Quantum Physics
Average value
Cont'd from "Dirac Notation: Intrinsic properties"
Average of a function
A function f(x) has an average value ⟨f(x)⟩ found by integration over the range R occupied by the function, divided by the range it occupies:
Hence, a function (infinite or otherwise) f(x) has an average value over the range a to b of
Example:
Consider the function f(x) = cos(x) + 1. What is its average value in the range -π ≤ x ≤ +π?
First, let’s look at the function and find what we expect the average to be.
From looking at the graph we can see the average is the mid-point between the maximum and minima. Since the difference between these two points is 2, we expect the average to be ⟨f(x)⟩ = 1. Now we can carry out the calculation to see if this is true.
Recall the equation for the average of a function between limits a = −π and b = +π
Applying the sum rule for integration, we get
We can integrate each part to get
and compute the limits
and so, we find that
as expected!
Although we could have found this by inspecting the graph, this method is powerful because it allows us to find the average of any integrable function between any valid limits – including those which would be near-impossible to find by eye.
Average of a discrete set
A discrete set S(n) of N total values sn, defined generally as
or
has an average ⟨S(n)⟩ given by
In turn, we can find the average of a range, from the ath value to the bth within this set:
where S(a ≤ n ≤ b) ⊂ S(n).
Example:
Consider the following set H of 4 discrete values
This could be the height (in feet) of 6 different people, for example. So, what is the average height ⟨H⟩ of these people?
We know that the average height of people is around 6 ft – so, if this is a wholly representative group of people we should expect the average to be around 6. However, by simply looking at the range of values within the set, we expect that the average will be slightly lower than this expectation.
Mathematically, we must hence find the average of the set H, found by
where hn is value of the nth element. So,
which is what we expected from looking at the set and also confirms our assumption that this group is not representative of the entire population due to their ’less-than-average’ mean height (5% lower than expected).
Measurement error:
It is worth noting that since we’re working with a physical quantity, ’height’, we should consider the measurement error on the value, which will be
where σ⟨H⟩ and σhn are the errors on the expectation value and measured values of the height, respectively. We can assume that the measuring device was accurate to the nearest 0.1 ft since the measurements are given to that accuracy, implying that σhn = 0.05.