Matter Waves and Uncertainty Principle UNIT 1 Matter Waves and Uncertainty Principle

WAVE-PARTICLE DUALITY OF LIGHT In 1924 Einstein wrote:- “ There are therefore now two theories of light, both indispensable, and … without any logical connection.” Evidence for wave-nature of light Diffraction and interference Evidence for particle-nature of light Photoelectric effect Compton effect Light exhibits diffraction and interference phenomena that are only explicable in terms of wave properties Light is always detected as packets (photons); if we look, we never observe half a photon

MATTER WAVES de Broglie relation We have seen that light comes in discrete units (photons) with particle properties (energy and momentum) that are related to the wave-like properties of frequency and wavelength. In 1923 Prince Louis de Broglie postulated that ordinary matter can have wave-like properties, with the wavelength λ related to momentum p in the same way as for light de Broglie relation Planck’s constant de Broglie wavelength NB wavelength depends on momentum, not on the physical size of the particle Prediction: We should see diffraction and interference of matter waves

Estimate some de Broglie wavelengths Wavelength of electron with 50eV kinetic energy Wavelength of Nitrogen molecule at room temperature Wavelength of Rubidium(87) atom at 50nK

ELECTRON DIFFRACTION The Davisson-Germer experiment (1927) The Davisson-Germer experiment: scattering a beam of electrons from a Ni crystal. Davisson got the 1937 Nobel prize. Davisson G.P. Thomson θi θi At fixed angle, find sharp peaks in intensity as a function of electron energy At fixed accelerating voltage (fixed electron energy) find a pattern of sharp reflected beams from the crystal G.P. Thomson performed similar interference experiments with thin-film samples

The Uncertainty Principle The more precisely the position is determined, the less precisely the momentum is known in this instant, and vice versa. Werner Heisenberg,1927

Precise Position Messes Momentum In Quantum Mechanics it is not possible to precisely specify a particle’s position and momentum (velocity) at the same time.

HEISENBERG UNCERTAINTY PRINCIPLE We cannot have simultaneous knowledge of such as position and momenta. Note, however, etc Arbitary precision is possible in principle for position in one direction and momentum in another

HEISENBERG UNCERTAINTY PRINCIPLE There is also an energy-time uncertainty relation Transitions between energy levels of atoms are not perfectly sharp in frequency. n = 3 n = 2 n = 1 An electron in n = 3 will spontaneously decay to a lower level after a lifetime of order Intensity Frequency There is a corresponding ‘spread’ in the emitted frequency

CONCLUSIONS Light and matter exhibit wave-particle duality Relation between wave and particle properties given by the de Broglie relations , Evidence for particle properties of light Photoelectric effect, Compton scattering Evidence for wave properties of matter Electron diffraction, interference of matter waves (electrons, neutrons, He atoms Heisenberg uncertainty principle limits simultaneous knowledge of conjugate variables

Waves: Phase and group velocities of a wave packet The velocity of a wave can be defined in many different ways, partly because there are different kinds of waves, and partly because we can focus on different aspects or components of any given wave. The wave function depends on both time, t, and position, x, i.e., where A is the amplitude. At any fixed location on the x axis the function varies sinusoidally with time. The angular frequency, , of a wave is the number of radians (or cycles) per unit of time at a fixed position. At any fixed instant of time, the function varies sinusoidally along the horizontal axis. The wave number, k, of a wave is the number of radians (or cycles) per unit of distance at a fixed time.

Waves: Phase and group velocities of a wave packet Here is the result of superposing two sine waves whose amplitudes, velocities and propagation directions are the same, but their frequencies differ slightly. We can write: While the frequency of the sine term is that of the phase, the frequency of the cosine term is that of the “envelope”, i.e. the group velocity.

Waves: Phase and group velocities of a wave packet The speed at which a given phase propagates does not coincide with the speed of the envelope. Note that the phase velocity is greater than the group velocity.

wave properties: phase velocity does not describe particle motion

Generic wave properties

Phase and Group velocities simple plane wave inadequate to describe particle motion problems with phase velocity and infinite wave train represent particle with wave packet (wave group) simplified version: superposition of two waves of slightly different wavelength -if wave velocity is independent of wavlength, each wave (and thus the packet) travel at the same speed -if wave velocity is depends upon wavlength, each wave travels at a different speed, in turn different from the wave packet speed.

de Broglie waves for massive particles