# Usefulness of the momentum wavefuction in solving Schrodinger's Equation

• piareround
In summary, the conversation discussed the use of position and momentum wavefunctions in solving differential quantum equations. It was mentioned that in most cases, position wavefunctions are preferred due to their simpler form. However, there are cases where momentum wavefunctions may be advantageous, such as in scattering calculations and relativistic field theory problems. Both representations have their own strengths and are important in different situations.

#### 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 Schrodinger'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?

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.

piareround said:
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 Schrodinger'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.

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.

maverick_starstrider said:
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.

maverick_starstrider said:
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.

## 1. How is the momentum wavefunction useful in solving Schrodinger's Equation?

The momentum wavefunction is useful in solving Schrodinger's Equation because it allows us to obtain information about the momentum of a particle in a quantum system. This information is crucial in understanding the behavior and dynamics of the particle.

## 2. Can the momentum wavefunction be used to determine the position of a particle?

No, the momentum wavefunction cannot be used to directly determine the position of a particle. It only provides information about the momentum of the particle. To determine the position, the position wavefunction must be used.

## 3. How is the momentum wavefunction related to the position wavefunction?

The momentum wavefunction and the position wavefunction are related through the Fourier transform. The momentum wavefunction is the Fourier transform of the position wavefunction, and vice versa.

## 4. Can the momentum wavefunction be used for any quantum system?

Yes, the momentum wavefunction can be used for any quantum system, as long as the system can be described by Schrodinger's Equation. It is a fundamental concept in quantum mechanics and is applicable to all quantum systems.

## 5. What other information can be obtained from the momentum wavefunction besides momentum?

Besides momentum, the momentum wavefunction can also provide information about the spread of momentum values, the average momentum, and the uncertainty in the momentum of a particle in a quantum system. This information is essential in understanding the behavior and properties of the particle.