by a superposition of momentum eigenfunctions, with coefficients given by the Fourier transform of the initial wavefunction: where the integral is over all k-space and (to ensure that the wave packet is a solution of the free particle Schrödinger equation). Here is the value of the wave function at time 0 and is the Fourier transform of . (The Fourier transform is essentially the momentum wave function of the position wave function , but written as a function of rather than .) The expectation value of the momentum p for the complex plane wave is and for the general wave packet it is The expectation value of the energy E is Group velocity and phase velocity The phase velocity is defined to be the speed at which a plane wave solution propagates, namely Note that is not the speed of a classical particle with momentum ; rather, it is half of the classical velocity. Meanwhile, suppose that the initial wave function is a wave packet whose Fourier transform is concentrated near a particular wave vector . Then the group velocity of the plane wave is defined as which agrees with the formula for the classical velocity of the particle. The group velocity is the (approximate) speed at which the whole wave packet propagates, while the phase velocity is the speed at which the individual peaks in the wave packet move. The figure illustrates this phenomenon, with the individual peaks within the wave packet propagating at half the speed of the overall packet. Spread of the wave packet The notion of group velocity is based on a linear approximation to the dispersion relation near a particular value of . In this approximation, the amplitude of the wave packet moves at a velocity equal to the group velocity without changing shape. This result is an approximation that fails to capture certain interesting aspects of the evolution a free quantum particle. Notably, the width of the wave packet, as measured by the uncertainty in the position, grows linearly in time for large times. This phenomenon is called the spread of the wave packet for a free particle. Specifically, it is not difficult to compute an exact formula for the uncertainty as a function of time, where is the position operator. Working in one spatial dimension for simplicity, we have: where is the time-zero wave function. The expression in parentheses in the second term