Chapter 1 - The Magic of Quantum Mechanics

Abstract

Classical mechanics was unable to explain certain phenomena: black body radiation, the photoelectric effect, the stability of atoms and molecules as well as their spectra. Quantum mechanics, created mainly by Werner Heisenberg and Erwin Schrödinger, explained these effects. The new mechanics was based on six postulates:•Postulate I says that all information about the system follows from the wave function ψ. The quantity ∣ψ∣2 represents the probability density of finding particular values of the coordinates of the particles, the system is composed of.•Postulate II allows mechanical quantities (e.g., energy) to be ascribed to operators. One obtains the operators by writing down the classical expression for the corresponding quantity, and replacing momenta (e.g., px) by momenta operators (here, pˆx=-iℏ∂∂x).•Postulate III gives the time evolution equation for the wave function ψ(time-dependent Schrödinger equation Hˆψ=iℏ∂ψ∂t), using the energy operator (HamiltonianHˆ). For time-independent Hˆ one obtains the time-independent Schrödinger equation Hˆψ=Eψ for the stationary states.•Postulate IV pertains to ideal measurements. When making a measurement of a quantity A, one can obtain only an eigenvalue of the corresponding operatorAˆ. If the wave function ψ represents an eigenfunction of Aˆ, i.e., (Aˆψ=aψ), then one obtains always as a result of the measurement the eigenvalue corresponding to ψ(i.e., a). If, however, the system is described by a wave function, which does not represent any eigenfunction ofAˆ, then one obtains also an eigenvalue of Aˆ, but there is no way to predict which eigenvalue. The only thing one can predict is the mean value of many measurements, which may be computed as a¯=ψ∣Aˆψψ∣ψ.•Postulate V says that an elementary particle has an internal angular momentum (spin). One can measure only two quantities: the square of the spin length ss+1ℏ2 and one of its components msℏ, where ms=-s,-s+1,…,+s, with spin quantum number s≥0 characteristic for the type of particle (integer for bosons, half-integer for fermions). The spin magnetic quantum number ms takes 2s+1 values, related to the 2s+1 values of the (granular) spin coordinate σ.•Postulate VI has to do with the symmetry of the wave function with respect to different labeling identical particles. If one exchanges the labels of two identical particles (we sometimes call it the exchange of all the coordinates of the two particles), then for two identical fermions the wave function has to change its sign (antisymmetric), while for two identical bosons the function does not change (symmetry). As a consequence, two identical fermions with the same spin coordinate cannot occupy the same point in space.