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Averages. There are many:
2. arithmetic mean
3. geometric mean
4. root mean square
5. harmonic mean
The probability density r = dp/dx. To understand why you compute averages the way the text says to, go back to the definition of probability in terms of numbers of experiments. Divide the whole domain into intervals, Dxi, and the average of x is then estimated by
where x*1 is some point in the first interval. The quotients N1 / N is Dp1, the probability to be in the first interval. In the limit that the intervals Dxi approach zero, this is an integral of x dp, denoted by < x >.
If you need to find the average value (arithmetic mean) of x2, you simply use x2 wherever you had x above.
3 Apr Means and standard deviations. If <x> denotes the mean of a set of data, the quantity
is a measure of how wide the distribution of data is. The square root of the above number is denoted Dx, the "standard deviation."
Compute both of these quantities for three different probability densities and see why the results behave as they do. Estimate the corresponding answer for one that I didn't compute.
Apply the same idea to the diffraction of light through a slit. It spreads out, and a simple analysis shows how the width of the central maximum in the intensity pattern is related to the width of the slit.
Translate the preceeding analysis of light into the language of photons and the probability density describing where the photons will hit the wall. By looking at the transverse component of the momentum of the photon, I came up with a simple relationship between the standard deviation in the values of py and the standard deviation in y, the position where the photon passed through the slit. The product is a constant, a multiple of Planck's constant.
The general relationship is Heisenberg's uncertainty principle
This applies not just to photons but to electrons, as they both obey p = h / l.
There is a similar relationship between the perceived frequency of a musical note and its duration. If it is too short it will sound like a click and will not have a discernible pitch. The corresponding relationship in this case is
5 Apr Spend the entire time on trying to understand these probability densities as applied to the hydrogen atom. Various types of means and even modes.