Uncertainty Principle

The uncertainty principle expresses a fundamental limit on the precision with which certain pairs of physical quantities can be simultaneously known. It arises from the wave nature of matter and the properties of the Fourier transform.

Fourier Uncertainty Principle

For any function $f(x)$ and its Fourier transform $F(\nu)$:

$$ \Delta x \cdot \Delta \nu \geq \frac{1}{4\pi} $$

where $\Delta x$ and $\Delta \nu$ are the root-mean-square widths (standard deviations) in position and frequency.

Interpretation: A signal cannot be simultaneously localized in both time/space and frequency. Narrow in one domain means wide in the other.

Heisenberg’s Uncertainty Principle

In quantum mechanics, for position $x$ and momentum $p$:

$$ \Delta x \cdot \Delta p \geq \frac{\hbar}{2} $$

where $\hbar = h/(2\pi)$ is the reduced Planck constant.

This is a direct consequence of the Fourier relationship between position and momentum wavefunctions:

$$ \psi(x) \xleftrightarrow{\mathcal{F}} \phi(p) $$

Minimum Uncertainty States

The Gaussian function achieves the minimum uncertainty product:

$$ \Delta x \cdot \Delta \nu = \frac{1}{4\pi} $$

This is why Gaussian beams and Gaussian wave packets are special — they are as localized as possible in both domains simultaneously.

Time-Frequency Uncertainty

For time $t$ and frequency $f$:

$$ \Delta t \cdot \Delta f \geq \frac{1}{4\pi} $$

Applications:

• Short pulses have broad spectra
• Narrow spectral lines require long coherence times
• Radar resolution limits
• Musical notes: short notes have less definite pitch

Time-Bandwidth Product

The time-bandwidth product $\Delta t \cdot \Delta f$ characterizes signals:

• Gaussian pulse: $\Delta t \cdot \Delta f = 0.44$
• Rectangular pulse: $\Delta t \cdot \Delta f = 0.89$
• Transform-limited pulses have the minimum product for their shape

In Optics

The uncertainty principle manifests as:

• Diffraction limits (small apertures → wide angular spread)
• Coherence length and bandwidth relationship
• Pulse broadening in dispersive media

See Also

• Fourier transform
• Gaussian beam
• Diffraction