Quantum Physics

1. Quantum Physics • Waves and particles • The Quantum Sun • Schrödinger’s Cat and the Quantum Code

2. Waves are continuous have poorly defined position diffract and interfere Particles are discrete have well-defined position don’t (classically) diffract or interfere Waves and Particles

3. Light is a wave • Thomas Young (1773–1829) • light undergoes diffraction and interference (Young’s slits) • (also: theory of colour vision, compressibility of materials (Young’s modulus), near-decipherment of Egyptian hieroglyphs—clever chap…) • James Clerk Maxwell (1831–79) • light as an electromagnetic wave • (and colour photography, thermo-dynamics, Saturn’s rings—incredibly clever chap…)

4. Light is particles • Blackbody spectrum • light behaves as if it came in packets of energy hf (Max Planck) • Photoelectric effect • light does come in packets of energy hf (Einstein) • used to measure h by Millikan in 1916

5. Photoelectric effect • Light causes emission of electrons from metals • energy of electrons depends on frequency of light, KE = hf – w • rate of emission (current) depends on intensity of light • this is inexplicable if light is a continuous wave, but simple to understand if it is composed of particles (photons) of energy hf

6. Millikan’s measurement of h h = (6.57± 0.03) x 10-27 erg s (cf h = 6.6260755x 10-27)

7. Electrons are particles • JJ Thomson (1856–1940) • “cathode rays” have well-defined e/m (1897) • RA Millikan • measured e using oil drop experiment (1909)

8. Electrons are waves • GP Thomson (1892–1975) • electrons undergo diffraction • they behave as waves with wavelength h/p • JJ Thomson won the Nobel Prize for Physics in 1906 for demonstrating that the electron is a particle. • GP Thomson (son of JJ) won it in 1937 for demonstrating that the electron is a wave. • And they were both right!

9. hydrogen lines in A0 star spectrum Electrons as waves & light as particles • Atomic line spectra • accelerated electrons radiate light • but electron orbits are stable • only light with hf = DE can induce transition • Bohr atom • electron orbits asstanding waves

10. The Uncertainty Principle • Consider measuring position of a particle • hit it with photon of wavelength l • position determined to precision Dx ~ ±l/2 • but have transferred momentum Dp ~ h/l • therefore, DxDp ~ h/2 (and similar relation between DE and Dt) • Impossible, even in principle, to know position and momentum of particle exactly and simultaneously

11. Wavefunctions • Are particles “really” waves? • particle as “wave packet” • but mathematical functions describing particles as waves sometimes give complex numbers • and confined wave packet will disperse over time • Born interpretation of “matter waves” • Intensity (square of amplitude) of wave at (x,t) represents probability of finding particle there • wavefunction may be complex: probability given by Y*Y • tendency of wave packets to spread out over time represents evolution of our knowledge of the system

12. Postulates of Quantum Mechanics • The state of a quantum mechanical system is completely described by the wavefunction Y • wavefunction must be normalisable: ∫Y*Ydt = 1 (particle must be found somewhere!) • Observable quantities are represented by mathematical operators acting on Y • The mean value of an observable is equal to the expectation value of its corresponding operator

13. The Schrödinger equation • non-relativistic quantum mechanics • classical wave equation • de Broglie wavelength • non-relativistic energy • put them together!

14. Barrier penetration • Solution to Schrödinger’s equation is a plane wave if E > V • If E < V solution is a negative exponential • particle will penetrate into a potential barrier • classically thiswould not happen

15. The Quantum Sun • Sun is powered by hydrogen fusion • protons must overcome electrostatic repulsion • thermal energy at core of Sun does not look high enough • but wavefunction penetrates into barrier (nonzero probability of findingproton inside) • tunnelling • also explains adecay

16. The Pauli Exclusion Principle • Identical particles are genuinely indistinguishable • if particles a and b are interchanged, either Y(a,b) = Y(b,a) or Y(a,b) = –Y(b,a) • former described bosons (force particles, mesons)latter describes fermions (quarks, leptons, baryons) • negative sign implies that two particles cannot have exactly the same quantum numbers, as Y(a,a) must be zero • Pauli Exclusion Principle

17. The Quantum Sun, part 2 • When the Sun runs out of hydrogen and helium to fuse, it will collapse under its own gravity • Electrons are squeezed together until all available states are full • degenerate electron gas • degeneracy pressure halts collapse • white dwarf star

18. Entangled states • Suppose process can have two possible outcomes • which has happened? • don’t know until we look • wavefunction of state includes both possibilities (until we look) • e.g. p0 gg • spin 0  1+1, so g spins must be antiparallel • measuring spin of photon 1 automatically determines spin of photon 2 (even though they are separated by 2cDt)

19. Quantum cryptography • existence of entangled states has been experimentally demonstrated • setup of Weihs et al., 1998 • could send encryption key from A to Bwith no possibility of eavesdropping • interception destroys entangled state

20. Origin of quantum mechanics: energy of light waves comes in discrete lumps (photons) other quantised observables: electric charge, angular momentum Interpretation of quantum mechanics as a probabilistic view of physical processes explains observed phenomena such as tunnelling Possible applications include cryptography and computing so, not as esoteric as it may appear! Summary