Wave packet

[4] He solved his wave equation for a quantum harmonic oscillator, introduced the superposition principle, and used it to show that a compact state could persist.

The year after Schrödinger's paper, Werner Heisenberg published his paper on the uncertainty principle, showing in the process, that Schrödinger's results only applied to quantum harmonic oscillators, not for example to Coulomb potential characteristic of atoms.

[4]: 829 The following year, 1927, Charles Galton Darwin explored Schrödinger's equation for an unbound electron in free space, assuming an initial Gaussian wave packet.

[4]: 830 Quantum mechanics describes the nature of atomic and subatomic systems using Schrödinger's wave equation.

Quantum wave packet profiles change while propagating; they show dispersion.

A monochromatic (single momentum) source produces convergence difficulties in the scattering models.

[9] From the basic one-dimensional plane-wave solutions, a general form of a wave packet can be expressed as

The nondispersive propagation of the real or imaginary part of this wave packet is presented in the above animation.

[12] For example, the solution to the one-dimensional free Schrödinger equation (with 2Δx, m, and ħ set equal to one) satisfying the initial condition

representing a wave packet localized in space at the origin as a Gaussian function, is seen to be

It is evident that this dispersive wave packet, while moving with constant group velocity ko, is delocalizing rapidly: it has a width increasing with time as √ 1 + 4t2 → 2t, so eventually it diffuses to an unlimited region of space.

The above dispersive Gaussian wave packet, unnormalized and just centered at the origin, instead, at t=0, can now be written in 3D, now in standard units:[13][14]

The inverse Fourier transform is still a Gaussian, but now the parameter a has become complex, and there is an overall normalization factor.

only changes in time in a simple way: its phase rotates with a frequency determined by the energy of η.

It varies quadratically with position, which means that it is different from multiplication by a linear phase factor

This linear growth is a reflection of the (time-invariant) momentum uncertainty: the wave packet is confined to a narrow Δx = √a/2, and so has a momentum which is uncertain (according to the uncertainty principle) by the amount ħ/√2a, a spread in velocity of ħ/m√2a, and thus in the future position by ħt /m√2a.

In contrast to the above Gaussian wave packet, which moves at constant group velocity, and always disperses, there exists a wave function based on Airy functions, that propagates freely without envelope dispersion, maintaining its shape, and accelerates in free space:[19]

There is no dissonance with Ehrenfest's theorem in this force-free situation, because the state is both non-normalizable and has an undefined (infinite) ⟨x⟩ for all times.

The three equalities demonstrate three facts: Note the momentum distribution obtained by integrating over all x is constant.

Since this is the probability density in momentum space, it is evident that the wave function itself is not normalizable.

The narrow-width limit of the Gaussian wave packet solution discussed is the free propagator kernel K. For other differential equations, this is usually called the Green's function,[22] but in quantum mechanics it is traditional to reserve the name Green's function for the time Fourier transform of K. Returning to one dimension for simplicity, with m and ħ set equal to one, when a is the infinitesimal quantity ε, the Gaussian initial condition, rescaled so that its integral is one,

Note that a very narrow initial wave packet instantly becomes infinitely wide, but with a phase which is more rapidly oscillatory at large values of x.

The factor involving ε is an infinitesimal quantity which is there to make sure that integrals over K are well defined.

Since the amplitude to travel from x to y after a time t+t' can be considered in two steps, the propagator obeys the composition identity,

For a particle which is randomly walking, the probability density function satisfies the diffusion equation[24]

for any test function f. The time-varying Gaussian is the propagation kernel for the diffusion equation and it obeys the convolution identity,

The exponential can be defined over a range of ts which include complex values, so long as integrals over the propagation kernel stay convergent,

As long as the real part of z is positive, for large values of x, K is exponentially decreasing, and integrals over K are indeed absolutely convergent.

The limit of this expression for z approaching the pure imaginary axis is the above Schrödinger propagator encountered,

holds for all complex z values, where the integrals are absolutely convergent so that the operators are well defined.

A looped animation of a wave packet propagating without dispersion: the envelope is maintained even as the phase changes
A wave packet without dispersion (real or imaginary part)
A wave packet with dispersion. Notice the wave spreads out and its amplitude reduces.
Position space probability density of an initially Gaussian state moving in one dimension at minimally uncertain, constant momentum in free space.
Superposition of 1D plane waves (blue) that sum to form a Gaussian wave packet (red) that propagates to the right while spreading. Blue dots follow each plane wave's phase velocity while the red line follows the central group velocity.
1D Gaussian wave packet, shown in the complex plane, for a =2 and k =4
1D Gaussian wave packet, shown in the complex plane, for . The group velocity is zero. At , the wavefunction has zero phase and minimal width. For , the wavefunction has quadratic phase, decreasing width. For , the wavefunction has quadratic phase, increasing width.
1D Gaussian wave packet, shown in the complex plane, for . The overall group velocity is positive, and the wave packet moves as it disperses.
A 2D gaussian quantum wave packet. The color (yellow green blue) indicates the phase of the wave function , its brightness indicates . ,
The Airy wave train in phase space. Its shape is a series of parabolas with the same axis, but oscillating according to the Airy function. Its time-evolution is a shearing along the -direction. Each parabola retains its shape under this shearing, and its apex performs a translation along another parabola. Thus, the Airy wave train does not disperse, and the group motion of the wave train undergoes constant acceleration.