Spread of Wave Function Under Potential

[boderam]

I have books (Quantum Theory by Bohm for example) with derivation of the spread of the wavefunction of a free particle in the Schrödinger equation. But does this spreading only happen as a free particle? What about under the general Schrödinger equation where there exist potentials that seem to confine it? Is there a more general spread relation including potentials. Related to this question is the general solution for a single particle Schrödinger Equation...is there a simple Fourier transform type solution? Is the phenomena of spreading wavefunction simply eliminated by assuming boundary conditions? For example, in the Hydrogen atom solution what is the spread term if any? What are the boundary conditions for the hydrogen atom?

 

[azntoon]

The spreading of the wavefunction of a particle is a fundamental property of quantum mechanics, and it occurs regardless of whether the particle is free or interacting with a potential. The general solution for the single particle Schrödinger equation involves solving for the wavefunction in terms of the eigenvalues and eigenvectors of the Hamiltonian operator. It is not possible to obtain a simple Fourier transform solution in this case, as the eigenvectors form an orthonormal basis which does not lend itself to a Fourier transform.In the case of the Hydrogen atom, the wavefunction is usually obtained using the separation of variables technique, where the wavefunction is written as a product of two parts: one part that depends on the radial distance from the nucleus, and another that depends on the angular coordinates. In this case, the boundary conditions are determined by the physical requirements that the wavefunction must be finite at the origin, and must tend to zero sufficiently fast as the radial distance approaches infinity. The spreading of the wavefunction is again a fundamental property of quantum mechanics, and is accounted for by the wavefunction's dependence on the angular coordinates.