Wave packet: Superposition principle Superposition of two waves the phenomenon of beats.

Phase velocity vs. group velocity Combined wave moves with group velocity Group velocity describes the speed of the envelope.

“localized region” Localized region: What do we learn from this?

In order to know precisely the position of the wave packet envelope (x small), we must have a large range of wave numbers (k large).

Wave packets and particles A plane wave describes a free particle with definite momentum. However, the particle is not localized at all. (The wave spreads all over the space.) A wave function describing a localized free particle? Fourier series and integral: superposition of large number of waves

Wave packets and particles

Wave packets and particles

Wave packets and particles Normalization requires Expectation value of the momentum? Direct calculation Use probability distribution of the momentum

Wave packets and particles What does (x) look like? (x) --peaks at x=0, --falls off to zero when xp/(2)=, --oscillates with a rapidly diminishing maximum height. The particle is localized.

Wave packets and particles The more precise the momentum, the more spread out the pulse is in space. A superposition of plane waves with a finite range of momenta is known as a wave packet.

Uncertainty Relations Heisenberg (1927) We cannot simultaneously measure the position and the momentum of a particle with arbitrary precision.

UncertaintyStandard deviation

Uncertainty Relations The meaning of the uncertainty relations The indeterminacy reflects the limits beyond which we can no longer use the intuitive classical notions of the trajectory x(t) of a particle and its momentum m dx/dt. The uncertainty relations play no roles on the macroscopic scale Ex. A dust particle, m=10-7kg, v=10m/s, v=10-6m/s, x=?

Uncertainty Relations The uncertainty relation plays a crucial role on the atomic scale. Zero-point energy of a physical system Atomic lifetime/natural line width

Uncertainty principle Estimating ground state energies of physical systems (Ex: SHO) Classical description? Quantum description?

Uncertainty principle Zero-point energy

Is it possible for an e to be confined within the nucleus? For a nucleus of several fms, the uncertainty of position is According to uncertainty relation, the uncertainty in momentum of an electron is Usually kinetic energy is smaller than 1eV…...

Superposition principle

Superposition principle: Atoms in transitions Oscillating charge: transition probability: lifetime of atomic states:

Homework assignment: 24.1, 24.3, 24.4, 24.7