Usefulness of the momentum wavefuction in solving Schrodinger's Equation

[piareround]

So my atomic physics professor was doing a review today of quantum mechanics. One of slightly odd things he mentioned was how most of the time we solve differential quantum equation, like Schrödinger's Time Dependent Equation, using a position wave-function rather a momentum wave-function because momentum version tends to be a higher order ODE than the position version.

Is there ever a case in Quantum or Atomic physics when it is more advantageous to use the momentum wavefunction rather than the position wavefunction? In other words it there any time where the momentum representation of the differential equation is simpler than the position representation?

 

[dextercioby]

The free, non-relativistic particle case is trivial in the momentum representation. The wavefunction is just a phase factor times a delta distribution.

The atomic potentials (e.g. H-atom), however, look awful in the momentum representation.

 

[A. Neumaier]

So my atomic physics professor was doing a review today of quantum mechanics. One of slightly odd things he mentioned was how most of the time we solve differential quantum equation, like Schrödinger's Time Dependent Equation, using a position wave-function rather a momentum wave-function because momentum version tends to be a higher order ODE than the position version.

Is there ever a case in Quantum or Atomic physics when it is more advantageous to use the momentum wavefunction rather than the position wavefunction? In other words it there any time where the momentum representation of the differential equation is simpler than the position representation?

What he said is really odd. Probably he referred to most of his time only.

In scattering calculations and for relativistic field theory problems, one generally prefers the momentum representation, whereas in bound state multiparticle calculations, the position representation is usually superior. Of course, in atomic physics (but only there), the latter dominate.

 

[maverick_starstrider]

In my experience most of quantum is done in either the momentum or energy basis. If you're concerned with any sort of perturbative force you are going to be concerned with how that perturbs the energy eigenstates not the position one. Second Quantization usually favors creation operators over field operators because it's simpler and any form of quantum on a lattice will deal with momentum (band theory and fermi surfaces and such). I'd say it's quite rare to see the POSITION basis.

 

[A. Neumaier]

In my experience most of quantum is done in either the momentum or energy basis. If you're concerned with any sort of perturbative force you are going to be concerned with how that perturbs the energy eigenstates not the position one. Second Quantization usually favors creation operators over field operators because it's simpler and any form of quantum on a lattice will deal with momentum (band theory and fermi surfaces and such).

There are creation operators both in position and in momentum representation (namely Fourier transforms of each other). The position representation of the creation operator for photons is widely used in quantum optics, where one often needs to do spatially resolved calculations.

I'd say it's quite rare to see the POSITION basis.

Your view is just the opposite extreme of that of the atomic physics professor, and equally one-sided - this time viewed solely from the quantum field theory perspective.

In the much used GAMESS package (http://www.msg.chem.iastate.edu/gamess/gamess.html) for nonperturbative atomic and molecular quantum computations (and in similar packages), all wave functions are represented in position space. The reason for using the position representation is that neutral atoms and molecules interact by short-range interactions, and are always strongly localized. Localization is very difficult to represent in momentum space.

It is only when working with perturbations of free fields (or particles) in perturbation theory that the momentum representation is essential, namely to compute the terms of the Born series.

Thus both representations are very important and have complementary uses.