Quantum mechanics/Wavepackets

Wavepackets

In quantum mechanics, when a particle is not subjected to any external force (i.e., it exists in a region of constant potential, which can always be chosen to be zero, because one has the freedom of assigning zero potential to all points in space), the free Schrödinger equation describing it is given by

- \frac{\hbar^2}{2m} \nabla^2 \ \psi(\mathbf{r}, t) = i\hbar\frac{\partial}{\partial t} \psi (\mathbf{r}, t) ~,

which, in one dimension, has the general solution of the form

\psi(x,t) = A e^{i(kx-\omega t)}~.

This represents plane waves. These plane wave solutions are not normalisable because the resulting probability density is uniform everywhere. The Hamiltonian commutes with the momentum operator $\hat{p}$ , whose eigenfunctions are $e^{ikx}$ . There exist two methods of constructing normalisable solutions (wave functions). First, one may restrict the wave functions within a region (see Particle in a box).

Another way is to take linear combinations of plane waves with several k (momenta), called wave packets,

\psi(x,0) = \frac{1}{\sqrt{2\pi}} \int^{\,\infty}_{-\infty} \phi(k) ~ e^{ikx} \,dk

where $\phi(k) \,$ is the wave function of the same particle in the momentum basis. The $\psi(x) \,$ can be normalised by choosing appropriate $\phi(k)$ s . $\phi(k) \,$ and $\psi(x) \,$ constitute a fourier transform pair for t=0 (we are given the wave packet at some t, which can be taken to be zero, and asked to determine the wave function at a later time). One may check that both are normalised by the application of Plancherel theorem.

If we require to localise the particle in some finite region, we need some of the coefficients $\phi(k) \,$ to be non-zero, i.e, a superposition of waves with different k's. Therefore, the more precise the location of particle, the more plane waves needed with a wide range of momenta, the less precise the momentum of the particle---the Heisenberg Uncertainty inequality.

Evolution of wave packet

Each energy eigenfunction evolves by acquiring a phase factor $e^{i\omega(k)t}$ , where $\omega(k) =\frac{\hbar^2k^2}{2m} \,$ corresponds to the energy eigenvalue associated with k. $\omega(k)\,$ can have a more complex form if the particle encounters a different potential, potential barrier, for example. Ascribing this time factor to each constituent eigenfunction, yields the travelling wave packet.

\psi(x,t) = \frac{1}{\sqrt{2\pi}} \int^{\,\infty}_{-\infty} \phi(k) ~ e^{i(kx-\omega(k)t)} \,dk

Suppose the wave packet is peaked around at some particular k; expanding $\omega(k)\,$ around its vicinity and retaining up to two terms, yields a moving wave packet (the envelope of plane waves) with a group velocity

v_g = \frac{\partial \omega}{\partial k}\,

By contrast, each plane wave travels with the corresponding phase velocity $v_p = \frac{\omega}{k}\,$ . The group velocity, i.e., the apparent velocity of wave packet with which it travels, is twice the phase velocity, the velocity of the constituents.

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