Capstone: modern physics in technology

The ideas of this course began as answers to puzzling experiments, but within a century they had become the working principles of everyday technology. A phone in your pocket contains billions of transistors that operate by the quantum mechanics of electrons in crystals, a GPS receiver that applies both special and general relativity, a camera sensor that works by the photoelectric effect, and a screen whose LEDs or quantum dots emit photons of energies set by quantum confinement. Hospitals image the body with positron annihilation, nuclear magnetic resonance and characteristic X-rays. This capstone lesson works through six technologies, each drawing on several earlier lessons, and ends with the quantum computers that the 2025 Nobel Prize helped make possible. For each one, ask which ideas from the course it uses, and which numbers you could now calculate yourself.

GPS: relativity every day

The Global Positioning System has 31 or so active satellites in orbits about 20,200 km above the Earth, each carrying atomic clocks and broadcasting the time and its position. A receiver measures the arrival times of signals from at least four satellites and solves for its own position and clock error. Since light travels 30 cm in a nanosecond, positions accurate to a few metres need timing accurate to about 10 ns. Two relativistic effects are much larger than that.

First, special relativity. The satellites move at about 3.87 km/s, so their clocks run slow by \(\tfrac12 \beta^2 = 8.3 \times 10^{-11}\), lesson 2, about 7.2 microseconds per day. Second, general relativity, which predicts that clocks run faster higher in a gravitational field. The fractional rate difference between a clock at radius \(r\) and one on the ground at \(R_E\) is \(\Delta f/f = GM(1/R_E - 1/r)/c^2\), about \(5.3 \times 10^{-10}\), or 45.7 microseconds per day. The net effect is a gain of about 38 microseconds per day. Uncorrected, it would produce position errors growing at about 12 km per day. The system handles it by setting the satellite clocks' frequency slightly low before launch, 10.229999995 MHz instead of 10.23 MHz, so that they tick at the right rate in orbit, and receivers apply a further small correction for each satellite's slightly elliptical orbit.

import math
GM = 3.986004418e14       # Earth's gravitational parameter, m^3/s^2
c = 299792458.0
R_E = 6.371e6             # mean Earth radius, m
r = 26.5616e6             # GPS orbit radius, m (altitude 20,190 km)
day = 86400
v = math.sqrt(GM / r)
sr = -0.5 * (v / c)**2                            # special relativity: moving clock slow
gr = GM / c**2 * (1 / R_E - 1 / r)                # general relativity: higher clock fast
# the ground clock also moves with the Earth's rotation (about 465 m/s at the equator)
v_ground = 2 * math.pi * R_E / 86164
sr_ground = -0.5 * (v_ground / c)**2
net = sr + gr - sr_ground
print(f"orbital speed {v / 1000:.3f} km/s, period {2 * math.pi * r / v / 3600:.2f} h")
print(f"special relativity: {sr * day * 1e6:+.2f} microseconds per day")
print(f"general relativity: {gr * day * 1e6:+.2f} microseconds per day")
print(f"ground clock's own rotation correction: {-sr_ground * day * 1e6:+.3f} microseconds per day")
print(f"net: {net * day * 1e6:+.2f} microseconds per day = fractional {net:.3e}")
print(f"factory frequency: {10.23e6 * (1 - net):.6f} Hz instead of 10,230,000 Hz")
print(f"uncorrected position error after one day: about {net * day * c / 1000:.1f} km")

orbital speed 3.874 km/s, period 11.97 h
special relativity: -7.21 microseconds per day
general relativity: +45.72 microseconds per day
ground clock's own rotation correction: +0.104 microseconds per day
net: +38.61 microseconds per day = fractional 4.469e-10
factory frequency: 10229999.995429 Hz instead of 10,230,000 Hz
uncorrected position error after one day: about 11.6 km

The numbers match the system's design: a net gain of about 38 microseconds per day, and a factory offset of 4.5 parts in \(10^{10}\). The same physics limits the best clocks on Earth. Optical atomic clocks at NIST in Boulder and elsewhere are now so precise that raising one by a few centimetres changes its rate measurably, so clocks can be used to measure height differences, a technique called relativistic geodesy.

Atomic clocks and the second

Since 1967 the second has been defined by an atomic transition: it is the duration of 9,192,631,770 periods of the radiation from the hyperfine transition of the cesium-133 ground state, a cousin of hydrogen's 21 cm line, lesson 9. In a cesium fountain clock, atoms are cooled by lasers to microkelvin temperatures, tossed upward through a microwave cavity, and probed on the way up and down, giving a long interaction time and a sharp resonance. The best cesium clocks are accurate to about one second in 300 million years. Optical clocks based on ions or neutral atoms in optical lattices tick at frequencies near \(10^{15}\) Hz, about a hundred thousand times faster, and reach about one second in the age of the universe. A redefinition of the second based on optical clocks is being prepared.

Semiconductors, LEDs and solar cells

In a single atom, electrons occupy discrete levels, lesson 10. In a crystal, the levels of trillions of neighbouring atoms overlap and spread into bands of allowed energies separated by gaps. In a metal the highest occupied band is partly full and electrons move freely. In an insulator it is full and the gap to the next band is large. A semiconductor such as silicon has a modest gap, 1.12 eV, so at room temperature a few electrons are thermally excited across it, and the conductivity can be controlled by adding small amounts of impurities, doping. Phosphorus atoms add electrons (n-type) and boron atoms add holes, missing electrons that behave as positive carriers (p-type). A junction between p-type and n-type material is a diode, and combinations of junctions make transistors, the switches of every computer. PHYS-131 introduced these devices; here the point is that every one of them works by the quantum physics of bands.

In a light-emitting diode, electrons and holes meet at the junction and recombine, each emitting a photon with energy close to the band gap. The colour is set by the material: gallium arsenide (1.42 eV) emits in the near infrared, used in remote controls; aluminium gallium indium phosphide gives red and amber; and indium gallium nitride gives blue. Blue LEDs were the hardest to make, and Isamu Akasaki, Hiroshi Amano and Shuji Nakamura shared the 2014 Nobel Prize for them, because a blue LED coated with a yellow phosphor gives white light. A solar cell runs the process backwards: a photon with energy above the gap creates an electron–hole pair, the junction's built-in field separates them, and a current flows. Photons below the gap pass through unabsorbed, and photons above it lose their excess energy as heat, which limits a single-junction silicon cell to about 33% efficiency, the Shockley–Queisser limit of 1961.

import math
hc = 1239.84198                              # eV nm
materials = [("silicon", 1.12), ("gallium arsenide", 1.42), ("AlGaInP (red)", 1.95),
             ("InGaN (blue)", 2.70), ("gallium nitride", 3.40)]
for name, Eg in materials:
    print(f"{name:18s} gap {Eg:4.2f} eV -> photon wavelength {hc / Eg:6.0f} nm")
# fraction of sunlight (6000 K blackbody) photons above each gap, and the ideal 'ultimate' efficiency
def planck_photons(x): return x * x / math.expm1(x)          # photon number per unit x = E/kT
def planck_energy(x): return x**3 / math.expm1(x)
kT = 8.617333e-5 * 5772
def integrate(f, a, b, n=4000):
    h = (b - a) / n
    return h * (f(a) / 2 + sum(f(a + i * h) for i in range(1, n)) + f(b) / 2)
total_E = integrate(planck_energy, 1e-6, 40)
for name, Eg in materials[:2]:
    xg = Eg / kT
    n_above = integrate(planck_photons, xg, 40)
    eff = xg * n_above / total_E            # each absorbed photon delivers only Eg
    print(f"{name:18s}: ideal fraction of solar energy delivered at the gap = {eff:.3f}")

silicon            gap 1.12 eV -> photon wavelength   1107 nm
gallium arsenide   gap 1.42 eV -> photon wavelength    873 nm
AlGaInP (red)      gap 1.95 eV -> photon wavelength    636 nm
InGaN (blue)       gap 2.70 eV -> photon wavelength    459 nm
gallium nitride    gap 3.40 eV -> photon wavelength    365 nm
silicon           : ideal fraction of solar energy delivered at the gap = 0.438
gallium arsenide  : ideal fraction of solar energy delivered at the gap = 0.410

The last two lines compute the ultimate efficiency, which assumes every photon above the gap delivers exactly the gap energy. It is about 44% for silicon, before the further losses from emission and recombination that bring the Shockley–Queisser limit down to about 33%. Commercial silicon panels convert 20 to 24% of sunlight; multi-junction cells that stack several gaps reach about 47% in the laboratory and power spacecraft.

PET: antimatter in the hospital

Positron emission tomography, lesson 6, uses annihilation photons to image where a tracer collects in the body. The commonest tracer is fluorodeoxyglucose, a sugar labelled with fluorine-18, half-life 110 minutes, which accumulates where cells use glucose rapidly, as in many tumours and in active regions of the brain. Each positron travels about a millimetre in tissue before annihilating with an electron into two 511 keV photons, which leave in almost exactly opposite directions. A ring of scintillation detectors records pairs arriving within a few nanoseconds of each other, and each pair defines a line of response. Modern time-of-flight scanners also measure the difference in arrival times; a difference of 200 picoseconds locates the annihilation along the line to within 3 cm, sharpening the image and reducing the dose needed.

import math
c = 299792458.0
for dt_ps in (500, 200, 100, 10):
    dx = c * dt_ps * 1e-12 / 2
    print(f"timing resolution {dt_ps:4d} ps -> position along the line {dx * 100:5.1f} cm")
# tracer decay between production and scan: fluorine-18 half-life 109.8 minutes
t_half = 109.8
for delay in (30, 60, 120, 240):
    print(f"after {delay:3d} min, {0.5**(delay / t_half) * 100:5.1f}% of the F-18 remains")
# photon energy check: E = m_e c^2 for each annihilation photon
me, eV = 9.1093837e-31, 1.602176634e-19
print(f"annihilation photon energy: {me * c**2 / eV / 1000:.1f} keV")

timing resolution  500 ps -> position along the line   7.5 cm
timing resolution  200 ps -> position along the line   3.0 cm
timing resolution  100 ps -> position along the line   1.5 cm
timing resolution   10 ps -> position along the line   0.1 cm
after  30 min,  82.7% of the F-18 remains
after  60 min,  68.5% of the F-18 remains
after 120 min,  46.9% of the F-18 remains
after 240 min,  22.0% of the F-18 remains
annihilation photon energy: 511.0 keV

The short half-life is both a blessing and a constraint: it keeps the patient's dose low, but tracers must be made in a cyclotron near the scanner and used within a few hours. TRIUMF in Vancouver and many hospitals in Canada and the United States run medical cyclotrons for this purpose.

MRI: nuclear spin in a magnetic field

Protons have spin one-half and a magnetic moment, lesson 10. In a magnetic field \(B\), the two spin states of a proton have energies differing by \(\Delta E = h f\), where the Larmor frequency \(f = \gamma_p B\) with \(\gamma_p = 42.58\ \mathrm{MHz/T}\). In the 1.5 T or 3 T field of a clinical MRI scanner, the frequency is 64 or 128 MHz, in the radio band, and the photon energy is only about \(5 \times 10^{-7}\) eV, far below \(kT\) at body temperature. So the population difference between the two states is tiny, about 10 in a million at 3 T, but a human body contains about \(10^{27}\) hydrogen nuclei, mostly in water and fat, and the small net magnetization is easily detected. A radio-frequency pulse tips the magnetization; as it precesses back it induces a signal in a coil. Magnetic field gradients make the frequency depend on position, so the signal can be decoded into an image, and the different relaxation times of tissues give contrast between them. Paul Lauterbur and Peter Mansfield received the 2003 Nobel Prize in Physiology or Medicine for the imaging methods. The strong fields come from superconducting magnets cooled by liquid helium, themselves a macroscopic quantum effect.

import math
h, k = 6.62607015e-34, 1.380649e-23
gamma_p = 42.577478e6          # Hz per tesla
T_body = 310.0
for B in (0.5, 1.5, 3.0, 7.0):
    f = gamma_p * B
    dE = h * f
    excess = math.tanh(dE / (2 * k * T_body))          # (N_lower - N_upper) / N
    print(f"B = {B:3.1f} T: Larmor {f / 1e6:7.2f} MHz, photon {dE / 1.602176634e-19:.2e} eV,"
          f" spin excess {excess * 1e6:5.2f} per million")
N_H = 6.6e27                  # hydrogen nuclei in a 70 kg person, roughly
print(f"net aligned protons at 3 T in a whole body: about {N_H * math.tanh(h * gamma_p * 3 / (2 * k * T_body)):.1e}")

B = 0.5 T: Larmor   21.29 MHz, photon 8.80e-08 eV, spin excess  1.65 per million
B = 1.5 T: Larmor   63.87 MHz, photon 2.64e-07 eV, spin excess  4.94 per million
B = 3.0 T: Larmor  127.73 MHz, photon 5.28e-07 eV, spin excess  9.89 per million
B = 7.0 T: Larmor  298.04 MHz, photon 1.23e-06 eV, spin excess 23.07 per million
net aligned protons at 3 T in a whole body: about 6.5e+22

Quantum computing

A classical bit is 0 or 1. A quantum bit, a qubit, can be in a superposition of both, and a register of \(n\) qubits can be in a superposition of all \(2^n\) bit strings at once. Quantum algorithms use interference among these possibilities, lesson 7, so that wrong answers cancel and right ones reinforce. Peter Shor showed in 1994 that a large enough quantum computer could factor large numbers efficiently, which would break widely used encryption, and quantum simulation of molecules and materials is a likely early application. In 2024 NIST published its first post-quantum cryptography standards so that encryption can be updated before such machines exist.

The leading hardware platform is the superconducting qubit, a tiny circuit of superconducting metal interrupted by Josephson junctions, thin insulating barriers that Cooper pairs of electrons tunnel through, lesson 8. Such a circuit has quantized energy levels, like an artificial atom, and its two lowest levels form a qubit with a transition frequency of about 5 GHz. The 2025 Nobel Prize in Physics, to John Clarke, Michel Devoret and John Martinis, recognized the 1984–85 Berkeley experiments that first showed a macroscopic electrical circuit tunnelling quantum mechanically and absorbing energy in discrete quanta. Martinis later led the Google team whose 53-qubit processor performed a sampling task in 2019 that would take a classical supercomputer far longer. The qubits must be kept at about 15 millikelvin in dilution refrigerators, because the qubit's energy quantum must be large compared with \(kT\) for the excited state not to be populated thermally, Planck's argument from lesson 5 in a new setting. Canadian companies have been active from the start: D-Wave, founded in 1999 near Vancouver, built the first commercial quantum annealers, and Xanadu in Toronto builds photonic quantum computers.

import math
h, k = 6.62607015e-34, 1.380649e-23
f_q = 5.0e9                         # qubit frequency, Hz
for T in (4.2, 0.1, 0.015):
    x = h * f_q / (k * T)
    p_excited = 1 / (1 + math.exp(x))      # two-level thermal population
    print(f"T = {T * 1000:7.1f} mK: hf/kT = {x:6.2f}, thermal excited-state probability {p_excited:.2e}")
print(f"qubit photon energy: {h * f_q / 1.602176634e-19 * 1e6:.1f} micro eV; wavelength {2.998e8 / f_q * 100:.1f} cm")
for n in (10, 53, 100, 300):
    print(f"{n:3d} qubits describe {2**n:.3e} amplitudes")

T =  4200.0 mK: hf/kT =   0.06, thermal excited-state probability 4.86e-01
T =   100.0 mK: hf/kT =   2.40, thermal excited-state probability 8.32e-02
T =    15.0 mK: hf/kT =  16.00, thermal excited-state probability 1.13e-07
qubit photon energy: 20.7 micro eV; wavelength 6.0 cm
 10 qubits describe 1.024e+03 amplitudes
 53 qubits describe 9.007e+15 amplitudes
100 qubits describe 1.268e+30 amplitudes
300 qubits describe 2.037e+90 amplitudes

At liquid helium temperature, 4.2 K, a 5 GHz qubit would be thermally excited about 49% of the time, useless for computing. At 15 mK the probability falls below one in a million. The exponential growth of the number of amplitudes is the source of the potential power: 300 qubits describe more amplitudes than there are atoms in the observable universe, though extracting useful answers requires carefully designed algorithms and error correction, which remains the central engineering challenge.

Looking back over the course

Every technology in this lesson draws on the same small set of ideas. The constancy of the speed of light gives time dilation and \(E = mc^2\), which appear in GPS, PET and nuclear power. Quantization of energy, \(E = hf\), appears in blackbody radiation, LEDs, solar cells, MRI and qubits. Matter waves and the Schrödinger equation give atoms, bands, tunnelling and the periodic table. And the nucleus, with its binding energy curve and exponential decay, gives power stations, medical isotopes and the light of the stars. Classical physics remains the right tool for bridges and orbits; modern physics is the right tool whenever speeds approach \(c\) or actions approach \(h\). Later courses in the physics program, Quantum Mechanics I and II, Statistical Mechanics, Nuclear and Particle Physics, Solid State Physics, Astrophysics and General Relativity, develop each of these threads in full.