Quantum mechanics

The Heisenberg uncertainty principle

The position and momentum spreads of a state cannot both be arbitrarily small: ΔxΔp ≥ ℏ/2.

The uncertainty principle is a structural limit on quantum states, not an imperfection of instruments or observer clumsiness. It concerns statistical spreads across repeated measurements on identically prepared systems.

\Delta x\,\Delta p_x\geq\frac{\hbar}{2},\qquad \Delta A\,\Delta B\geq\frac12\left|\langle[A,B]\rangle\right|

Definition: Standard deviation

For an observable $A$ , uncertainty is its standard deviation $\Delta A=\sqrt{\langle A^2\rangle-\langle A\rangle^2}$ . The general relation uses the commutator $[A,B]=AB-BA$ . Since $[x,p_x]=i\hbar$ , the position–momentum inequality follows.

Adjust the Gaussian packet width: tighter localization in position broadens the wave-number/momentum distribution, consistent with ΔxΔp ≥ ℏ/2.

Mathematical origin

For any two observables, the general relation is $\Delta A\Delta B\geq |\langle[A,B]\rangle|/2$ . Commuting operators have no positive lower bound from this relation; position and momentum do not commute and therefore have one. A minimum-uncertainty Gaussian saturates the $x,p$ bound.

Example: Wavelength and momentum

An electron packet has $\Delta x=0.10$ nm. Find the minimum possible $\Delta p_x$ .

Solution

$\Delta p_x\geq\hbar/(2\Delta x)=1.055\times10^{-34}/(2.0\times10^{-10})=5.28\times10^{-25}$ kg·m/s.

Consider a Gaussian wave packet with position standard deviation $\sigma_x$ . Fourier analysis shows that narrowing the packet increases its momentum spread; a Gaussian saturates $\Delta x\Delta p=\hbar/2$ . Translating the packet can give arbitrary means $\langle x\rangle$ and $\langle p\rangle$ without changing the standard-deviation product. This is a property of the state, not an instrument defect. More generally, for operators $A,B$ , the Robertson inequality is $\Delta A\Delta B\ge |\langle[A,B]\rangle|/2$ .

Quick check

What is the standard lower bound for the position–momentum uncertainty product?

What usually happens to momentum spread when a wave packet is more localized in position?

References

David J. Griffiths, Darrell F. Schroeter (2018). Introduction to Quantum Mechanics