The Generalised Uncertainty Relation

The Intuition Behind the Bound

Imagine you are trying to describe a wave packet. If the wave is very localized in space (small position uncertainty), it must be composed of many different wavelengths (large momentum uncertainty). Conversely, a pure sine wave has a perfectly defined wavelength (zero momentum uncertainty) but is spread out infinitely in space (infinite position uncertainty). This is a Fourier transform property of waves. In quantum mechanics, position and momentum are not just coordinates; they are operators that act on the state vector. When two operators do not commute, no state can give sharp values of both; this is a statement about the spread of outcomes in one prepared state, not about one measurement disturbing the next. The Generalised Uncertainty Relation quantifies this trade-off. It tells us that the product of the "widths" of the probability distributions for two observables cannot be smaller than half the magnitude of the expectation value of their commutator. If the commutator is zero, the observables are compatible, and their uncertainties can both be zero. If the commutator is non-zero, there is a fundamental limit to how precisely we can define both simultaneously.

Mathematical Formulation

Let \(\hat{A}\) and \(\hat{B}\) be two Hermitian operators representing observables. The uncertainty (standard deviation) of \(\hat{A}\) in a state \(|\psi\rangle\) is defined as \(\Delta A = \sqrt{\langle \psi | (\hat{A} - \langle A \rangle)^2 | \psi \rangle}\). Similarly for \(\hat{B}\). We define the deviation operators \(\hat{a} = \hat{A} - \langle A \rangle\) and \(\hat{b} = \hat{B} - \langle B \rangle\). Note that \(\langle \psi | \hat{a} | \psi \rangle = 0\) and \(\langle \psi | \hat{b} | \psi \rangle = 0\). The expectation value of the product of these deviation operators is \(\langle \psi | \hat{a} \hat{b} | \psi \rangle\). This complex number can be decomposed into its real and imaginary parts. The real part is related to the covariance, and the imaginary part is related to the commutator. Specifically, \(\langle \psi | \hat{a} \hat{b} | \psi \rangle = \frac{1}{2} \langle \psi | \{ \hat{a}, \hat{b} \} | \psi \rangle + \frac{1}{2} \langle \psi | [ \hat{a}, \hat{b} ] | \psi \rangle\). Since \(\hat{a}\) and \(\hat{b}\) are Hermitian, the anticommutator \(\{ \hat{a}, \hat{b} \}\) is Hermitian, and the commutator \([ \hat{a}, \hat{b} ]\) is anti-Hermitian. The expectation value of an anti-Hermitian operator is purely imaginary. Thus, \(\text{Im} \langle \psi | \hat{a} \hat{b} | \psi \rangle = \frac{1}{2i} \langle \psi | [ \hat{a}, \hat{b} ] | \psi \rangle\), a real number because \( \langle [\hat{a}, \hat{b}] \rangle \) is purely imaginary. Note that \([ \hat{a}, \hat{b} ] = [ \hat{A}, \hat{B} ]\) because the constants \(\langle A \rangle\) and \(\langle B \rangle\) commute with everything.

Using the Cauchy-Schwarz inequality on the vectors \(\hat{a}|\psi\rangle\) and \(\hat{b}|\psi\rangle\), we have \(\langle \psi | \hat{a}^2 | \psi \rangle \langle \psi | \hat{b}^2 | \psi \rangle \geq |\langle \psi | \hat{a} \hat{b} | \psi \rangle|^2\). The left-hand side is \((\Delta A)^2 (\Delta B)^2\). The right-hand side is the modulus squared of the complex number \(\langle \psi | \hat{a} \hat{b} | \psi \rangle\). For any complex number \(z = x + iy\), \(|z|^2 = x^2 + y^2 \geq y^2\). Therefore, \((\Delta A)^2 (\Delta B)^2 \geq (\text{Im} \langle \psi | \hat{a} \hat{b} | \psi \rangle)^2\). Substituting the expression for the imaginary part, we get \((\Delta A)^2 (\Delta B)^2 \geq \left( \frac{1}{2i} \langle \psi | [ \hat{A}, \hat{B} ] | \psi \rangle \right)^2 = \frac{1}{4} \left| \langle [ \hat{A}, \hat{B} ] \rangle \right|^2\). Taking the square root of both sides yields the Generalised Uncertainty Relation.

Application to Position and Momentum

The most famous application is the Heisenberg Uncertainty Principle for position \(x\) and momentum \(p\). The canonical commutation relation is \([ \hat{x}, \hat{p} ] = i\hbar\). This is a constant, independent of the state. Therefore, the expectation value is simply \(i\hbar\). The magnitude is \(\hbar\). Substituting this into the GUR, we get \(\Delta x \cdot \Delta p \geq \frac{1}{2} \hbar\). This result holds for any quantum state. It is not a statement about the disturbance caused by measurement, but about the intrinsic spread of the wavefunction. If a particle is in a momentum eigenstate, \(\Delta p = 0\), which implies \(\Delta x \to \infty\). If a particle is localized in a small region, \(\Delta x\) is small, which forces \(\Delta p\) to be large. This has profound implications for bound states, such as the electron in a hydrogen atom. If the electron were to collapse into the nucleus, \(\Delta x\) would become very small, causing \(\Delta p\) (and thus the kinetic energy) to become very large, preventing the collapse.

Saturation and Minimum Uncertainty States

The inequality becomes an equality when the vectors \(\hat{a}|\psi\rangle\) and \(\hat{b}|\psi\rangle\) are linearly dependent and, in addition, the real (covariance) part of \( \langle \hat{a}\hat{b} \rangle \) vanishes; together these require \( \hat{b}|\psi\rangle = i\lambda\, \hat{a}|\psi\rangle \) with \( \lambda \) real. Such states are called minimum uncertainty states. For position and momentum, the minimum uncertainty states are Gaussian wave packets. These states have the property that \(\Delta x \cdot \Delta p = \frac{\hbar}{2}\). They are the closest quantum analogues to classical particles. Under free evolution, however, a Gaussian spreads and stops being a minimum uncertainty state; in a harmonic oscillator the ground state, and the coherent states built from it, keep \( \Delta x \cdot \Delta p = \hbar/2 \) for all time. For \( x \) and \( p \), the saturation condition is the first-order equation \( (\hat{p} - \langle p \rangle)\psi = i\lambda(\hat{x} - \langle x \rangle)\psi \), whose normalisable solutions are exactly these Gaussians. This connection between the algebraic structure of the commutator and the specific form of the wavefunction is a key insight in quantum mechanics.