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Physics 614
As a prelude to the quantum mechanical formalism which we shall explore, it is useful to examine the basic tenets as outlined by Feynman in his pedagogical approach to the subject. The following material is extracted from Feynman's Lectures in Physics, Volume III, Chapter 1, 5. The reader is invited to consult the original source for more details.
Electron Diffraction
Consider electrons passing through a two slit system. If we had particles passing through we know the answer of what would occur - a probability pattern would emerge as shown in Fig. 1:.
(
WALL
Figure 1
'. PI: as shown is just the sum of the probabilities PI and P2 which would result if either slit #1 or slit #2 were blocked off, as indicated in Fig. 2. This is equivalent to saying that the particle went through either slit #1 or slit #2.
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Suppose now, however, that S is a light source. We know here too what would occur. If the slit separation is of the order of the incident wavelength we find an interference pattern, as shown in Fig. 3. x x
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Figure 3
The existence of such a pattern is assured since light is a wave and the field amplitudes E and B add rather than the intensities. Of course, if we blocked off either slit #1 or slit #2 we would find curves PI and P2 similar to those indicated in Fig. 2. However, if both slits are open the light passes through both slits and interferes as seen on the screen.
We can easily analyze the situation as follows. Assume the slit separation a is much less than the slit-screen distance L. Then
~-L
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Now consider what happens when the dipole is placed in a magnetic field which does not vary with y and is symmetric with respect to the yz plane, as drawn below.
s
Figure 7
Consider the force acting on a dipole placed in this field. The interaction energy is given by U = -p. B Thus
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Since there is no variation with y, Fy = O. Now since if a moment which enters the system is not aligned along one of the B lines it will precess, J.i is not fixed and must be averaged over. For a moment traveling along the symmetry axis, we have then ilx = ily = O. Then
- ( . 8Bz". 8Bz-: 8Bz ~ F=J.iz -J+-2+-k8y 8x 8z )
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i.e. particles are deflected in the z direction in amounts depending onA.
Although classically if an atom with a magnetic moment is passed through the device, there would be a continuum of possible values of fiz corresponding to differing angles of alignment with the z axis, quantum mechanically only discrete values for fiz are allowed. Hence atoms passing through such a device (called by the way a Stern-Gerlach apparatus) are separated into a discrete number of beams. It turns out that there are certain kinds of atoms (called spin 1 atoms) for which
n=+l,O,-l
Thus when such an atom passes through a Stern-Gerlach machine it must take one of three paths .as shown, depending on its value of n.
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Now imagine we build an improved Stern-Gerlach apparatus as shown below.
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It consists of three high gradient magnets. The first (on the left) is an ordinary Stern-Gerlach machine and splits the incoming beam into three separate beams (for a spin 1 particle). The second magnet is twice as long and has its polarity reversed. It binds the paths of the particles back toward the axis as shown. The third magnet
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