# An example of using the Uncertainty Principle

Suppose that we create a beam of electrons, all flying in the x-direction with some speed vx; let's make this speed small compared to the speed of light, so that we can ignore relativistic effects. That will simplify the math a bit.

We shine this beam of electrons at a wall which has a slit of width d in it.

What happens when the electrons pass through the slit? There are two ways we can approach this problem:

• use deBroglie's hypothesis to treat the electrons as waves, and then apply the classical diffraction formula
• use Heisenberg's ideas about uncertainties

Let's try the uncertainty method first. Before the electrons reach the wall, they are moving purely in the x-direction. That means that their velocity in the y-direction must be zero; and, if their motion is truly in the x-direction alone, the uncertainty in their velocity in the y-direction must also be zero. Thus, the uncertainty in their y-momentum must also be zero.

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Q:  If the uncertainty in y-momentum is zero,
what do we know about the y-position
of each electron before it strikes the wall?

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The electrons reach the wall. Most hit the wall and are absorbed or reflected, but some pass through the slit and continue to the right.

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Q:  What do we know about the y-position of
any electron which passed through the slit?

What constraints does this place on the
y-momentum of the electrons which pass through
the slit?

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Looking at the electrons which passed through the slit, we see that although the average y-momentum is zero, there is now a small range of values around zero.

There are several different ways one can describe the width of this distribution. Let's pick a simple one: the typical magnitude of the momentum in the y-direction is just

That means we can now draw the direction of an electron which is at a typical angle from the center:

```

Q:  What is the momentum in the x-direction for these
electrons?

Q:  Assume that the angle theta is small, so that
the y-momentum is much smaller than the x-momentum.
Using deBroglie's formula, express the electron's
momentum in terms of its wavelength.

Q:  Derive an expression for the typical angle, theta,
by which the beam of electrons will spread.

```

Okay, that's one way to look at it. Suppose that we instead look at this experiment using deBroglie's formula to turn the electrons into waves, and then apply classical diffraction theory.

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Q:  What is the wavelength of the electrons
before they strike the wall?

Q:  At what angle theta will the first dark
spot appear in a classical diffraction pattern?

```

Aha! The answer is basically the same, whether we approach the problem using uncertainty, or using diffraction theory. That's a good sign that the uncertainty principle is correct.