\epsilon}{k T}\right)^{2} \frac{1}{\left(1+e^{\epsilon / k T}\right)^{2}}$

C. $N_{0} k\left(\frac{\epsilon}{k T}\right)^{2} e^{-\epsilon / k T}$

D. $\frac{N_{0} k}{2}\left(\frac{\epsilon}{k T}\right)^{2}$

E. $\frac{3}{2} N_{0} k$


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A system in thermal equilibrium at temperature $T$ consists of a large number $N_{0}$ of subsystems, each of which can exist only in two states of energy $E_{1}$ and $E_{2}$, where $E_{2}-E_{1}=\epsilon>0$. In the expressions that follow, $k$ is the Boltzmann constant.
Which of the following is true of the entropy of the system?

A. It increases without limit with $T$ from zero at $T=0$.

B. It decreases with increasing $T$.

C. It increases from zero at $T=0$ to $N_{0} k \ln 2$ at arbitrarily high temperatures.

D. It is given by $N_{0} k\left[\frac{5}{2} \ln T-\ln p+\right.$ constant $]$.

E. It cannot be calculated from the information given.

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Two circular hoops, $X$ and $Y$, are hanging on nails in a wall. The mass of $X$ is four times that of $Y$, and the diameter of $X$ is also four times that of $Y$. If the period of small oscillations of $X$ is $T$, the period of small oscillations of $Y$ is
A. $T$
B. $T / 2$
C. $T / 4$
D. $T / 8$
E. $T / 16$

$$
{ }_{92}^{235} \mathrm{U} \rightarrow{ }_{90}^{231} \mathrm{Th}+{ }_{2}^{4} \mathrm{He}
$$


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A uranium nucleus decays at rest into a thorium nucleus and a helium nucleus, as shown above. Which of the following is true?

A. Each decay product has the same kinetic energy.

B. Each decay product has the same speed.

C. The decay products tend to go in the same direction.

D. The thorium nucleus has more momentum than the helium nucleus.

E. The helium nucleus has more kinetic energy than the thorium nucleus.


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The configuration of three electrons $1 \mathrm{~s} 2 \mathrm{p} 3 \mathrm{p}$ has which of the following as the value of its maximum possible total angular momentum quantum number?

A. $\frac{7}{2}$

B. 3

C. $\frac{5}{2}$

D. 2

E. $\frac{3}{2}$


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A: 1
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C: 0
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Consider a heavy nucleus with spin $\frac{1}{2}$. The magnitude of the ratio of the intrinsic magnetic moment of this nucleus to that of an electron is

A. zero, because the nucleus has no intrinsic magnetic moment

B. greater than 1, because the nucleus contains many protons

C. greater than 1, because the nucleus is so much larger in diameter than the electron

D. less than 1, because of the strong interactions among the nucleons in a nucleus

E. less than 1, because the nucleus has a mass much larger than that of the electron



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One ice skater of mass $m$ moves with speed $2 v$ to the right, while another of the same mass $m$ moves with speed $v$ toward the left, as shown in Figure I. Their paths are separated by a distance $b$. At $t=0$, when they are both at $x=0$, they grasp a pole of length $b$ and negligible mass. For $t>0$, consider the system as a rigid body of two masses $m$ separated by distance $b$, as shown in Figure II. Which of the following is the correct formula for the motion after $t=0$ of the skater initially at $y=b / 2$ ?

A. $x=2 v t, y=b / 2$

B. $x=v t+0.5 b \sin (3 v t / b), \quad y=0.5 b \cos (3 v t / b)$

C. $x=0.5 v t+0.5 b \sin (3 v t / b), \quad y=0.5 b \cos (3 v t / b)$

D. $x=v t+0.5 b \sin (6 v t / b), \quad y=0.5 b \cos (6 v t / b)$

E. $x=0.5 v t+0.5 b \sin (6 v t / b), \quad y=0.5 b \cos (6 v t / b)$



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The dispersion curve shown above relates the angular frequency $\omega$ to the wave number $k$. For waves with wave numbers lying in the range $k_{1}<k<k_{2}$, which of the following is true of the phase velocity and the group velocity?

A. They are in opposite directions.

B. They are in the same direction and the phase velocity is larger.

C. They are in the same direction and the group velocity is larger.

D. The phase velocity is infinite and the group velocity is finite.

E. They are the same in direction and magnitude.

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A: 1
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C: 0
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A beam of electrons is accelerated through a potential difference of 25 kilovolts in an x-ray tube. The continuous $x$-ray spectrum emitted by the target of the tube will have a short wavelength limit of most nearly

A. $0.1 \AA$

B. $0.5 \AA$

C. $2 \AA$

D. $25 \AA$

E. $50 \AA$



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B: 1
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In the $R L C$ circuit shown above, the applied voltage is

$$
\varepsilon(t)=\varepsilon_{\mathrm{m}} \cos \omega t .
$$

For a constant $\varepsilon_{\mathrm{m}}$, at what angular frequency $\omega$ does the current have its maximum steady-state amplitude after the transients have died out?

A. $\frac{1}{R C}$

B. $\frac{2 L}{R}$

C. $\frac{1}{\sqrt{L C}}$

D. $\sqrt{\frac{1}{L C}-\left(\frac{R}{2 L}\right)^{2}}$

E. $\sqrt{\left(\frac{1}{R C}\right)^{2}-\left(\frac{L}{R}\right)^{2}}$



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A thin plate of mass $M$, length $L$, and width $2 d$ is mounted vertically on a frictionless axle along the $z$-axis, as shown above. Initially the object is at rest. It is then tapped with a hammer to provide a torque $\tau$, which produces an angular impulse $\mathbf{H}$ about the $z$-axis of magnitude $H=\int \tau d t$. What is the angular speed $\omega$ of the plate about the $z$-axis after the tap?

A. $\frac{H}{2 M d^{2}}$

B. $\frac{H}{M d^{2}}$

C. $\frac{2 H}{M d^{2}}$

D. $\frac{3 H}{M d^{2}}$

E. $\frac{4 H}{M d^{2}}$



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Two pith balls of equal mass $M$ and equal charge $q$ are suspended from the same point on long massless threads of length $L$ as shown in the figure above. If $k$ is the Coulomb's law constant, then for small values of $\theta$, the distance $d$ between the charged pith balls at equilibrium is

A. $\left(\frac{2 k q^{2} L}{M g}\right)^{\frac{1}{3}}$

B. $\left(\frac{k q^{2} L}{M g}\right)^{\frac{1}{3}}$

C. $\left(\frac{2 k q^{2} L}{M g}\right)^{\frac{1}{2}}$

D. $\left(\frac{k q^{2} L}{M g}\right)^{\frac{1}{2}}$

E. $\frac{L}{4}$


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An electron oscillates back and forth along the + and $-x$-axes, consequently emitting electromagnetic radiation. Which of the following statements concerning the radiation is NOT true?

A. The total rate of radiation of energy into all directions is proportional to the square of the electron's acceleration.

B. The total rate of radiation of energy into all directions is proportional to the square of the electron's charge.

C. Far from the electron, the rate at which radiated energy crosses a perpendicular unit area decreases as the inverse square of the distance from the electron.

D. Far from the electron, the rate at which radiated energy crosses a perpendicular unit area is a maximum when the unit area is located on the + or $-x$-axes.

E. Far from the electron, the radiated energy is carried equally by the transverse electric and the transverse magnetic fields.

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A free electron (rest mass $m_{e}=0.5 \mathrm{MeV} / \mathrm{c}^{2}$ ) has a total energy of $1.5 \mathrm{MeV}$. Its momentum $p$ in units of $\mathrm{MeV} / \mathrm{c}$ is about

A. 0.86

B. 1.0

C. 1.4

D. 1.5

E. 2.0



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The circuit shown above is used to measure the size of the capacitance $C$. The $y$-coordinate of the spot on the oscilloscope screen is proportional to the potential difference across $R$, and the $x$-coordinate of the spot is swept at a constant speed $s$. The switch is closed and then opened. One can then calculate $C$ from the shape and the size of the curve on the screen plus a knowledge of which of the following?

A. $V_{0}$ and $R$

B. $s$ and $R$

C. $s$ and $V_{0}$

D. $R$ and $R^{\prime}$

E. The sensitivity of the oscilloscope


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A particle of mass $M$ moves in a circular orbit of radius $r$ around a fixed point under the influence of an attractive force $F=\frac{K}{r^{3}}$, where $K$ is a constant. If the potential energy of the particle is zero at an infinite distance from the force center, the total energy of the particle in the circular orbit is

A. $-\frac{K}{r^{2}}$

B. $-\frac{K}{2 r^{2}}$

C. 0

D. $\frac{K}{2 r^{2}}$

E. $\frac{K}{r^{2}}$


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A parallel-plate capacitor is connected to a battery. $V_{0}$ is the potential difference between the plates, $Q_{0}$ the charge on the positive plate, $E_{0}$ the magnitude of the electric field, and $D_{0}$ the magnitude of the displacement vector. The original vacuum between the plates is filled with a dielectric and then the battery is disconnected. If the corresponding electrical parameters for the final state of the capacitor are denoted by a subscript $f$, which of the following is true?

A. $V_{f}>V_{0}$

B. $V_{f}<V_{0}$

C. $Q_{f}=Q_{0}$

D. $E_{f}>E_{0}$

E. $D_{f}>D_{0}$



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The energy levels for the one-dimensional harmonic oscillator are $h v\left(n+\frac{1}{2}\right), n=0,1,2 \ldots$ How will the energy levels for the potential shown in the graph above differ from those for the harmonic oscillator?

A. The term $\frac{1}{2}$ will be changed to $\frac{3}{2}$.

B. The energy of each level will be doubled.

C. The energy of each level will be halved.

D. Only those for even values of $n$ will be present.

E. Only those for odd values of $n$ will be present.


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The spacing of the rotational energy levels for the hydrogen molecule $\mathbf{H}_{2}$ is most nearly

A. $10^{-9} \mathrm{eV}$

B. $10^{-3} \mathrm{eV}$

C. $10 \mathrm{eV}$

D. $10 \mathrm{MeV}$

E. $100 \mathrm{MeV}$


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B: 1
C: 0
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The particle decay $\Lambda \rightarrow p+\pi^{-}$must be a weak interaction because

A. the $\pi^{-}$is a lepton

B. the $\Lambda$ has spin zero

C. no neutrino is produced in the decay

D. it does not conserve angular momentum

E. it does not conserve strangeness



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A flat coil of wire is rotated at a frequency of 10 hertz in the magnetic field produced by three pairs of magnets as shown above. The axis of rotation of the coil lies in the plane of the coil and is perpendicular to the field lines. What is the frequency of the alternating voltage in the coil?

A. $\frac{10}{6} \mathrm{~Hz}$

B. $\frac{10}{3} \mathrm{~Hz}$

C. $10 \mathrm{~Hz}$

D. $30 \mathrm{~Hz}$

E. $60 \mathrm{~Hz}$



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The figure above shows a small mass connected to a string, which is attached to a vertical post. If the mass is released when the string is horizontal as shown, the magnitude of the total acceleration of the mass as a function of the angle $\theta$ is

A. $g \sin \theta$

B. $2 g \cos \theta$

C. $2 g \sin \theta$

D. $g \sqrt{3 \cos ^{2} \theta+1}$

E. $g \sqrt{3 \sin ^{2} \theta+1}$

Answer scores (1 = correct):
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B: 0
C: 0
D: 0
E: 1
Which of the following is a Lorentz transformation? (Assume a system of units such that the velocity of light is 1 .)

A. $x^{\prime}=4 x$

$y^{\prime}=y$

$z^{\prime}=z$

$t^{\prime}=.25 t$

B. $x^{\prime}=x-.75 t$

$y^{\prime}=y$

$z^{\prime}=z$

$t^{\prime}=t$

C. $x^{\prime}=1.25 x-.75 t$

$y^{\prime}=y$

$z^{\prime}=z$

$t^{\prime}=1.25 t-.75 x$

D. $x^{\prime}=1.25 x-.75 t$

$y^{\prime}=y$

$z^{\prime}=z$

$t^{\prime}=.75 t-1.25 x$

E. None of the above


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B: 0
C: 1
D: 0
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A beam of $10^{12}$ protons per second is incident on a target containing $10^{20}$ nuclei per square centimeter. At an angle of 10 degrees, there are $10^{2}$ protons per second elastically scattered into a detector that subtends a solid angle of $10^{-4}$ steradians. What is the differential elastic scattering cross section, in units of square centimeters per steradian?

A. $10^{-24}$

B. $10^{-25}$

C. $10^{-26}$

D. $10^{-27}$

E. $10^{-28}$



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B: 0
C: 1
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E: 0