Relevance to Quantum Mechanics
As described inChapter 4, the projection operator is useful in determining the probability of measurement outcomes for a quantum observable by projecting the state of a system onto the eigenstates of the operator for that observable.
average over many systems, not an average over time (and it is certainly not the value you expect to get from a single measurement).
If that sounds unusual, think of the average score of all the soccer matches played on a given day. The winning sides might have an average of 2.4 goals and the losing sides an average of 1.7 goals, but would you ever expect to see a final score of 2.4 to 1.7? Clearly not, because in an individual match each side scores an integer number of goals. Only when you average over multiple matches can you expect to see non-integer values of goals scored.
This soccer match analogy is helpful in understanding why the expectation value is not the value you expect to get from an individual measurement, but it lacks one feature that’s present in all quantum-mechanical observations. That feature is probability, which is the reason most quantum texts use examples such as thrown dice when introducing the concept of the expectation value. So instead of thinking about averaging a set of scores from completed matches, consider the way you might determine the expected value for the average number of goals scored by the winning side over a large number of matches if you’re given a set of probabilities. For example, you might be told that the probability of the winning side scoring zero goals or more than six goals is negligible, and the probabilities of scoring one to six goals are shown in this table:
Winning side total goals 0 1 2 3 4 5 6
Probability (%) 0 22 43 18 9 5 3
Given this information, the expected number of goals (g) for the winning team can be determined simply by multiplying each possible score (call itλn) by its probability (Pn) and summing the result over all possible scores:
g = N n=1
λnPn, (2.56)
so in this case
g =λ0P0+λ1P1+λ2P2+ · · · +λ6P6
=0(0)+1(0.22)+2(0.43)+3(0.18)+4(0.09)+5(0.05)+6(0.03)
=2.4.
To use this approach, you must know all of the possible outcomes and the probability of each outcome.
This same technique of multiplying each possible outcome by its proba- bility to determine the expectation value can be used in quantum mechanics.
To see how that works, consider a Hermitian operator O and normalized wavefunction represented by the ket|ψ. As explained inSection 1.6, this ket can be written as the weighted combination of the kets representing the eigenvectors of operatorO:
|ψ =c1|ψ1 +c2|ψ2 + · · · +cN|ψN = N n=1
cn|ψn, (1.35) in whichc1throughcN represent the amount of each orthonormal eigenfunc- tion|ψnin|ψ. Now consider the expression
ψ|O|ψ,
which, as described previously, can be represented as the inner product of|ψ with the result of applying operatorOto ket|ψ. Applying the operatorOto
|ψas given byEq. 1.35yields O|ψ =O
N n=1
cn|ψn = N n=1
cnO|ψn = N n=1
λncn|ψn, (2.57) in whichλnrepresents the eigenvalue of operatorOapplied to eigenket|ψn.
Now find the inner product of|ψwith this expression forO|ψ. The bra ψ|corresponding to|ψis
ψ| = ψ1|c∗1+ ψ2|c∗2+ · · · + ψN|c∗N= N m=1
ψm|c∗m,
in which the index m is used to differentiate this summation from the summation ofEq. 2.57. This means that the inner product(|ψ,O|ψis
ψ|O|ψ = N m=1
ψm|c∗m N n=1
λncn|ψn
= N m=1
N n=1
c∗mλncnψm|ψn.
But if the eigenfunctions ψn are orthonormal, only the terms with n = m survive, so this becomes
ψ|O|ψ = N n=1
λnc∗ncn= N n=1
λn|cn|2= o. (2.58)
This has the same form asEq. 2.56, with|cn|2 in place of the probability Pn. So the expressionψ|O|ψwill produce the expectation valueoas long as the square magnitude ofcnrepresents the probability of obtaining resultλn. As you’ll see inChapter 4, that’s exactly what|cn|2represents.
The expressions for the expectation values presented in this section can be extended to apply to situations in which the outcomes may be represented by a continuous variablexrather than discrete valuesλn. In such situations, the discrete probabilitiesPnfor each outcome are replaced by the continuous probability density function P(x), and the sum becomes an integral over infinitesimal incrementsdx. The expectation value of the observablexis then
x = ∞
−∞xP(x)dx. (2.59)
Using the inner product, the expectation value can be written in Dirac notation and integral form as
x = ψ|X|ψ = ∞
−∞[ψ(x)]∗X[ψ(x)]dx, (2.60) in whichXrepresents the operator associated with observablex.
In quantum mechanics, expectation values play an important role in the determination of the uncertainty of a quantity such as position, momentum or energy. Calling the uncertainty in positionx, the square of the uncertainty is given by
(x)2= x2
− x2, (2.61)
in which x2
represents the expectation value of the square of position (x2) and x2represents the square of the expectation value ofx.
Taking the square root of both sides ofEq. 2.61gives x=
x2
− x2. (2.62)
As you can see on the book’s website, for a distribution of position valuesx, (x)2is equivalent to the variance ofx. That variance is defined as the average of the square of the difference between each value ofxand the average value ofx(that average is the expectation valuex):
Variance ofx=(x)2≡,
(x− x)2 -
, (2.63)
which means thatx, the square root of the variance, is the standard deviation of the distributionx. Thus the uncertainty in position xmay be determined
using the expectation value of the square ofxand the expectation value ofxin Eq. 2.62. Similarly, the uncertainty in momentumpis given by
p= p2
− p2, (2.64)
and the uncertainty in energyEmay be found using E=
E2
− E2. (2.65)
Main Idea of This Section
The expression ψ|O|ψ gives the expectation value of the observable associated with operatorOfor a system in quantum state|ψ.
Relevance to Quantum Mechanics
When Schr¨odinger published his equation in 1926, the meaning of the wavefunction ψ became the subject of debate. Later that year, German physicist Max Born published a paper in which he related the solutions of the Schr¨odinger equation to the probability of measurement outcomes, stating in a footnote that “A more precise consideration shows that the probability is proportional to the square” of the quantities we’ve calledcn
inEq. 2.58. You can read more about the “Born rule” inChapter 4.