interference | prob amplitude | complex amplitude | wavefn / orbital
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The electron diffraction experiments described previously show that predictions about electrons must be stated in terms of probabilities. Now we use a slightly modified form of these experiments to show that the rules for calculating these probabilities is not at all what we expect for a "particle", but more like what we expect for a "wave".
The new experiment uses the apparatus shown below. Notice that the screen now contains two small holes separated by a small distance.
Given the fact that diffraction through a single hole must be described using probabilities, the same must be true for diffraction from two holes. Since there are two independent paths by which the electron can arrive at the detector (diffraction from the "upper" hole, or diffraction from the "lower" hole), we expect the number of electrons reaching the detector to be higher. Also, we expect the probability, P, of an electron reaching a particular detector position to be the sum of two diffraction probabilities, P(diffraction from lower hole) and P(diffraction from upper hole):
P = P(lower) + P(upper)
This prediction is displayed in the following graph. The dashed lines show what is actually observed when ONLY the upper or lower hole is open, and the heavy solid line represents what we expect to observe when both holes are open.
Of course, so far we have been only been describing "expectations", and not the real behavior. What do we really observe in a "two small hole" experiment?
The real observations are shown in the following graph. The number of electrons oscillates with detector position; large numbers of electrons are detected at some positions, while no electrons are detected at intervening positions. The latter result ("no electrons") is impossible to explain in classical terms because electrons are detected at these positions when only ONE hole is open. Somehow, opening BOTH holes lowers the probability of an electron reaching these positions.
Like the simpler diffraction experiments, this behavior was first observed in analogous "two hole" experiments involving light beams. The oscillating pattern is known as an "interference" pattern, and it was taken as conclusive evidence that light was a "wave" (the "no light detected" result was explained by saying that waves coming out of the two holes simultaneously cancel each other out at certain detector positions). Electrons are not "waves" (and cannot go through the two holes simultaneously), but it would appear that the mathematical laws for predicting electron behavior must be similar to those used by classical physics to predict wave behavior.
Quantum mechanics is able to predict the result of the two hole experiment by introducing two new rules for calculating probabilities. These rules are not the ones based on everyday experience (see above), but rely instead on something called a "probability amplitude".
The probability amplitude, which I will denote using the Greek letter "phi" (ø), is mathematically analogous to a classical wave. The value of ø is similar to the "height" of a wave, and can be either positive or negative. Also, it is possible for different ø to cancel each other out, just like two waves.
Quantum mechanics states that P is related to ø by:
These rules correctly predict the results of the two hole experiment because they allow P(two hole) to be less than or greater than P(one hole) (recall that our "classical" expectation had been that P(two hole) would be greater than P(one hole) regardless).
P(two hole) is given by:
P(two hole) = [ø(upper) + ø(lower)]2
where ø(upper) and ø(lower) are the probability amplitudes for the two ways the electron can reach the detector, i.e., by diffraction from the upper or lower holes respectively. If ø(upper) and ø(lower) have opposite signs they will cancel each other out, and P(two hole) will be small, or even zero. On the other hand, if ø(upper) and ø(lower) have the same sign, P(two hole) will be larger than P(one hole).
The next step is to introduce the quantum mechanical rules for calculating ø. This is covered on the next few pages, but first a few more properties of ø must be introduced.
I have stated that probability amplitudes can be both positive and negative numbers, and that the amplitudes are converted into probabilities by squaring them. These statements are only true, however, if the probability amplitudes are real numbers.
It is also possible for a probability amplitude to be a complex number, such as ø = A + iB, where "i" is the square root of -1. In this case, we define the "complex conjugate" of ø to be ø*, where ø* = A - iB (note: all you have to do to make a complex conjugate is flip the sign of the imaginary part; leave everything else alone).
When ø is complex, the correct quantum mechanical rules become:
Note that only the first rule is affected. Also, note that P is always real and positive:
P = ø(ø*)
= (A + iB)(A - iB)
= A2 + B2 = real
Finally, note that our earlier rule, which was based on "real" ø, is just a special case of the rule for "complex" ø. That is, if ø is real then ø* = ø, so P = ø2.
Chemists refer to probability amplitudes as "wavefunctions" or "orbitals", and I will now adopt this practice.
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(last updated 6/7/97)