# Wavefunction of a single particle

If I fire a stream of electrons at a CRT screen, the electrons go pretty much where I point them (this is how CRT Tv's work).
But from reading popular science books I am led to believe that until the electron hits the screen, it has a range of possible positions and can in theory appear in any number of positions, defined by it's wavefunction.
Why is it then that they appear where they are fired and not spread out randomly?

Even if you fire electrons at the CRT screen, even then there is a range of possible positions. Engineers are taking care about their "range of positions" to make sure that it is such that the screen works according to the specifications. If they would fire electrons too slow, it could stop working.

Matterwave
Gold Member
The spread is small enough that you don't really notice.

The reason I ask is that in the double slit experiment, the interference pattern is quite wide on the screen. And the pattern is caused by two wavefunctions overlapping. But why, if one slit is closed, is the width of the image on the screen not the same, albeit without the interference pattern? I think I am right in saying that we just get a dot on the screen.

Because in a CRT, there is no diffraction taking place.

The narrowness of the slit through which the electrons pass through determines the wideness of the diffraction pattern. If the slit were very narrow, the electrons would leave a very wide diffraction pattern.

The slit itself is what somehow removes information about momentum -- and part of momentum is direction. In a diffraction experiment, this uncertainty in momentum is amplified by the distance to the screen to produce a visible effect.

In a CRT, on the other hand, the uncertainty is kept largely minimal:

1) The initial velocity (including direction) of the ejected electron is within certain reasonable bounds.
2) The initial position of the ejected electron is within very small bounds.
3) By using an electromagnetic "funnel", the final position of the electron can be ensured to within tight bounds, regardless of reasonable variations in the velocity.

You can imagine how a magnifying glass can bring light from many different points on the sun to a very, very small point on a piece of paper. You could make that point arbitrarily small, if you like, but the smaller you make it, the less you know about where the light will go after it passes through the hole in the paper. You know, for example, that all the light from the sun striking the (large) magnifying glass will end up at that one single point -- this is because you know exactly the momentum of each photon (as the sun is very far away)! but light from that one (small) single point (if it keeps going) could end up anywhere in a very large region - this is basically because of an uncertainty in the momentum.

[PLAIN]http://www.antonine-education.co.uk/Physics_A2/options/Module_5A/Topic_1/Ray_1.gif [Broken]

Thus the "funneling" of the CRT takes advantage of the very sure initial position of the electron to ensure almost as well that the electron ends up at the proper final position. What is lost, however, is information about momentum. Electrons leaving one way or another will be guided by the electromagnetic field to land at the same point. But we don't care about that!

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We may consider the uncertainty relation:

$${\Delta}{x}{\Delta}{p}{\geq}{\hbar}{/}{2}$$

If the spread in the momentum values is large, the spread in the position coordinate will be very small l.It would be interesting to consider the travel of the electron/electrons from the source to the target through vacuum in the absence of photons. To ensure a pointed beam, the standard deviation in the values of momentum has to be large.
If the uncertainty in momentum is zero[as it happens in Bose Einstein Condensation,of course at very low temperatures]the uncertainty in position will become infinitely large.Of course in this case we have fermions--but uncertainty in position would become infinitely large if a constant momentum beam is considered.
For a single particle moving with constant momentum, the wave speed should exceed the speed of light.

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Observations:
1) For a constant momentum beam along the x-axis, x coordinate can be anything,considering the large value of delta_x.
2)If delta_y and delta_z are zero, p(y) and p(z) can assume any value, considering the unboundedness in the values of delta_px and delta_py.

If a small range of momentum values is considered[in the x-direction], one may view the beam as a wave stretching between the source and the target covering a huge number of values in the x direction. This conforms to the notion that larger the extent of a function wrt to the independent variable the smaller is its spread on the Fourier domain.
A large range of momentum values should favor the particulate nature of the beam while a small range favors the wave nature of the beam. The second formulation is more suitable for the problem at hand.

It would be convenient to assume a small thickness of the beam perpendicular to the x axis.This thickness should be macroscopically small but microscopically large enough to make delta_py and delta_pz sufficiently small.

[The momentum range in the x direction should be much smaller than the corresponding ranges along the y and the z directions]

uncertainty principle doesn't mean you can not determine the position of something infinitely precise but it says if you do so you will perturb momentum infinitely large.

it should also be noted that the particle will be a wave packet and that these packets follow classical laws on average. though I do find this question rather intriguing.

You didn't understand this theory about quantum physics very well. Suppose the point from where you fire the electron is point A, and the point on the screen where you aim the electron is point B. This theory states that the particle takes an infinite number of routes from point A to B. But it always lands on point B. So when you fire the electron, it takes every possible path to the screen. Besides there are no slits like there were in the double-slit experiment. So there is no interference pattern. This theory doesn't state that the particle (in this case the electron) may hit any point. The point from where the particle is released and the point from where the electron hits is defined. But the particle takes every possible path to get there. So from the point the particle is fired and the screen, the particle takes every path to the screen and ultimately reaches the point towards which the particle was aimed at.