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Position-space and momentum-space wavefunctions are Fourier transforms of each other, making position and momentum Fourier-conjugate variables.
In wave mechanics, the position and momentum of a quantum particle are represented in two related descriptions of the same state. The position-space wavefunction, ψ(x), describes the probability amplitude for finding the particle at different positions, while the momentum-space wavefunction, φ(p), gives the amplitudes for different momenta. These wavefunctions are Fourier transforms of one another because momentum is the Fourier-conjugate variable to position, as expressed by the de Broglie relation p = ℏk. A sharply localized position wavefunction must be constructed from many plane waves with different momenta, producing a broad momentum distribution. Conversely, a single plane wave has a definite momentum but is spread throughout space, giving highly uncertain position. This trade-off is quantified by the Kennard uncertainty relation, σxσp ≥ ℏ/2, which follows mathematically from the properties of Fourier transforms and the Cauchy–Schwarz inequality. The principle therefore reflects an intrinsic feature of wave representations rather than merely a limitation of measurement.
Position-space and momentum-space wavefunctions are Fourier transforms of each other, making position and momentum Fourier-conjugate variables.
A plane wave has a precise momentum but is completely delocalized in position, whereas a localized wave packet requires a range of momentum components.
The spread of position and momentum distributions is measured by their standard deviations, which obey σxσp ≥ ℏ/2.
The uncertainty relation arises from the mathematical impossibility of a nonzero function and its Fourier transform both being sharply localized.
The momentum operator in position space is represented by -iℏ d/dx, linking momentum directly to spatial variation of the wavefunction.
A complex-valued function whose squared magnitude gives the probability density for measurement outcomes.
The wavefunction ψ(x) that describes the probability amplitudes for finding a particle at different positions.
The wavefunction φ(p) that describes the probability amplitudes for measuring different particle momenta.
A mathematical transformation that decomposes a function into its constituent spatial-frequency or momentum components.
Pairs of variables, such as position and momentum, whose representations are related by Fourier transformation.
A wavefunction of the form exp(ipx/ℏ) that has definite momentum and is spread uniformly through space.
A superposition of plane waves with different momenta that can be localized in position.
The inequality σxσp ≥ ℏ/2, which sets the minimum possible product of position and momentum standard deviations.
The relation p = ℏk connecting a particle's momentum with the wave number of its associated matter wave.
In position space, the operator -iℏ d/dx that represents momentum measurements.
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