Photons carry discrete energy quanta E=hνE=h\nuE=hν.
In the photoelectric effect, increasing the intensity (at fixed frequency above cutoff) primarily increases
A Stopping potential
B Number of emitted electrons (photoelectron current)
C Kinetic energy of each electron
D Work function
More photons → more photoelectrons; KE depends on frequency.
Threshold frequency in photoelectric effect depends on
A Intensity of incident light
B Work function of the material
C Electron charge only
D Distance to the source
(Careful: threshold frequency ν0=ϕ/h \nu_0 = \phi/hν0=ϕ/h depends on work function φ; but the option list: A B C D — correct is B. Correction: Answer should be B.) Threshold frequency ν0=ϕ/h \nu_0=\phi/hν0=ϕ/h set by work function.
Einstein’s photoelectric equation is
A KE=hνKE = h\nuKE=hν
B KEmax=hν−ϕKE_{max} = h\nu – \phiKEmax=hν−ϕ
C ϕ=h/ν\phi = h/\nuϕ=h/ν
D KEmax=ϕ−hνKE_{max} = \phi – h\nuKEmax=ϕ−hν
The Compton effect demonstrates that X-rays scatter off electrons with
A No change in wavelength
B Shift to longer wavelength depending on scattering angle
C Shorter wavelength always
D Total absorption only
(Careful: correct is B.) Compton shift: Δλ=hmec(1−cosθ) \Delta\lambda = \frac{h}{m_ec}(1-\cos\theta)Δλ=mech(1−cosθ), wavelength increases.
Compton scattering treated X-ray as a particle colliding with an electron and uses
A Conservation of charge
B Conservation of mass only
C Conservation of energy and momentum
D Maxwell’s equations alone
(Correct is C.) Energy and momentum conservation give Compton formula.
The de Broglie wavelength of a particle is given by
A λ=h/p\lambda = h/pλ=h/p
B λ=p/h\lambda = p/hλ=p/h
C λ=hE\lambda = hEλ=hE
D λ=2πp\lambda = 2\pi pλ=2πp
Electrons accelerated through a potential V acquire de Broglie wavelength approximately
A λ=h/2meeV\lambda = h/\sqrt{2m_e eV}λ=h/2meeV
B λ=2meeV/h\lambda = \sqrt{2m_e eV}/hλ=2meeV/h
C λ=heV\lambda = h eVλ=heV
D λ=0\lambda = 0λ=0
(Careful arithmetic: correct is A.) λ=h/2meeV \lambda = h/\sqrt{2m_e eV}λ=h/2meeV (nonrelativistic).
Wave–particle duality implies that matter displays
A Only wave properties at macroscopic scales
B Both wave and particle behavior depending on experiment
C Particle behavior always
D No measurable quantum effects
(Correct is B.) Behavior depends on the measurement setup.
Heisenberg uncertainty principle relates uncertainties Δx\Delta xΔx and Δp\Delta pΔp by
A Δx Δp≥0\Delta x\,\Delta p \ge 0ΔxΔp≥0
B Δx Δp≤ℏ/2\Delta x\,\Delta p \le \hbar/2ΔxΔp≤ℏ/2
C Δx Δp≥ℏ/2\Delta x\,\Delta p \ge \hbar/2ΔxΔp≥ℏ/2
D Δx Δp=0\Delta x\,\Delta p = 0ΔxΔp=0
(Correct is C.) Fundamental lower bound: Δx Δp≥ℏ/2 \Delta x\,\Delta p \ge \hbar/2ΔxΔp≥ℏ/2.
A consequence of the uncertainty principle is that a particle localized very tightly in space has
A Very small momentum spread
B Very large momentum spread
C Definite momentum
D Infinite position uncertainty
(Correct is B.) Tight localization → large Δp\Delta pΔp.
The time-dependent Schrödinger equation (TDSE) for a particle is linear and first-order in
A Time derivative and second-order in spatial derivatives
B Second-order in time only
C Nonlinear in ψ
D Independent of potential
The stationary (time-independent) Schrödinger equation applies when
A The potential is time-dependent
B Looking for energy eigenstates in time-independent potential
C For relativistic particles only
D Only for photons
The wavefunction ψ(x,t)\psi(x,t)ψ(x,t) is normalized so that
A ∫∣ψ∣2 dx=0\int |\psi|^2 \,dx = 0∫∣ψ∣2dx=0
B ∫∣ψ∣2 dx=1\int |\psi|^2 \,dx = 1∫∣ψ∣2dx=1
C ψ=0\psi = 0ψ=0 everywhere
D ∫ψ dx=1\int \psi \,dx = 1∫ψdx=1
(Correct is B.) Probabilistic interpretation: total probability = 1.
For an infinite square well of width aaa, allowed energies vary as
A nnn
B n2n^2n2
C 1/n1/n1/n
D Continuous
(Correct is B.) En∝n2E_n \propto n^2En∝n2 (quantized levels).
Boundary conditions for wavefunctions at an infinite potential wall require
A ψ finite but derivative infinite
B ψ = 0 at the wall
C ψ derivative zero at the wall
D No restriction
The probability current density j\mathbf{j}j measures
A Rate of change of probability in space (flow of probability)
B Electric current only
C Magnetic field
D Energy density
Expectation value of an observable O^\hat{O}O^ in state ψ is given by
(Correct is B.) Hermitian operators have real eigenvalues (observables).
If two operators commute, their observables can be
A Measured simultaneously with definite values (common eigenstates)
B Never measured together
C Only measured approximately
D Non-physical
(Correct is A.)
The angular momentum operator L^2\hat{L}^2L^2 eigenvalues are ℏ2l(l+1)\hbar^2 l(l+1)ℏ2l(l+1). The quantum number l can be
A Any real number
B Integer ≥ 0
C Only 0 or 1
D Negative integers only
(Correct is B.)
The z-component LzL_zLz has eigenvalues mℏm\hbarmℏ where m takes values
A From −l to +l in integer steps
B Any real number
C Only ±l
D Fractional multiples of ℏ only
(Correct is A.)
Spin is an intrinsic form of angular momentum; electron spin quantum number s equals
A 0
B 1/2
C 1
D 3/2
(Correct is B.)
Magnetic moment associated with electron spin gives rise to
A No interaction with magnetic field
B Interaction μ⋅B\mu\cdot Bμ⋅B causing energy splitting
C Only electric effects
D Infinite energy
(Correct is B.)
Stern–Gerlach experiment demonstrated quantization of
A Charge
B Angular momentum (space quantization of magnetic moment)
C Mass
D Light intensity
(Correct is B.)
In a Stern–Gerlach apparatus with silver atoms, the beam splits into two because
A Silver atoms are neutral overall
B Silver has a single unpaired electron (spin-1/2) giving two spin states
C Magnetic field is uniform
D Atoms ionize
(Correct is B.)
Pauli exclusion principle states that no two electrons in an atom can have the same set of
A Energy only
B Four quantum numbers (n, l, m, s)
C Spin only
D Position only
(Correct is B.)
The Zeeman effect is the splitting of spectral lines in presence of
A Electric field only
B Magnetic field
C Gravitational field
D Temperature change
In weak-field (normal) Zeeman effect, spectral lines split into
A A single line only
B Three components (triplet) with spacing proportional to B
C Infinite components
D Two components only
(Correct is B.)
Spin–orbit coupling arises from interaction between electron spin and its motion around nucleus, effectively producing
A An electric field at electron rest frame
B A magnetic field in electron rest frame that couples with spin magnetic moment
C No interaction
D Nuclear spin flip only
Fine structure in atomic spectra arises due to combination of relativistic correction to kinetic energy, spin–orbit coupling, and
A Photoelectric effect
B Lamb shift only
C Darwin term and other small corrections (including relativistic)
D Classical thermal effects
(Correct is C.)
The spin quantum number s for an electron allows two possible msm_sms values:
A −1, 0, +1
B −1/2 and +1/2
C 0 only
D Any integer
The hydrogen atom energy levels from Schrödinger equation (non-relativistic) depend on principal quantum number n as
A En∝nE_n \propto nEn∝n
B En∝1/n2E_n \propto 1/n^2En∝1/n2
C En∝n2E_n \propto n^2En∝n2
D Continuous spectrum only
(Correct is B.)
Radial wavefunctions for hydrogen are characterized by nodes; number of radial nodes equals
A n
B n − l − 1
C l
D 0 always
The ladder (raising and lowering) operators L+,L−L_+ , L_-L+,L− act on spherical harmonics to change
A n quantum number
B m quantum number by ±1
C l by ±1
D Energy by large amounts
A normalized wavefunction squared gives
A Probability density of finding particle at a point
B Potential energy
C Momentum directly
D Temperature distribution
Quantum tunneling is a phenomenon where a particle
A Is reflected with probability 1 if barrier > energy
B Has finite probability to penetrate and cross a classically forbidden potential barrier
C Can never escape bound states
D Always loses energy crossing barrier
In alpha decay, tunneling through Coulomb barrier explains
A Why alpha particles are never emitted
B The finite lifetime and decay rates despite classically forbidden escape
C Conservation of momentum violation
D Creation of alpha particles from vacuum
Nodes in standing matter waves correspond to positions where probability density is
A Maximum
B Zero
C Negative
D Undefined
Time evolution of a stationary state ψn(x)e−iEnt/ℏ\psi_n(x)e^{-iE_nt/\hbar}ψn(x)e−iEnt/ℏ gives observables that are
A Time-dependent oscillating
B Time-independent expectation values
C Divergent with time
D Randomly varying
(Correct is B.)
The Born interpretation assigns ∣ψ∣2|\psi|^2∣ψ∣2 as
A Energy density
B Probability density of finding the particle
C Electric field intensity
D Momentum density
(Correct is B.)
Spin–orbit coupling causes splitting within otherwise degenerate levels because it depends on
A Electron charge only
B Relative orientation of spin and orbital angular momenta (j = l ± 1/2)
C Nuclear spin only
D External temperature
(Correct is B.)
The total angular momentum J=L+S\mathbf{J} = \mathbf{L} + \mathbf{S}J=L+S; possible j values for electron are
A l only
B l ± 1/2
C S only
D Any integer
(Correct is B.)
In LS coupling (Russell–Saunders), spin–orbit interactions are treated as
A Negligible always
B Weak compared to electrostatic interactions between electrons (good for light atoms)
C Dominant over all interactions
D Only for free electrons
Pauli exclusion principle explains the structure of periodic table because it prohibits two electrons in an atom from occupying
A The same orbital only if both are excited
B The same set of four quantum numbers simultaneously
C Same nucleus
D Different energy levels
Identical fermions (e.g., electrons) have a total wavefunction that is
A Symmetric under particle exchange
B Antisymmetric under exchange
C Neither symmetric nor antisymmetric
D Always real-valued
(Correct is B.)
Bosons differ from fermions in that boson many-particle wavefunction is
A Antisymmetric under exchange
B Symmetric under exchange allowing multiple occupancy of same state
C Non-normalizable
D Forbidden by Pauli principle
Fine structure constant α\alphaα is dimensionless and approximately equal to
A 1/137
B 137
C 0
D π
The Lamb shift (not predicted by Dirac theory) arises due to
A Strong nuclear forces
B Radiative corrections from quantum electrodynamics (vacuum fluctuations)
C Classical electron orbits
D Thermal motion
Measurement in quantum mechanics causes wavefunction collapse (per Copenhagen interpretation), which implies that after a precise position measurement, the particle’s momentum becomes