How is a momentum space wavefunction related to the corresponding coordinate space wavefunction? To answer this question, let us consider the representative of the momentum eigenkets in the Schrödinger representation for a system with a single degree of freedom. This representative satisfies
where use has been made of Equation (169) (for the case of a system with one degree of freedom). The solution of the above differential equation is
where . It is easily demonstrated that
The well-known mathematical result
This is consistent with Equation (171), provided that . Thus,
Consider a general state ket whose coordinate wavefunction is , and whose momentum wavefunction is . In other words,
It is easily demonstrated that
where use has been made of Equations (118), (172), (185), and (187). Clearly, the momentum space wavefunction is the Fourier transform of the coordinate space wavefunction.
Consider a state whose coordinate space wavefunction is a wavepacket. In other words, the wavefunction only has non-negligible amplitude in some spatially localized region of extent . As is well-known, the Fourier transform of a wavepacket fills up a wavenumber band of approximate extent . Note that in Equation (190) the role of the wavenumber is played by the quantity . It follows that the momentum space wavefunction corresponding to a wavepacket in coordinate space extends over a range of momenta . Clearly, a measurement of is almost certain to give a result lying in a range of width . Likewise, measurement of is almost certain to yield a result lying in a range of width . The product of these two uncertainties is
This result is called the Heisenberg uncertainty principle.
Actually, it is possible to write the Heisenberg uncertainty principle more exactly by making use of Equation (83) and the commutation relation (138). We obtain
for any general state. It is easily demonstrated that the minimum uncertainty states, for which the equality sign holds in the above relation, correspond to Gaussian wavepackets in both coordinate and momentum space.
- Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)