Introduction: Saturating the Bound
The uncertainty principle gives a lower bound: \(\Delta x \cdot \Delta p \geq \hbar/2\). Which states actually achieve equality? These minimum uncertainty states are as "classical" as quantum mechanics allows, and they're Gaussian wavepackets.
Condition for Minimum Uncertainty
The inequality is saturated when:
\[(\hat{a} + i\lambda\hat{b})|\psi\rangle = 0\]for some real \(\lambda\), where \(\hat{a} = \hat{x} - \langle x\rangle\) and \(\hat{b} = \hat{p} - \langle p\rangle\).
This leads to:
\[\left(\hat{x} - \langle x\rangle + \frac{i\hbar}{2(\Delta x)^2}(\hat{p} - \langle p\rangle)\right)|\psi\rangle = 0\]The Gaussian Solution
Solving this differential equation gives:
\[\psi(x) = \left(\frac{1}{2\pi(\Delta x)^2}\right)^{1/4} \exp\left(-\frac{(x - \langle x\rangle)^2}{4(\Delta x)^2}\right) \exp\left(\frac{i\langle p\rangle x}{\hbar}\right)\]A Gaussian envelope with center \(\langle x\rangle\), width \(\Delta x\), and average momentum \(\langle p\rangle\).
Worked Examples
Example 1: The Ground State of the Harmonic Oscillator
The harmonic oscillator ground state is:
\[\psi_0(x) = \left(\frac{m\omega}{\pi\hbar}\right)^{1/4} e^{-m\omega x^2/2\hbar}\]This is a Gaussian with \(\Delta x = \sqrt{\hbar/2m\omega}\) and \(\Delta p = \sqrt{m\omega\hbar/2}\).
Check: \(\Delta x \cdot \Delta p = \frac{\hbar}{2}\) exactly! ✓
Example 2: Coherent States
Coherent states \(|\alpha\rangle\) (eigenstates of the lowering operator) are minimum uncertainty states that remain minimum uncertainty under time evolution. They're the "most classical" quantum states.
Example 3: Squeezed States
Squeezed states also saturate the bound but trade off:
- \(\Delta x < \sqrt{\hbar/2m\omega}\) (squeezed in position)
- \(\Delta p > \sqrt{m\omega\hbar/2}\) (stretched in momentum)
Product still equals \(\hbar/2\).
The Quantum Connection
Minimum uncertainty states are crucial in quantum optics and quantum computing. Coherent states describe laser light. Squeezed states enable precision measurements beyond the "quantum limit." Understanding these states shows that the uncertainty principle is not just a restriction—it's a design principle for engineering quantum systems.