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 $\mbox{particle } \longrightarrow \Psi(x,t)$.
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. $P(x,t) \propto \vert\Psi(x,t)\vert^2$ .
We need to respect energy quantisation as prompted by the study of black body radiation.	For quantum particles $E = h\nu$.
The wave nature of particles is evidenced by the de Broglie relation.	For quantum particles $\lambda = h/p$.

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 $+x$-direction. A logical way to postulate wave function for a free particle is then :

$$\Psi(x,t) = Ae^{-(i/\hbar)(Et-px)}$$ (22)

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

$$E=h\nu = \frac{h}{2\pi} 2\pi \omega = \hbar\omega$$ (23)

which expresses the quantisation of energy and

$$\frac{h}{2\pi}\frac{2\pi}{\lambda} = \hbar k = p$$ (24)

which is de Broglie's relation linking particle and wave properties. The energy $E$ and momentum $p$ 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

$$\frac{\hbar}{i} \frac{\partial \Psi(x,t)}{\partial x} = p \Psi(x,t)$$ (25)

and

$$i\hbar \frac{\partial \Psi(x,t)}{\partial t} = E \Psi(x,t) .$$ (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

$$\mbox{Momentum operator} \qquad \mathbf{\mathsf{p}} = \frac{\hbar}{i} \frac{\partial }{\partial x}$$ (27)

and

$$\mbox{Energy operator} \qquad \mathbf{\mathsf{E}} = i \hbar \frac{\partial }{\partial t}$$ (28)

Note that the operator for momentum squared is

$$\mathbf{\mathsf{p^2}} = \left( \frac{\hbar}{i} \frac{\partial }{\partial x} \right) ^2$$

$$= - \hbar ^2 \frac{\partial^2 }{\partial x^2}$$ (29)