Lecture_09: Outline Matter Waves  Wave packets  Phase and group velocities  Uncertainty relations  Problems

Wave packets “Constructing” Particles From Waves: Particles are localized in space Ideal wave is unlocalized in space. It is possible to built “ localized ” entity from waves –Such an entity must have a non-zero amplitude only within a small region of space Such wave is called “ wave packet ”

Wave packets Wave addition:

Wave packets Sum of two waves:

Wave packets Sum of many waves: Multiple waves are superimposed so that one of its crests is at x = 0 The result is that all the waves add constructively at x = 0 There is destructive interference at every point except x = 0 The small region of constructive interference is called a wave packet

Wave packets Sum of many waves:

Wave packets General Case: Amplitudes A(k, ω) determine how much each wave contributes to the packet and thus they determine the shape of the wave packet Fixed moment of time t 0 :

Wave packets 1.A(k) is a very strong spike at a given k 0, and zero everywhere else only one wave with k = k 0 (λ = λ 0 ) contributes; thus one knows momentum exactly, and the wavefunction is a traveling wave – particle is delocalized 2.A(k) is the same for all k No distinctions for momentums, so particle’s position is well defined - the wavefunction is a “spike”, representing a “very localized” particle 3.A(k) is shaped as a bell-curve Gives a wave packet – “partially” localized particle

Phase and group velocities

Wave velocity: EM wave: Two velocities: Phase velocity Group velocity

Wave packets Sum of two waves:

Phase and group velocities Group velocity: Free particle: Photon:

Uncertainty relations We want to know Coordinate and Momentum of a particle at time t = 0 –If we know the forces acting upon the particle than, according to classical physics, we know everything about a particle at any moment in the future –The answer in quantum mechanics is different and is presented by the Heisenberg’s Uncertainty Principle: An experiment cannot simultaneously determine a component of the momentum of a particle (e.g., p x ) and the exact value of the corresponding coordinate, x. The best one can do is

Uncertainty relations 1.The limitations imposed by the uncertainty principle have nothing to do with quality of the experimental equipment 2.The uncertainty principle does not say that one cannot determine the position or the momentum exactly –It refers to exact knowledge about both: e.g. if Δx = 0, then Δ p x is infinity, and vice versa 3.The uncertainty principle is confirmed by experiment, and is a direct consequence of the de Broglie’s hypothesis

Problems Velocity of a 100 g bullet is measured to be 1000 m/s with an accuracy of 0.01%. What is the uncertainty of its position?

Problems Kinetic energy of an electron is measured to be 4.9 eV with an accuracy of 0.01%. What is the uncertainty of its position?

Uncertainty relations Heisenberg-Bohr thought experiment: It shows that a measurement itself introduces the uncertainty When we “look” at an object we see it through the photons that are detected by the microscope –These are the photons that are scattered in the angle < 2θ –Momentum of electron is changed

Uncertainty relations Heisenberg-Bohr thought experiment: Trying to determine the electron position, we introduce the uncertainty of the momentum

Uncertainty relations Heisenberg-Bohr thought experiment: Image is not the point, but the diffraction pattern The uncertainty of the position is approximately the width of the central maximum

Uncertainty relations Uncertainty energy-time: Free particle: Energy conservation law can be violated but for the short time interval!

Problems How long is the time interval during which you disappear but nobody notices?

Problems What is the linewidth for the emission from the electron level with 5 ns lifetime?