Uncertainty: Fourier Width Bounds for Wave Packets

01What width does a localized wave packet have?

A localized material core and its accompanying field do not have the same spatial distribution. The object here is the effective envelope: which wave-number components are required to form a field concentrated in a finite region? Reducing its spatial extent changes its phase gradients and spectrum. An uncertainty relation first constrains these two widths quantitatively; their connection to material measurements comes next.

Take a one-dimensional envelope $\psi(x)$, where $x$ is position along the direction considered. Normalize it and use the symmetric Fourier convention of Chapter 5, (5.3.23):

Here $k$ is the wave number conjugate to $x$, and $\widetilde\psi(k)$ is its amplitude. Their squared moduli have equal integral normalization and define mathematical weights in position and wave number. This does not yet identify them with experimental probabilities for core positions or momentum readings.

To give “width” a definite meaning, define standard deviations about the means:

The second position moment must be finite, the first weak derivative of $\psi$ must be square-integrable, and the decaying boundary conditions used below must hold. A weak derivative allows envelopes that need not be classically differentiable everywhere but still have a suitable derivative in the integral sense. A specified slit width, half-maximum width or first-zero separation is not automatically this standard deviation.

02How integration by parts produces a lower bound

To bring spatial and spectral width into one calculation, define $F=(x-x_0)\psi$ and $G=(\partial_x-ik_0)\psi$, with inner product $\langle F,G\rangle=\int F^*G\,dx$. The function $F$ weights displacement from the center. The function $G$ removes the mean wave number's common linear phase from the derivative.

Taking the real part of the inner product removes the contribution from $-ik_0$ and leaves the derivative of the squared envelope modulus. The decaying boundary conditions make $[(x-x_0)|\psi|^2]_{-\infty}^{\infty}$ vanish, so

The inner product therefore cannot become arbitrarily small. Cauchy–Schwarz bounds its modulus by the product of the two function norms, while Parseval converts the position-domain derivative norm into the wave-number variance:

Taking nonnegative square roots gives $\Delta x\Delta k\ge1/2$. This follows from normalization, Fourier structure and the boundary conditions, without assuming stochastic detector response.

03From wave number to momentum requires material matching

Turning wave-number bandwidth into momentum width requires the phase–momentum matching $p=\hbar k$ discussed on the matter-wave page. With the same constant $\hbar$ throughout this matching, $\Delta p=\hbar\Delta k$. The existing equation registry records the mathematical proof and this physical connection together:

Thus $\Delta x\Delta k\ge1/2$ is an envelope relation, while $\Delta x\Delta p\ge\hbar/2$ initially constrains the position and canonical-momentum distributions associated with that envelope. ASD must connect the measured core momentum distribution to the wave-number distribution through specified preparation, core–field coupling and detection protocols. Substituting $\hbar$ does not remove the conditions on mode-action matching or experimental mapping.

04Why a Gaussian width and a slit width cannot be interchanged

The lower bound is attainable. Chapter 5, (5.3.26), gives a Gaussian envelope with linear phase:

Here $\sigma_x>0$ is the position standard deviation, $x_0$ is the envelope center and $k_0$ is the mean wave number. Its wave-number standard deviation is $1/(2\sigma_x)$, giving a width product of exactly $1/2$. Translation, a common phase or a linear phase does not change that minimum. Additional nonlinear phase, hard truncation or a non-Gaussian amplitude generally increases the product.

Slit experiments often use a different quantity: the width of a diffraction main lobe. The far-field amplitude of an ideal hard rectangular aperture is a sinc function, with sinc² intensity. Its tails make the second wave-number moment diverge. The first-zero separation can describe the main lobe but cannot be inserted as the standard deviation in this inequality. Real edge smoothing, finite material response and detector bandwidth alter the distribution and must be included in the same variance calculation.

This distinction keeps the aperture formulas in interference and diffraction consistent with the width bound. Geometric slit width, diffraction first-zero width and root-mean-square width answer different measurement questions.

05Duration and spectrum of a finite process

Time here is the metrological parameter recording the evolution of a process. A finite-duration amplitude also requires more than one frequency, but its width relation must follow the actual signal definition. For example, a rectangular time window of duration $T$ and carrier frequency $\omega_c$ gives

Here $\omega$ is angular frequency. The first-zero half-width of the rectangular-window main lobe is $\Delta\omega_0=2\pi/T$. With the adopted energy–frequency matching, $\Delta E_0=\hbar\Delta\omega_0$ gives the exact width identity $\Delta E_0T=2\pi\hbar=h$, as stated after final equation (5.3.27). This is a first-zero width, not a variance: the rectangular-window spectrum has a divergent second moment, and $T$ is not a time standard deviation.

For a square-integrable time amplitude with finite root-mean-square duration and angular-frequency standard deviation, the same Fourier proof gives $\Delta t\Delta\omega\ge1/2$. Another familiar process is $\theta(t)e^{-t/(2\tau)}e^{-i\omega_ct}$, where $\theta(t)$ is the step function and $\tau$ is the intensity lifetime. Its Lorentz-shaped power spectrum has angular-frequency full width at half maximum $1/\tau$. Lifetime, time window, half-maximum width and standard deviation retain their distinct definitions.

06How wave-packet structure enters measurement

ASD connects the bandwidth of a finite accompanying structure to the field's phase and gradient structure. Apparatus back-action describes how preparation or detection coupling changes that structure. The envelope's Fourier constraint, the dynamics of measurement back-action and individual-record statistics are related questions with different calculation objects.

This page establishes an envelope inequality with defined widths and its material connection conditional on momentum matching. It does not thereby specify microscopic core trajectories or a universal time operator. Discrete observables such as spin need their own generators and algebra. Phase–momentum matching remains associated with OP-QM-002, statistical universality with OP-QM-004, and the conditional detector calculation is developed in detection and statistics.