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Light and Matter Wave mechanics Tim Freegarde School of Physics & Astronomy University of Southampton Wave mechanics • response to action of neighbour • delayed reaction f x, t f x x, t t e.g. f x, t f vt x • waves are bulk motions, in which the displacement is a delayed response to the neighbouring displacements 2 Gravitational waves • delay may be due to propagation speed of force (retarded potentials) m1 a m1m2 F G 2 r m2 • vertical component of force m1G F t m2 3 at r c r 3 Wave mechanics • waves result when the motion at a given position is a delayed response to the motion at neighbouring points • derivatives with respect to time and position are related by the physics of the system, which lets us write differential equations e.g. 2 2 y y 2 2 2 v y v p p 2 2 t x • in certain circumstances, a wave may propagate without distortion: f x, t f v pt x f x, t f v pt r f r, t • a surface of constant phase, with the phase velocity, v p x, t , is known as a wavefront, and propagates • the solutions depend upon whether the system shows linearity or dispersion 4 Linearity and superpositions • if the system is linear, then the wave equation may be split into separate equations for superposed components; i.e., if y1 and y2 are wave solutions, then so is any superposition of them • if sinusoidal solutions are allowed, then the wave shape at any time may be written as a superposition of sinusoidal components yx ak coskx k Fourier analysis k • complex coefficients allow waves which are complex exponentials: cos kx i sin kx exp ikx 5 Dispersion • linear systems may show dispersion – that is, the wave speed varies with frequency • if sinusoidal solutions are allowed, then the wave shape may still be written as a superposition of sinusoidal components • dispersion causes the components to drift in phase as the wave propagates • the wave may no longer be written as f x, t f v pt x 6 Dispersion • 2 10sinusoidal sinusoidalcomponents: components: • spreading of wavepacket vg 2v p • this illustration corresponds to the wavepacket evolution of a quantum mechanical particle, described by the Schrödinger equation 7 Plane wave solutions to wave equations • linear, non-dispersive exp it kx sin t kx f vt x • linear, dispersive exp it kx sin t kx • non-linear solitons: f vt x 8 Alternative solutions • show that spherical waves of the form exp it kr r , t r are valid solutions to the Schrödinger equation of a free particle i 2 t 9 Wave mechanical operators • an operator is a recipe for determining an observable from a wave function e.g. an operator Ô could yield the parameter o from the wave ykx t o Oˆ ykx t • for convenience, to avoid the observable depending upon the magnitude of the wavefunction, we instead define the general operator i.e. Oˆ ykx t o ykx t o ykx t Oˆ ykx t • the square brackets are commonly omitted 10 Messenger Lecture • Richard P. Feynman (1918-1988) Nobel prize 1965 • Messenger series of lectures, Cornell University, 1964 • Lecture 6: ‘Probability and Uncertainty – the quantum mechanical view of nature’ • see the later series of Douglas Robb memorial lectures (1979) online at http://www.vega.org.uk/series/lectures/feynman/ 11