Reference: Griffiths, David J. (2005), Introduction to Quantum Mechanics, 2nd Edition; Pearson Education – Problems 1.1 – 1.2.
Having reached the end of the problems in Griffiths’s book, I’ll rewind and return to chapter 1 to fill in the gaps that I left in my rush to get into the meat of quantum mechanics.
Probability is at the heart of quantum mechanics, so it’s a good idea to be sure we understand some of the basic concepts. Variables in quantum mechanics come in both discrete and continuous forms, so we’ll do a quick review of the average, variance and standard deviation in these cases.
In the discrete case, the average of a set of values is just the sum of all the values divided by the number of values. To make this precise, suppose that value occurs times; then the average of is
The average (also called the mean) is distinct from the median value, which is the value the divides the set into two equal subsets, with one subset containing values less than the median and the other containing values greater than the median. (Obviously this works exactly only if we have an even number of values, but you get the idea.)
Having calculated the average, we can then calculate the deviation from that average for each value . It turns out that the square of the deviation is more useful, so we have
The variance is the mean of these squares:
and the standard deviation is the (positive) square root of the variance:
The probability of a value being chosen at random from the set is
Example 1 Suppose we have a group of people with ages as follows:
age 14 1 15 1 16 3 22 2 24 2 25 5
For this set, we have
The median age is between 22 and 24 (say, 23) since there are 7 people with ages greater than 23 and 7 with ages less than 23.
We can now add a column for :
age 14 1 15 1 16 3 22 2 1 24 2 3 25 5 4
We can check equation 5
For continuous distributions, we need to use integrals instead of sums to calculate the various averages. To calculate the average of a function , we need the probability density which gives the probability that the value of lies in the interval . Given this, we then have
Some special cases of this formula are used to calculate the average and variance:
A probability density must be normalized so that if :
Example 2 Extending Griffiths’s example 1.1. A rock is dropped off a cliff of height . Neglecting air resistance, the position of the rock at time (using as the top of the cliff and as the bottom) is
The time taken for the rock to hit the bottom of the cliff is
Now suppose that the position of the rock is randomly sampled a large number of times during its fall. If the sampling times are truly random, the probability of a sample being taken in any time interval is constant and since the probability that a given sample occurs sometime between and must be 1, we have
The average location of the rock over all the samples is then
The standard deviation is
To get the probability density for position instead of time, we have
The probability of a sample being taken when is more than away from the average is
We could also work this out without first finding by finding the times at which from 19 and then integrating over the corresponding time range.