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

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

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…)

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

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

Millikan’s measurement of h h = (6.57± 0.03) x erg s (cf h = x )

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

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!

Electrons as waves & light as particles Atomic line spectra –accelerated electrons radiate light –but electron orbits are stable –only light with hf =  E can induce transition Bohr atom –electron orbits as standing waves hydrogen lines in A0 star spectrum

The Uncertainty Principle Consider measuring position of a particle –hit it with photon of wavelength –position determined to precision  x ~ ± /2 –but have transferred momentum  p ~ h/ –therefore,  x  p ~ h/2 (and similar relation between  E and  t) Impossible, even in principle, to know position and momentum of particle exactly and simultaneously

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  *  tendency of wave packets to spread out over time represents evolution of our knowledge of the system

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

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

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 this would not happen

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 finding proton inside) –tunnelling –also explains  decay

The Pauli Exclusion Principle Identical particles are genuinely indistinguishable –if particles a and b are interchanged, either  (a,b) =  (b,a) or  (a,b) = –  (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  (a,a) must be zero –Pauli Exclusion Principle

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

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.     spin 0  1+1, so  spins must be antiparallel measuring spin of photon 1 automatically determines spin of photon 2 (even though they are separated by 2c  t)

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

Summary 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!