# Electron Wave Function

1. May 24, 2014

### Mikeal

I have read a number of books on quantum mechanics and I am now at peace with the idea that the wave-function of an electron instantaneously populates the universe with finite probabilities that the electron will be detected at a given point, if a measurement is conducted at that point. However, going back to the double-slit experiment, it would seem that, as the wave-function is instantaneous (i.e. greater than the speed of light), then an electron would be detected at the screen instantaneously. This seems to conflict with the concept that the electron cannot transit from the source to the screen faster than the speed of light. What am I missing?

2. May 24, 2014

### Matterwave

I wouldn't say the wave function instantaneously "populates the whole universe". The electron's wave function evolves according to the Schroedinger equation.

That being said, normal quantum mechanics is not Lorentz invariant, that a particle's wave function vanishes outside its lightcone is not a built in part of the theory. The Lorentz invariant version of quantum mechanics is quantum field theory.

3. May 24, 2014

### Staff: Mentor

Mate what you need to see is the QM explanation of the double slit experiment:
http://arxiv.org/ftp/quant-ph/papers/0703/0703126.pdf

The issue here is its often used to motivate the quantum formalism but then once that formalism is developed the texts do not go back and show how that formalism explains it. Its got nothing to do with wave functions travelling instantaneously etc etc.

Note the above paper is slightly simplified - I can post the link to a paper explaining those simplifications but as a beginner its not germane to the main point which is its got nothing to do with wave particle duality, particles going through both slits simultaneously etc etc. Its a simple application of the quantum formalism.

What is the rock bottom essence of that formalism. The answer depends on your mathematical sophistication, but at the beginner level the following is a good start:
http://www.scottaaronson.com/democritus/lec9.html

Basically its an extension of probability theory that allows continuous transformations between pure states. The argument goes something like this. Suppose we have a system in 2 states represented by the vectors [0,1] and [1,0]. These states are called pure. These can be randomly presented for observation and you get the vector [p1, p2] where p1 and p2 give the probabilities of observing the pure state. Such states are called mixed. Now consider the matrix A that say after 1 second transforms one pure state to another with rows [0, 1] and [1, 0]. But what happens when A is applied for half a second. Well that would be a matrix U^2 = A. You can work this out and low and behold U is complex. Apply it to a pure state and you get a complex vector. This is something new. Its not a mixed state - but you are forced to it if you want continuous transformations between pure states. What QM basically is is the theory that makes sense out of pure states that are complex vectors.

Thanks
Bill

Last edited: May 25, 2014
4. May 26, 2014

### Mikeal

Thanks. I am begining to get there by considering particular solutions to the time-dependent version of the Schroedinger equation. I also checked the references and that helped.