Quantum Mechanics 101 · Unit 4: Atoms, Spin & Identity

Quantum Academy · Chapter 19: Bosons, Fermions & Quantum Matter

Lasers, superconductors and Bose–Einstein condensates

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Instructions. Work through these questions with the lesson closed first, then check yourself against it. For calculations, show every step, keep three significant figures, and use h = 6.626×10⁻³⁴ J·s, ħ = 1.055×10⁻³⁴ J·s, c = 2.998×10⁸ m/s, k_B = 1.381×10⁻²³ J/K, e = 1.602×10⁻¹⁹ C and 1 u = 1.661×10⁻²⁷ kg. One point for each vocabulary match and each question; the challenge is a bonus.

Part A Vocabulary matching

Write the letter of the matching definition in the blank beside each term.

1.Indistinguishable particles
2.Exchange symmetry
3.Fermion
4.Boson
5.Spin-statistics theorem
6.Occupation number
7.Fermi energy
8.Bose–Einstein condensate
9.Stimulated emission
10.Cooper pair
11.Meissner effect
12.Superfluid

A.The result of relativistic quantum field theory (Fierz 1939, Pauli 1940) that half-integer spin means fermion and integer spin means boson.
B.A particle with integer spin whose many-particle wavefunction is symmetric; any number may share one quantum state.
C.A bound pair of electrons in a superconductor, held together by a lattice-mediated attraction and carrying integer total spin.
D.The energy of the highest filled state in a fermion system at absolute zero; about 7 eV for the conduction electrons of copper.
E.A particle with half-integer spin whose many-particle wavefunction is antisymmetric; it obeys the Pauli exclusion principle.
F.Emission of a photon triggered by an incoming photon, producing an exact copy in the same mode; the basis of the laser.
G.How a many-particle wavefunction responds when two identical particles are swapped: unchanged (symmetric) or sign-flipped (antisymmetric).
H.Particles of the same species that no measurement, even in principle, can tell apart; swapping them changes no observable.
I.The average number of particles found in one single-particle quantum state of a given energy at a given temperature.
J.A liquid that flows with no measurable viscosity because its atoms move as one coherent quantum state, as helium-4 does below 2.17 K.
K.A state of matter in which a macroscopic fraction of bosons occupies the single lowest-energy quantum state.
L.The active expulsion of magnetic field from the interior of a material as it becomes superconducting.

Part B Questions

1.
Which single property determines whether a particle is a boson or a fermion?Circle one
1. AIts mass
2. BIts spin
3. CIts electric charge
4. DIts size
2.
With a sufficiently precise camera, you could follow two electrons through a collision and say which electron came out where.Circle one
TrueFalse
3.
A helium-3 atom contains two protons, one neutron and two electrons. It is therefore:Circle one
1. Aa boson, because every neutral atom is a boson
2. Ba boson, because it has an even number of protons
3. Ca fermion, because it contains an odd number (five) of fermions
4. Dneither, because composite particles have no exchange symmetry
4.
At absolute zero, the conduction electrons of a metal fill every state up to a sharp energy called the energy.Fill in the blank
5.
Bose–Einstein condensation was predicted in 1924–25 but not achieved in a dilute atomic gas until 1995.Circle one
TrueFalse
6.
Which process is responsible for the coherence of laser light?Circle one
1. AThermal radiation following Planck's law
2. BAbsorption of photons by the cavity mirrors
3. CSpontaneous emission at random times and directions
4. DStimulated emission, which copies the incoming photon's mode
7.
In a conventional superconductor, electrons near the Fermi energy bind into pairs.Fill in the blank
8.
A 1 mW laser pointer emits more total light power than a 60 W incandescent bulb.Circle one
TrueFalse
9.
Using the two-particle wavefunction, explain why two identical fermions cannot occupy the same single-particle state.Short answer
10.
Why is the Meissner effect evidence that a superconductor is more than a perfect conductor?Short answer
11.
In a metal at 300 K, what is the Fermi–Dirac occupation of a state 0.10 eV above the Fermi energy (take μ = E_F)?Show your working
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Answer:
12.
A 5.0 mW red laser pointer emits at 650 nm. How many photons does it emit each second?Show your working
Working
Answer:
13.
Magnetic flux through a superconducting ring is quantised in units of h/2e. Calculate this flux quantum, and say what value would have been found if the current carriers were single electrons.Show your working
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Answer:
14.
Ketterle's group cooled sodium-23 atoms (mass 22.99 u) to the microkelvin range. Find the thermal de Broglie wavelength at 2.0 microkelvin and the density at which an ideal gas would condense.Show your working
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Answer:

Part C Mad scientist challenge bonus

Repeat Fritz London's 1938 estimate. Liquid helium-4 has a density of 145 kg per cubic metre and atomic mass 4.0026 u. (a) Treating it as an ideal Bose gas, compute the condensation temperature from the condition n λ³ = 2.612, which rearranges to T_c = (2πħ² ÷ m k_B) × (n ÷ 2.612) to the two-thirds power. (b) Show that at that temperature the thermal wavelength exceeds the average spacing between atoms. (c) Compare with the measured lambda point of 2.17 K, and explain why liquid helium-3, whose atoms are nearly identical in size and chemistry, shows no such transition near 2 K.

Working