momentum representation

The momentum representation describes a quantum state as a function of momentum rather than of position. Instead of ψ(x), one works with a momentum-space wavefunction, often written φ(p), whose squared magnitude |φ(p)|² gives the probability density for measuring each value of momentum. It is the exact counterpart of the position representation, simply laying out the same abstract state along momentum 'axes' instead of position ones.

The two representations are joined by the Fourier transform, the mathematical operation that decomposes any function into the pure waves of definite wavelength that make it up. Because a definite momentum corresponds, by de Broglie's relation, to a pure wave of definite wavelength, asking 'how much of each momentum is in ψ?' is precisely a Fourier analysis. The momentum-space wavefunction is the recipe of wavelengths baked into the position-space one.

This dual viewpoint makes the uncertainty principle almost visible. A wavefunction sharply peaked in position must, as a mathematical fact about Fourier transforms, be spread broadly in momentum, and vice versa — you cannot make both narrow at once. The momentum representation is not a different theory but a different and often more convenient lens; problems awkward in position space, such as those for a free particle, frequently become simple when viewed through momentum.