Wave Packet - Gaussian Wavepackets in Quantum Mechanics

Gaussian Wavepackets in Quantum Mechanics

The above Gaussian wavepacket, unnormalized and just centered at the origin, instead, can now be written in 3D:

where a is a positive real number, the square of the width of the wavepacket, a = 2⟨r·r⟩/3⟨1⟩ = 2 (Δx)2.

The Fourier transform is also a Gaussian in terms of the wavenumber, the k-vector, (with inverse width, 1/a = 2⟨k·k⟩/3⟨1⟩ = 2 (Δpx/ħ)2, so that Δx Δpx=ħ/2, i.e. it saturates the uncertainty relation),

Each separate wave only phase-rotates in time, so that the time dependent Fourier-transformed solution is:

\begin{align}\Psi(\bold{k}, t) &= (2\pi a)^{3/2} e^{- a \bold{k}\cdot\bold{k}/2 }e^{-iEt/\hbar} \\ &= (2\pi a)^{3/2} e^{- a \bold{k}\cdot\bold{k}/2 - i(\hbar^2 \bold{k}\cdot\bold{k}/2m)t/\hbar} \\ &= (2\pi a)^{3/2} e^{-(a+i\hbar t/m)\bold{k}\cdot\bold{k}/2} .\end{align}

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

The integral of Ψ over all space is invariant, because it is the inner product of Ψ with the state of zero energy, which is a wave with infinite wavelength, a constant function of space. For any energy eigenstate η(x), the inner product:

,

only changes in time in a simple way: its phase rotates with a frequency determined by the energy of η. When η has zero energy, like the infinite wavelength wave, it doesn't change at all. The integral ∫|Ψ|2d3r is also invariant, which is a statement of the conservation of probability. Explicitly,

The width of the Gaussian is the interesting quantity which can be read off from |Ψ|2:

.

The width eventually grows linearly in time, as ħt /m√a, indicating wave-packet spreading.

This linear growth is a reflection of the momentum uncertainty: the wavepacket is confined to a narrow width √a, and so has a momentum which is uncertain (according to the uncertainty principle) by the amount ħ/2√a, a spread in velocity of ħ/2m√a, and thus in the future position by ħt /m√a. (The uncertainty relation is then a strict inequality, far from saturation.)

Read more about this topic:  Wave Packet

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