Understanding Destructive Interference in Electron Wave Functions

[jalebi]

Hi guys,

I have a question regarding electron wave functions. If an electron's wave function describes the probability of it being at a specific position, how can two wave functions destructively interfere (as occurs in the formation of anti-bonding atomic orbitals)? I can understand how the wave functions (essentially probability density functions, right?) can interfere constructively since the individual probabilities are summed together. But in destructive interference, does one wave function have negative probability? I can't see how else destructive intereference of electron wave functions occurs.

 

[Bill_K]

The wave function is not a probability, it's a probability *amplitude*. The crucial fact about quantum mechanics is that it must be described in terms of a complex quantity , which indeed can be negative, causing destructive interference. The probability itself is |Ψ|², which cannot be negative.

 

[jalebi]

Ah okay, so the wavefunction squared is the actual probability.

I suppose it's probably best to just consider electron = wave function. The only problem I have with that is that sometimes you have to stop considering electron = wave function and go back to electron = particle (e.g. instantaneous dipoles). At what point do you go from the quantum approach to the "an electron is a particle that moves like a wave" approach?

 

[StevieTNZ]

So every particle 'darkens' an area of the screen in a double-slit experiment? Because if the particle were a wave and it destructively interfered, wouldn't that destroy the particle?

 

[ZapperZ]

Ah okay, so the wavefunction squared is the actual probability.

I suppose it's probably best to just consider electron = wave function. The only problem I have with that is that sometimes you have to stop considering electron = wave function and go back to electron = particle (e.g. instantaneous dipoles). At what point do you go from the quantum approach to the "an electron is a particle that moves like a wave" approach?

It depends on what the electron is interacting with, and how close are the electrons to other electrons. In particle accelerators, for example, the charged particles are often far apart enough that their individual wavefunctions don't overlap. So in such a case, they are often treated as classical particles.

A QM treatment of this interference can be found in the Marcella paper.

http://arxiv.org/ftp/quant-ph/papers/0703/0703126.pdf

Zz.

 

[PhilDSP]

So every particle 'darkens' an area of the screen in a double-slit experiment? Because if the particle were a wave and it destructively interfered, wouldn't that destroy the particle?

It would indeed destroy the particle if the only thing the particle consisted of was the wave in the equation. One major criticism directed at the sole use of the wave equations in association with particles which arose when the de Broglie relations were considered was that the wave very rapidly disperses (according to the equation) and the energy becomes non-localized. In other words, there's nothing in the model to account for the fact that the particle's energy is held together in a localized region.

 

[alxm]

So every particle 'darkens' an area of the screen in a double-slit experiment? Because if the particle were a wave and it destructively interfered, wouldn't that destroy the particle?

No, why? Neither light nor quantum mechanical waves are destroyed by "destructive interference". It's the intensity/probability that is 'destroyed' in that particular location - and compensated for by a greater intensity/probability somewhere else.