   Next: Schrödinger's Time-Dependent Wave Equation Up: Introduction to Quantum Mechanics Previous: Young's double slit experiment

## Wave Functions, Operators

Experiments have guided our intuition in developing a new theory for quantum objects.

 Intuition Implementation Young's double slit experiment for particles and for waves yield similar interference patterns. Postulate a wave function associated with quantum particles . The interference pattern is observed via intensity variations on a screen. The intensity of a wave is its amplitude squared. Postulate that the probability of finding a quantum particle at a given position and time is related to the mod squared of its wave function. . We need to respect energy quantisation as prompted by the study of black body radiation. For quantum particles . The wave nature of particles is evidenced by the de Broglie relation. For quantum particles .

The simplest wave function is the wave function of a free-particle (absence of forces acting on the particle). The particle can be considered to be moving in the -direction. A logical way to postulate wave function for a free particle is then : (22)

where we have used our previous expression for wave motion in the -direction. To emphasize the applicability of this expression to a quantum particle, we have also incorporated the identities (23)

which expresses the quantisation of energy and (24)

which is de Broglie's relation linking particle and wave properties. The energy and momentum are now part of the description of the wave function for the free particle. Energy and momentum are termed observables in Quantum Mechanics, because we can measure'' or observe'' them.

We need to develop a mathematical procedure that allows us to calculate observables from the wave function. The first step towards this is to note that (25)

and (26)

Exercise 2.7
Verify these equations.

Because the momentum and energy can be extracted'' from the wave function by operating'' on it appropriately, we define the energy and momentum operators'' as (27)

and (28)

Note that the operator for momentum squared is     (29)   Next: Schrödinger's Time-Dependent Wave Equation Up: Introduction to Quantum Mechanics Previous: Young's double slit experiment
Simon Connell 2006-02-21