Probability of an event that is certain to occur equals
A 0
B 0.5
C 1
D Undefined
Certain event has probability 1.
Two events A and B are mutually exclusive means
A P(A∩B)=P(A)P(B)P(A\cap B)=P(A)P(B)P(A∩B)=P(A)P(B)
B P(A∩B)=0P(A\cap B)=0P(A∩B)=0
C P(A∣B)=P(A)P(A|B)=P(A)P(A∣B)=P(A)
D P(A∪B)=0P(A\cup B)=0P(A∪B)=0
Mutually exclusive means they cannot both occur.
A microstate of a system is
A The macroscopic temperature only
B A detailed specification of every particle’s state consistent with constraints
C The total energy only
D Number of accessible states
Microstate = full microscopic configuration.
A macrostate is characterized by
A Exact positions of particles
B Macroscopic quantities (E, V, N) that many microstates share
C Only momentum information
D None of these
Macrostate = set of microstates with same macroscopic observables.
The multiplicity Ω of a macrostate is the
A Probability of that macrostate
B Number of microstates corresponding to it
C Temperature times volume
D Entropy squared
For distinguishable classical particles distributed among energy levels, the appropriate statistics is
A Fermi–Dirac
B Bose–Einstein
C Maxwell–Boltzmann
D None
Fermions obey the Pauli exclusion principle and follow which distribution at equilibrium?
A Maxwell–Boltzmann
B Bose–Einstein
C Fermi–Dirac
D Classical Gaussian
Bosons at low temperature can occupy the same quantum state and follow
A Fermi–Dirac statistics
B Bose–Einstein statistics
C Maxwell–Boltzmann statistics
D Poisson statistics
In classical limit (low occupancy), MB, BE and FD distributions all reduce to
A Distinct forms
B Maxwell–Boltzmann form (classical)
C Bose–Einstein form only
D Fermi–Dirac form only
For an ideal gas in equilibrium, the Maxwell–Boltzmann speed distribution gives the most probable speed vpv_pvp proportional to
A T\sqrt{T}T
B 1/T1/T1/T
C T2T^2T2
D Constant
The entropy S in statistical mechanics (Boltzmann) is given by
A S=kBlnΩS = k_B \ln \OmegaS=kBlnΩ
B S=Ω/kBS = \Omega/k_BS=Ω/kB
C S=ln(kBΩ)S = \ln(k_B \Omega)S=ln(kBΩ)
D S=Ω2S = \Omega^2S=Ω2
Entropy is an extensive property which means it
A Does not change with system size
B Scales with system size (additive for independent subsystems)
C Is independent of energy
D Is always zero
Second law of thermodynamics in statistical form implies that an isolated system tends to evolve towards
A Lower multiplicity
B Higher multiplicity / maximum entropy
C Minimum temperature
D Deterministic microstates only
Irreversibility arises in macroscopic processes because
A Microscopic laws are irreversible
B Probability overwhelmingly favors evolution to higher entropy macrostates
C Energy is not conserved
D Time-reversal symmetry is broken at microscopic level
Thermodynamic potential appropriate when (T,V,N) are independent variables is
A Helmholtz free energy F = U − TS
B Gibbs free energy G = H − TS
C Enthalpy H = U + PV
D Grand potential Ω = U − TS − μN
Gibbs free energy G is natural potential for processes at
A Constant T and P
B Constant T and V
C Constant S and V
D Constant μ and V
Maxwell relations are derived from equality of mixed partial derivatives of thermodynamic potentials and depend on which mathematical property?
A Non-differentiability
B Exact differentials (Schwarz theorem)
C Nonlinearity only
D Dimensionless units
One Maxwell relation derived from Helmholtz free energy F(T,V) is
A (∂S∂V)T=(∂P∂T)V\left(\frac{\partial S}{\partial V}\right)_T = \left(\frac{\partial P}{\partial T}\right)_V(∂V∂S)T=(∂T∂P)V
B (∂T∂V)S=(∂S∂P)T\left(\frac{\partial T}{\partial V}\right)_S = \left(\frac{\partial S}{\partial P}\right)_T(∂V∂T)S=(∂P∂S)T
C (∂U∂V)T=0\left(\frac{\partial U}{\partial V}\right)_T = 0(∂V∂U)T=0
D (∂P∂V)T=(∂S∂T)V\left(\frac{\partial P}{\partial V}\right)_T = \left(\frac{\partial S}{\partial T}\right)_V(∂V∂P)T=(∂T∂S)V
The grand canonical ensemble is most convenient when which variable can fluctuate?
A Volume V
B Particle number N (exchange with reservoir)
C Temperature T
D Entropy S
In canonical ensemble the probability of a microstate with energy E is proportional to
A e+E/kTe^{+E/kT}e+E/kT
B e−E/kTe^{-E/kT}e−E/kT
C E^2
D 1/E
Partition function Z encodes thermodynamic information; internal energy U can be obtained from Z by
A U=−∂lnZ∂βU = -\frac{\partial \ln Z}{\partial \beta}U=−∂β∂lnZ where β=1/kT\beta = 1/kTβ=1/kT
B U=lnZU = \ln ZU=lnZ
C U=Z2U = Z^2U=Z2
D U=kTZU = kT ZU=kTZ
For distinguishable particles without quantum degeneracy, the Gibbs correction factor 1/N!1/N!1/N! in partition function corrects for
A Quantum entanglement
B Overcounting of identical permutations of particles
C Energy non-conservation
D Relativistic effects
Classical ideal gas entropy given by Sackur–Tetrode equation depends on which of the following?
A Volume, energy (or temperature), and particle mass
B Only temperature
C Only pressure
D Only number of particles
The Michelson–Morley experiment attempted to detect Earth’s motion through the luminiferous aether by measuring
A Changes in light speed with direction (fringe shifts)
B Gravitational redshift
C Electron charge
D Thermal expansion
The null result of Michelson–Morley experiment provided evidence that
A Aether exists
B Speed of light is the same in all inertial frames (one of inputs to special relativity)
C Light speed depends strongly on medium only
D Newtonian absolute time is correct
Einstein’s two postulates of special relativity are: (i) laws of physics same in all inertial frames; (ii)
A Speed of light in vacuum is constant (c) in all inertial frames
B Time is absolute
C Mass conserved separately
D Energy depends on velocity linearly
Relativity abolishes the concept of absolute simultaneity: two events simultaneous in one inertial frame may be
A Simultaneous in all frames
B Non-simultaneous in other inertial frames (relativity of simultaneity)
C Impossible to occur
D Always causally disconnected
Lorentz transformation reduces to Galilean transformation when
A c → 0
B velocities ≪ c (v/c → 0)
C v → c
D massless particles only
Time dilation formula for moving clock measured by stationary observer is Δt=γΔt0\Delta t = \gamma \Delta t_0Δt=γΔt0 where γ = 1/√(1−v²/c²). Here Δt₀ is proper time. Which interval is longer?
A Proper time Δt₀
B Dilated time Δt (moving clock appears to run slower so Δt > Δt₀)
C Both equal
D Depends on frame orientation
Length contraction formula for object moving at speed v relative to observer is L=L0/γL = L_0/\gammaL=L0/γ. Which is true?
A Moving rod appears longer
B Moving rod appears contracted (shorter) along direction of motion
C Length independent of motion
D Length increases with speed
Relativistic velocity addition law ensures that
A Speeds simply add (u+v)
B Resultant speed never exceeds c and reduces to u+v at low speeds
C Speeds multiply
D c is no longer a limit
Proper time is defined as the time interval measured between two events by an observer for whom the events occur at
A Different locations
B Same location (clock at same place)
C Any two inertial frames simultaneously
D Remote frames only
Invariance of spacetime interval s2=c2Δt2−Δx2−Δy2−Δz2s^2 = c^2 \Delta t^2 – \Delta x^2 – \Delta y^2 – \Delta z^2s2=c2Δt2−Δx2−Δy2−Δz2 means that s² is the same in all inertial frames. For time-like separation s² is
A Positive
B Zero
C Negative
D Undefined
Proper length of rod L₀ is measured in rod’s rest frame. Observers moving relative to rod measure L = L₀/γ. Which measurement is largest?
A L (contracted)
B L₀ (proper length)
C Both equal
D Depends on acceleration only
Relativistic momentum is p=γmv\mathbf{p} = \gamma m \mathbf{v}p=γmv. At low speeds (v ≪ c) this reduces to
A p≈γ2mvp ≈ \gamma^2 m vp≈γ2mv
B p≈mvp ≈ mvp≈mv (classical)
C p≈mcp ≈ m cp≈mc
D p≈0p ≈ 0p≈0
Relativistic energy of a particle is E=γmc2E = \gamma mc^2E=γmc2. The kinetic energy is K=(γ−1)mc2K = (\gamma – 1)mc^2K=(γ−1)mc2. As v → 0, K →
A Infinity
B Zero (classical)
C mc^2
D −mc^2
Mass–energy equivalence is expressed by
A E=mvE = mvE=mv
B E=mc2E = mc^2E=mc2 (rest energy)
C E=12mv2E = \frac{1}{2}mv^2E=21mv2
D E=mghE = mghE=mgh
Photon has energy E=hνE = h\nuE=hν and momentum magnitude p=E/c=hν/c=h/λp = E/c = h\nu/c = h/\lambdap=E/c=hν/c=h/λ. Photon rest mass is
A Non-zero finite mass
B Zero (no rest mass)
C Infinite
D Negative
In relativistic collisions, both energy and momentum conservation must be applied using
A Classical formulas only
B Relativistic energy and momentum (four-vectors)
C Energy only
D Momentum only
Lorentz factor γ is always
A Less than 1
B Equal to 0
C ≥ 1 and increases without bound as v → c
D Negative for v>c
Twin paradox resolution rests on the fact that
A Both twins are inertial all the time
B The traveling twin undergoes acceleration and changes inertial frames, breaking symmetry
C Time travel occurs
D Exchange of clocks is impossible
Relativistic Doppler shift for light approaching the observer (source moving toward observer) leads to
A Redshift (longer wavelength)
B Blueshift (shorter wavelength)
C No change
D Frequency halved
Muons created high in Earth’s atmosphere reach surface because of
A Classical mechanics only
B Time dilation (their lifetime in lab frame is extended) and/or length contraction as seen in muon frame
C They are stable particles
D Magnetic fields guide them
Lorentz transformations mix which quantities between frames?
A Mass and charge
B Space and time coordinates (x and t)
C Energy and entropy
D Temperature and pressure
Simultaneity is relative: two spatially separated events that are simultaneous in one frame may not be simultaneous in another if they are
A Light-like separated only
B Space-like separated
C Always simultaneous in all frames
D Causally connected only
Proper distance between two simultaneous events in a given frame can be spacelike; for spacelike separations there exists a frame in which the time order of events can be
A Only forward
B Reversed (order can change)
C Undefined always
D Fixed absolutely
Rest energy mc² for a mass m is the energy measured in the frame where the particle’s velocity is
A c
B Zero (particle at rest)
C Infinite
D Arbitrary non-zero
Energy–momentum relation for a particle is E2=(pc)2+(mc2)2E^2 = (pc)^2 + (mc^2)^2E2=(pc)2+(mc2)2. For a massless particle (m=0) this reduces to
A E=0E = 0E=0
B E=pcE = pcE=pc
C E=mc2E = mc^2E=mc2 still
D p=0p = 0p=0
In special relativity simultaneity, events connected by light signals have interval s² = 0 and are called
A Time-like
B Space-like
C Light-like (null)
D Imaginary
The classical limit of relativistic formulas is recovered when v/c → 0, meaning relativistic corrections become