Wave Packets and Uncertainty

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

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

Note that $\vert\psi\vert^2=1$ 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).

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

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

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

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

$$\bgroup\color{black}P(p)=\left\vert\phi(p)\right\vert^2\egroup$$

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

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

$$\bgroup\color{black}\Delta p\Delta x\geq {\hbar\over 2}\egroup$$

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.