Wave Packets and Uncertainty

The probability amplitude for a free particle with momentum \bgroup\color{black}$\vec{p}$\egroup and energy \bgroup\color{black}$E=\sqrt{(pc)^2+(mc^2)^2}$\egroup is the complex wave function

\begin{displaymath}\bgroup\color{black}\psi_{\mathrm{free particle}}(\vec{x},t)=e^{i(\vec{p}\cdot\vec{x}-Et)/\hbar}.\egroup\end{displaymath}

Note that \bgroup\color{black}$\vert\psi\vert^2=1$\egroup everywhere so this does not represent a localized particle. In fact we recognize the wave property that, to have exactly one frequency, a wave must be spread out over space.

We can build up localized wave packets that represent single particles by adding up these free particle wave functions (with some coefficients).

\begin{displaymath}\bgroup\color{black}\psi (x,t)={1\over \sqrt{2\pi\hbar}}\int\limits_{-\infty}^{+\infty}\phi(p)e^{i(px-Et)/\hbar}dp\egroup\end{displaymath}

(We have moved to one dimension for simplicity.) Similarly we can compute the coefficient for each momentum

\begin{displaymath}\bgroup\color{black}\phi(p)={1\over \sqrt{2\pi\hbar}}\int\limits_{-\infty}^\infty \psi(x) e^{-ipx/\hbar} dx.\egroup\end{displaymath}

These coefficients, \bgroup\color{black}$\phi(p)$\egroup, are actually the state function of the particle in momentum space. We can describe the state of a particle either in position space with \bgroup\color{black}$\psi(x)$\egroup or in momentum space with \bgroup\color{black}$\phi(p)$\egroup. We can use \bgroup\color{black}$\phi(p)$\egroup to compute the probability distribution function for momentum.


We will show that wave packets like these behave correctly in the classical limit, vindicating the choice we made for \bgroup\color{black}$\psi_{\mathrm free particle}(\vec{x},t)$\egroup.

The Heisenberg Uncertainty Principle is a property of waves that we can deduce from our study of localized wave packets.

\begin{displaymath}\bgroup\color{black}\Delta p\Delta x\geq {\hbar\over 2}\egroup\end{displaymath}

It shows that due to the wave nature of particles, we cannot localize a particle into a small volume without increasing its energy. For example, we can estimate the ground state energy (and the size of) a Hydrogen atom very well from the uncertainty principle.

The next step in building up Quantum Mechanics is to determine how a wave function develops with time - particularly useful if a potential is applied. The differential equation which wave functions must satisfy is called the Schrödinger Equation.

Jim Branson 2013-04-22