Wave Packet

A wave packet is a concentrated train of (quantum) waves of various wavelengths or momenta with the property that the packet is confined within a small region of space. Such a packet can be constructed by adding a very large number of waves so chosen that their sum interferes destructively everywhere except in a small region. If harmonic waves of different momenta are superposed, the packet can be expressed in the form ψ(x) = f A(k)e ^ikx dk where k = p/ħ and A(k) is the amplitude corresponding to the wave number k.

Although speculative attempts to identify atoms with systems of standing waves can be found back in the nineteenth century, in a quantum context it was Schrödinger who invented wave packets and related them to atomic particles. In his second communication on ► wave mechanics Schrödinger discussed the possibility of constructing a wave group or packet equivalent to a pointlike particle, such as an electron, and in a subsequent paper of 1926 he provided a more elaborate discussion in which he introduced the ► superposition principle. Analyzing the case of a one-dimensional harmonic oscillator, Schrödinger constructed for the first time a wave packet as an exact solution of the ► Schrodinger equation. Making use of the superposition principle, he constructed a wave packet of the form ψ = Σ a ⁿψ_n/n!, where a is a large number, ψ_n are the eigenstates, and 0 ≤ n ≤ ∞. The resulting wave packet, he showed, remains compact as time goes on and it has an energy which is exactly the same as the one of the classical oscillator. Schrödinger's wave packet was a “minimum uncertainty wave packet,” the first example of what later became known as “► coherent states.” He believed that this result would be valid also for electrons moving in atomic orbits and, if so, that it indicated that perhaps electrons and other particles are wave packets. At the end of his paper he foresaw that it was only a matter of time until “the representation by wave mechanics of the hydrogen atom” ► Bohr's atom model would be achieved.