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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 :

\begin{displaymath}
\Psi(x,t) = Ae^{-(i/\hbar)(Et-px)}
\end{displaymath} (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
\begin{displaymath}
E=h\nu = \frac{h}{2\pi}  2\pi \omega = \hbar\omega
\end{displaymath} (23)

which expresses the quantisation of energy and
\begin{displaymath}
\frac{h}{2\pi}\frac{2\pi}{\lambda} = \hbar k = p
\end{displaymath} (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

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

and
\begin{displaymath}
i\hbar   \frac{\partial \Psi(x,t)}{\partial t} = E   \Psi(x,t) .
\end{displaymath} (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

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

and
\begin{displaymath}
\mbox{Energy operator} \qquad \mathbf{\mathsf{E}} = i \hbar  \frac{\partial }{\partial t}
\end{displaymath} (28)

Note that the operator for momentum squared is

$\displaystyle \mathbf{\mathsf{p^2}}$ $\textstyle =$ $\displaystyle \left( \frac{\hbar}{i}   \frac{\partial }{\partial x} \right) ^2$  
  $\textstyle =$ $\displaystyle - \hbar ^2   \frac{\partial^2 }{\partial x^2}$ (29)


next up previous
Next: Schrödinger's Time-Dependent Wave Equation Up: Introduction to Quantum Mechanics Previous: Young's double slit experiment
Simon Connell 2006-02-21