Learning Path: Quantum Uncertainty Principle

Position–momentum uncertainty principle

The position–momentum uncertainty principle (also called Heisenberg’s indeterminacy principle) states that there is a fundamental limit to how precisely a quantum system’s position and momentum can be simultaneously known. More formally, for a particle with position uncertainty (standard deviation) \(\sigma_x\) and momentum uncertainty \(\sigma_p\), the Kennard inequality gives \(\sigma_x\sigma_p \ge \hbar/2\), where \(\hbar\) is the reduced Planck constant. Thus, improving the precision of one quantity necessarily worsens the precision of the other. In the wave-mechanics picture, this tradeoff arises because the position- and momentum-space wavefunctions are Fourier transforms of each other: position and momentum are conjugate variables. A wave packet that is highly localized in position must be composed of many plane-wave components with a wide spread of momenta, and vice versa. In the matrix-mechanics picture, the same limitation is linked to non-commuting observables: if two Hermitian operators do not commute, their measurement outcomes cannot be simultaneously sharp. The principle can be generalized beyond position and momentum using the Robertson and Robertson–Schrödinger uncertainty relations for arbitrary Hermitian operators, expressed in terms of commutators (and also anticommutators when correlations are included).

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Wave mechanics interpretation via Fourier conjugates

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