How do i visualize probability in double slit

[jimmylegss]

I know an electron when not interfered with after leaving it's source is not really in any space. It just has a probability of being in various places, it isn't real yet.

But how do i visualize how that happens? Let's say over 500 million Planck lengths in a vaccuum.

Does the probability move like a wave from where it left to the wall? Let's say it moves 90% light speed, so it would take 450 million Planck time units to get to the other side (this is average right?, almost never precise, even when distance would be precise. It would be purely random how many Planck time units it would take, if it would take longer it would arrive with more energy, and if it would take shorter then it would arrive with less energy?).

Would the most likely place to find after 225 million time Planck units exactly (or roughly, but still relatively precisely) be in the middle of this box with the length of 500 million Planck lenghts (so at 250 million Planck lenghts)?

Would the peak of this probability wave (so where it is most likely to be) move from where it left to the other side of the wall in a linear fashion, sort of like a regular sound or water wave? How do i visualize how these probabilities move in time?

Same with after you would 'intercept' this electron half way through. The superposition would collapse, and it would move as a particle. Does that mean the quantum weirdness dissapears, or simply that the probability wave becomes very narrow? And after the observation the electron goes back to it's quantum weirdness? Except in a very narrow wave form so it looks as if it behaves as a particle now.

Finally how much would the width of this space matter (still with the same 500 million Planck length unit length as before)? Would it make a big difference if it was also 500 million Planck lengths wide, or let's say 100 billion Planck lenghts wide?

Im still really bad at math, so i find it hard to extract this from the equations yet.:)

Thanks in advance.

 

[Simon Bridge]

I know an electron when not interfered with after leaving it's source is not really in any space. It just has a probability of being in various places, it isn't real yet.

But how do i visualize how that happens?

you don't, its just maths.
also, "not really in any place" does not make sense... you are certain to detect it in some place so it is really there.

Plank units are note useful for your discussion, they are a destraction.

Does the probability move like a wave from where it left to the wall?

No. The probability distribution is always everywhere. The wavefunction changes shape like a wave does.

Would the peak of this probability wave (so where it is most likely to be) move from where it left to the other side of the wall in a linear fashion, sort of like a regular sound or water wave? How do i visualize how these probabilities move in time

You are thnking of like the electron is confined between two walls?
How the wavefunction changes shape depends on the details of the electron state. It can look like a pulse boyncing back and forth, or a traveling wave, or it may be a stationary state so the peeak does not change position.

Same with after you would 'intercept' this electron half way through. The superposition would collapse, and it would move as a particle. Does that mean the quantum weirdness dissapears, or simply that the probability wave becomes very narrow? And after the observation the electron goes back to it's quantum weirdness? Except in a very narrow wave form so it looks as if it behaves as a particle now.

No. The electron always obeys quantum mechamics.
it never moves as a wave or as a particle but as itself.
the wave stuff is how we predict the results of experiments, the particle stuff is what we measure.

Same with after you would 'intercept' this electron half way through. The superposition would collapse, and it would move as a particle. Does that mean the quantum weirdness dissapears, or simply that the probability wave becomes very narrow? And after the observation the electron goes back to it's quantum weirdness? Except in a very narrow wave form so it looks as if it behaves as a particle now

the electron always obeys quantum mechanics, this may or may not be "weird" depending on how you look at it.
The wavefunction always occupies all space.

finally how much would the width of this space matter (still with the same 500 million Planck length unit length as before)? Would it make a big difference if it was also 500 million Planck lengths wide, or let's say 100 billion Planck lenghts wide?

The dimensions of the container determines the available energy states. The use of plank lengths here is probably misleading, but a container too small may not fit the electton.

 

[jimmylegss]

Thanks for the response.

No. The probability distribution is always everywhere. The wavefunction changes shape like a wave does.

What determines it shape? I guess i don't use the Planck lenghts, instead I just use time units and a cube of one by one metre.

I shoot an electron from the middle of one of the walls with speed of 90% light speed of this hollow cube to the other side , and aim at the exact middle at the other side. It is a vacuum. If I split this experiment up in 10 even units of time between leaving and hitting the other side, what would the wave function look like? Is there some website you can insert values to kind of see for yourself?

You are thnking of like the electron is confined between two walls?
How the wavefunction changes shape depends on the details of the electron state. It can look like a pulse boyncing back and forth, or a traveling wave, or it may be a stationary state so the peak does not change position.

how would you get these different looking shapes in time? Does the electron state means how much energy you give the state, and does it depend on the environment it is 'shot' in from whatever atom it left? Later in your post you say that the size of the container influences how the probability wave will look. Is that determined instantaneously after the electron leaves the atom? Sort of like with quantum entanglement?

 

[Simon Bridge]

I shoot an electron from the middle of one of the walls with speed of 90% light speed of this hollow cube to the other side , and aim at the exact middle at the other side. It is a vacuum. If I split this experiment up in 10 even units of time between leaving and hitting the other side, what would the wave function look like? Is there some website you can insert values to kind of see for yourself?

I don't knownof any such website.
The setup above would be well described by classical mechanics ànd is similar to what you get inside a crt ... only bigger.
In qm we say that the electron is prepared with a well defined initial position and momentum, the exact distribution determined by the source, which you have not described.

We would be able to represent the initial state mathematically as a superposition of energy eigenstates (or whatever seems convenient for the calculation we need to do) for the box. We'd then use the time dependent Schrödinger equn to time-evolve the total wavefunction. The likely result would have the principle max propagate much as you'd think the classical particle would. More precisely, the expectation value of position would propagate as the classical particle.

how would you get these different looking shapes in time?

carefully. A common approach is to use electrons confined to a metal and use a laser to excite particular states.

Does the electron state means how much energy you give the state, and does it depend on the environment it is 'shot' in from whatever atom it left? Later in your post you say that the size of the container influences how the probability wave will look. Is that determined instantaneously after the electron leaves the atom? Sort of like with quantum entanglement?

the state of a particle is the minimum description of what you need to know about it ... this will include the energy.
It is determined by the totality of the influences on it... so you are not actually being specific enough. In an atom, the state is often described by 3 or 4 quantum numbers.
How the probability waves looks is mathematics and is independent of the presence of the electron itself. The wavefunction is not physical so it does not need a cause.

 

[robphy]

This might be a good place to start:

Teaching Feynman’s sum-over-paths quantum theory
Edwin F. Taylor, Stamatis Vokos, John M. O’Meara and Nora S. Thornber
Comput. Phys. 12, 190 (1998); http://dx.doi.org/10.1063/1.168652
http://scitation.aip.org/content/aip/journal/cip/12/2/10.1063/1.168652

http://www.eftaylor.com/download.html#quantum

 

[Simon Bridge]

Or Feynmans own description... but these are not wave mechanics are they?
Per the title, the double slit experiment, the probability distribution does not change with time during the experiment.
The idea that some electron wave travels through the slits is a common misunderstanding.

 

[jimmylegss]

Thanks for the response, allthough that last part is not entirely clear. Let's say the electron leaves location A, and before it arrives, location B (where the electron was aimed at) is moved to a different location further away (or closer to) A. Does this affect the probability distribution of where to find the electron between A and B before it actually arrives at B?

 

[Simon Bridge]

The probability of detecting an electron depends on the position of the detector, type of detector, and the geometry of the setup... moving the detector about adds a dynamic component to the calculation. If you were to move the detector to a new location before the electron could possibly have reached the old one, there would be no difference to the way the probability was calculated... it would be whatever the probability at the new location would have been anyway, if the detector was there all along. The distribution is not usually thought of as affected by the detector... unless you put the detector inside one of the slits - which changes the geometry of the experiment.