Waves, Light & Quanta Tim Freegarde Web Gallery of Art; National Gallery, London

2 Light and optics RAYS straight propagation paths least time (Fermat’s principle) reflection, refraction, lenses, telescopes, microscopes WAVES Huygens’ description of propagation, reflection, refraction polarization, colour (wavelength, frequency) diffraction, interference, beats, interferometers directrix focus Maxwell’s electromagnetism, Einstein’s relativity PHOTONS energy quantized in units of ( h = Planck’s constant) momentum quantized in units of angular momentum quantized in units of

3 Quantum theory PHOTONS energy quantized in units of ( h = Planck’s constant) momentum quantized in units of angular momentum quantized in units of blackbody radiation photoelectric effect Compton scattering PARTICLES frequency determined by energy de Broglie wavelength determined by momentum electron diffraction angular momentum quantized in units of atomic theory discrete energy levels for bound particlesatomic theory Stern-Gerlach

4 Michelson stellar interferometer

5 The experiment with the two holes x y fringe maxima when  fringe spacing smallest visible feature size   illumination wavelength  illumination momentum equivalent to change in illumination angle and hence by

6 Visibility and coherence x y total field  visibility average intensity maximize:

7 Coherence NON-POINT OBJECTS many independent emitters, e.g. different atoms in star spatial extent (angular width) determines fringe visibility partial visibility when phase difference partially defined coherence is a measure of extent to which the phase difference is known degree of coherence (approximate expression – requires complex numbers) complex degree of coherence

8 Single slit diffraction x amplitude intensity

9 Uncertainty HEISENBERG’S UNCERTAINTY PRINCIPLE certain pairs of parameters may not simultaneously be exactly determined {position, momentum} {time, energy} {orientation, angular momentum} {intensity, phase} {x, y}, {x, z}, {y, z} components of angular momentum {position, wavelength} {time, frequency} conjugate parameters cannot be simultaneously definite QUANTUM MEASUREMENT measurement changes observed system so that parameter measured is subsequently definite process measure A, measure B not the same as measure B, measure A measure A, measure B are not commutative / do not commute commutator [measure A, measure B]  0

10 Uncertainty BEATING OF TWO DIFFERENT FREQUENCIES

11 Bandwidth theorem

12 Bandwidth theorem

13 Bandwidth theorem

14 Terminology UNCERTAINTY IN MEASUREMENT repeated experiment yields range of results expectation value = mean uncertainty = standard deviation before measurement, system was in a superposition probability of given result given by

15 Wave-particle duality + WHAT SORT OF WAVE? transverse/longitudinal motion? + transverse density density? QUANTUM WAVEFUNCTION amplitude 2 describes probability phase has no classical analogue amplitude and phase combined to form complex number ? phase matters! rate of phase variation defines frequency and wavelength

16 Diffracting molecules S Gerlich et al, Nature Physics (2007) MOLECULE DIFFRACTION molecules behave like waves molecule wavelength molecular wavefunction

17 Ramsauer-Townsend effect S G Kukolich, Am. J. Phys (1968) A anomalous dip in scattering probability at low energy Ar proves to be interference from front and rear ‘reflections’ from Ar atom

18 Particle interference MOLECULE DIFFRACTION and RAMSAUER-TOWNSEND give particle two or more routes through experiment interference depends upon relative phases of contributions phase depends upon path difference and wavelength STATIONARY PARTICLES give particle two or more routes through experiment interference depends upon relative phases of contributions phase depends upon frequency difference and duration

19 Atomic clock energy 0 Cs atom electron density depends upon relative phase of superposition components  = GHz

20 Atomic clock x/a 0 electron density depends upon relative phase of superposition components atomic wavefunction

21 Quantum measurement allowed energies energy 0 n = 1 n = 3 n =  n = 2 1.measured energy must be one of allowed values 2.…but until measurement, any energy possible 3.after measurement, subsequent measurements will give same value THE HYDROGEN ATOM QUANTUM MEASUREMENT

22 Quantum mechanics 1.particles behave like waves, and vice-versa 2.energies and momenta can be quantized, ie measurements yield particular results 3.all information about a particle is contained within a complex wavefunction, which determines the probabilities of experimental outcomes 4.80 years of experiments have found no inconsistency with quantum theory 5.explanation of the ‘quantum measurement problem’ – the collapse of the wavefunction upon measurement – remains an unsolved problem