Are wavefunctions specifically describing electrons?

[lonewolf219]

Hello,

Wave functions and energy levels... What particles are being described by Schrödinger's equation? If we are talking about energy levels, do we mean the energy levels in an atom? Other particles, such as electrons or quarks... do they have energy levels, or are they too small for us to know?

 

[Jolb]

In Quantum Mechanics, Schrödinger's equation describes any particle. (Remember though that Schrödinger's equation is a nonrelativistic equation, so without going to relativistic quantum mechanics--e.g. the Dirac equation--the particles that are described are the nonrelativistic ones like electrons, protons, neutrons, quarks, etc. but not photons.)

Schrödinger's equation dictates the behavior of a particle in the presence of a potential. The energy levels, or the different energy eigenstates of a potential, depend on the form of the potential and the intrinsic properties of the particle in question (mass, charge, etc.) Each energy eigenstate has a given energy and is associated with a wavefunction which gives the probability amplitude for finding a particle in any region of space.

As an archetypal example, the Hydrogen atom's energy levels can be calculated from the Schrödinger equation by 1: reducing the proton-electron system (a two-body problem) to an "reduced mass" in a static potential, just like in orbital mechanics and then 2: solving for the energy eigenstates of this static potential (for a particle with mass equal to the reduced mass) using the Schrödinger equation.

In other words, a particle only gets energy levels available to it when it's in some kind of potential. The energy levels can be discrete or continuous depending on the form of the potential. A free particle, or one in a trivial potential, has energy eigenstates which are plane waves of any energy. Only a particle in a potential well, e.g. a particle in a 1/r attractive potential, takes on a discrete spectrum of energy levels.

In elementary quantum mechanics, the potential that appears in the Schrödinger equation is a classical potential--exactly like the ones in classical mechanics. Once you start trying to deal with the quantum details of a real interaction--for example trying to account for the fact that an electron in a hydrogen atom is actually interacting with the proton via the exchange of photons--you are forced into Quantum Field Theory.

 

[lonewolf219]

Thanks, Jolb, I appreciate your detailed explanation! That was very helpful!

 

[aim1732]

Just a small technical detail but the wave function, being governed by the potential field,should describe a system(as potential energy is always mutual to a system).So the correct statement is that the wave function describes the electron-proton system in the Coulomb interaction.Actually the reduced mass approach is exactly about reducing the two body problem with a one body problem.

 

[Jorriss]

Electrons aren't even 'described' by the Schrödinger equation in entirety. Electrons have an extra property associated with them called their spin and the Schrödinger equation doesn't include this - we have to patch it on after the fact.

 

[Jolb]

Well Joriss, the spin does enter the Schrödinger equation if there is any coupling to the spin. If there isn't coupling, then the spin does little besides adding copies of each energy eigenstate. But in general the spin does couple and the Schrödinger equation gets a spin term, and the eigenenergies depend on the spin (which I tried to imply when I mentioned the 'intrinsic properties of the particle in question.') But yes, the electron's state consists of the wavefunction (the spatial part) together with the spin state or spinor (the intrinsic part.)Aim, there are often experiments where thinking of a system is rather contrived, and it's good enough to just say the potential. Imagine a Zeeman effect experiment: an unpaired electron pinned in a magnetic field. Is there any use in saying that's really a system of one electron interacting with ~10²³ electrons confined to a solenoid consisting of ~10²³ copper atoms connected with all the atoms in the power grid? Why not just say it's an external field giving an external potential?

Edit: On second thought, I can see why for philosophical reasons, one might stress always keeping in mind the idea of a "system" rather than the idea of a particle in some "external" potential/field when thinking about Quantum Mechanics. It might be very important to some people that you could always include a system's surroundings into the system and make the problem into to an "isolated" system, and maybe likewise for the entire universe--you might be very impressed with the idea that the universe as a whole and everything in it follows some sort of Schrödinger evolution... But at that level you'd better be talking about quantum field theory rather than quantum mechanics.

 

[Salman2]

Here is the first observed image of the electron wavefunction for the hydrogen atom:

http://io9.com/the-f...struc-509684901

The paper published in Phys Rev Lett, vol 110, issue 21, 2013: abstract:
http://prl.aps.org/abstract/PRL/v110/i21/e213001

 

[lonewolf219]

Wow, that's amazing... Thanks for posting Salman2!

 

[Khashishi]

The wavefunction describes anything and everything. You can have a wavefunction for a particle, or a wavefunction for two particles, or an indeterminate number of particles. It's a very general, powerful tool.