More Quantum Mechanics: Expectation Values

As I noted in a blog rebutting a fundie’s claim for a “precision universe”, that assumption fails because the cosmos at root is quantum mechanical in nature and not classical mechanical. This makes a world of difference, in particular, that we can’t hope to obtain exact or precise measurements but must use probabilities and uncertainties to get: “expectation values” (what we expect a position to be for example, in a given probability wave function configuration) or uncertainty computations.

In the case of expectation values, if the eigenfuction of an ensemble of particles is known (See Fig. 1), it may be used to calculate the average value that one would obtain when repeated measurements of that observable are made on the system. For any observable Q the expectation value, denoted as is given by the general formula (b) in Figure 1(which includes time,t - though sometimes dimension only is considered) . For the specific expectation value for a particle position in one dimension (say x- no dependence on time t) we need the form (c). Which we will soon compute.

Because the infinite square well (a) is bounded from 0 to L, we can replace the integral going from plus to minus infinity, with an easier to manage integral going from 0 to L.
This working to obtain the expectation value is given in Fig. 2 complete with the appropriate integrals, etc.

Note we take U = sin(kx) where k = pi/a, and U* = sin(kx) where k = pi/a (for n = 1 energy level) so:

UU* = sin^2(kx)

Quantum uncertainty and its computation, which we will get to in the next quantum instalment, measures the inherent spread in the results of measurements of an observable about the expectation value. The uncertainty in an observable (e.g. Q) with operator [Q] is given by the expression in Fig. 3. The key point is that if the uncertainty in the observable is zero, then the observable is said to be “sharp”. In other words, every measurement of Q made on the ensemble yields the same value. Thus there is no spread or distribution in the measurement.

Problem for next time: Consider the first excited state for an infinite square well, and obtain the expectation value in position (x) for it. Hint: the wave function U = sin (2pix/a).

Lastly, the energy level diagram for the solution to the previous problem is given in Fig. 4. The key step is to obtain the lowest energy state first, by getting: l’(max), s’(max) and j’(min). Thus: l’(max)= l1 + l2 = 0 + 1 = 1; s’(max) = ½ + ½ = 1 and j’(min) = [s’- l’] = 0. Then the lowest energy state is 3P_o. From LS coupling the other states of the multiplicity can easily be obtained (using s’ = 1, l’ =1)