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apparent that it will be necessary to perform infinite sums when calculating probabilities or expectation values of measurement outcomes. The following example indicates that this is frequently possible to actually do this.
Example: Suppose that for a quantum mechanical system the energies are
En = E 1 2 n 2 and that the system is in the state
∑^ ∞
n=
A
3 n^ |φn〉
where
|φn〉
are the energy eigenstates. Determine A, the probability with which an energy measurement yield the outcome En for any n and the expectation value of the energy measurements.
Answer: The normalization condition gives:
∑^ ∞
n=
A
3 n
2 = 1.
Thus
1 = |A|^2
∑^ ∞
n=
32 n
= |A|^2
∑^ ∞
n=
9 n
= |A|^2
[
]
The term in brackets is a geometric series and the following applies
1 + r + r^2 =... =
1 − r provided that |r| < 1. Thus
1 = |A|^2
= |A|^2
which leaves |A| =
Answer: and the most convenient choice is
A =
Thus the normalized version of the state is
∑^ ∞
n=
3 n−^1 |φn〉.
Now the probability of any given outcome for an energy measurement is
Pr (En) = |〈φn|Ψ〉|^2
To this end
〈φn|Ψ〉 = 〈φn|
∑^ ∞
m=
3 m−^1
|φm〉
∑^ ∞
m=
3 m−^1
〈φn|φm〉
∑^ ∞
m=
3 m−^1 δnm
3 n−^1
Thus Pr (En) =
9 n−^1
The expectation value is
〈E〉 =
∑^ ∞
n=
En Pr (En)
∑^ ∞
n=
E 1 2 n 2
9 n−^1
=
E 1
∑^ ∞
n=
)n
E 1
[
]
E 1
E 1
Note that the expectation value could also be computed using the energy observable,
∆x ∆x ∆x ∆x ∆x ∆x ∆x
x− 3 x− 2 x− 1 x 0 x 1 x 2 x 3
Figure IV.1.2: Discrete position measurement. The axis is divided into bins of length ∆x. The bins can be labeled by counting from the centermost bin (centered at x = 0) outwards. Bins to the right will be labeled by positive integers, j = 1, 2 ,... , and those to the left by negative integers, j = − 1 , − 2 ,.... Note that the center of the bin labeled j is at xj = j∆x.
forms the basis for assigning position values for this type of discretized position measure- ment. If the particle is located in bin j during a position measurement, then it’s position value is said to be xj. Thus the possible outcomes for a classical position measurement are
... x− 1 , x 0 , x 1.... A quantum mechanical description of this measurement begins with a description of the position measurement outcomes and the associated states. These are listed in Table IV.1.2. As with other types of measurements on quantum mechanical systems, the state |xj 〉 has
Position Associated State .. .
x− 2 |x− 2 〉
x− 1 |x− 1 〉
x 0 |x 0 〉
x 1 |x 1 〉
x 2 |x 2 〉 .. .
Table IV.1.2: Discretized position measurement outcomes and associated states.
the physical interpretation that if the position of the particle is measured then, the outcome will be that the particle is located in bin j with certainty. In terms of position values the outcome will fall in the range (xj − ∆x/ 2 , xj + ∆x/2) and we refer to this by saying that that the outcome of the position measurement will be xj with certainty although this really just refers to the label at the center of the bin. These states are referred to as position
states. Outcomes that lie within distinct bins will not be confused and thus the position states are orthonormal, i.e. 〈xi|xj 〉 = δij. (IV.1.15)
The position states form a basis for the vector space consisting of all states and this is indicated by the completeness relation
∑^ ∞
j=−∞
|xj 〉 〈xj | = I.ˆ (IV.1.16)
For a general state |Ψ〉 ,
|Ψ〉 = Iˆ |Ψ〉
=
∑^ ∞
j=−∞
|xj 〉 〈xj | |Ψ〉
∑^ ∞
j=−∞
|xj 〉 〈xj |Ψ〉. (IV.1.17)
The complex numbers Ψ(xj ) := 〈xj |Ψ〉 (IV.1.18)
form the components of |Ψ〉 in the basis
... , |x− 2 〉 , |x− 1 〉 , |x 0 〉 , |x 1 〉 , |x 2 〉 ,...
, i.e.
∑^ ∞
j=−∞
Ψ(xj ) |xj 〉. (IV.1.19)
Again the state |Ψ〉 dictates the probabilities for discretized position measurement out- comes. Thus Pr (xj ) = |〈xj |Ψ〉|^2. (IV.1.20)
Exercise: Show that if |Ψ〉 =
∑^ ∞
j=−∞
Ψ(xj ) |xj 〉
then Pr (xj ) = |Ψ(xj )|^2. (IV.1.21)
One consequence of Eq. (IV.1.21) is that
∑^ ∞
j=−∞
|Ψ(xj )|^2 = 1
which is equivalent to the fact that the state must be normalized, i.e.
〈Ψ|Ψ〉 = 1. (IV.1.22)
Thus
Pr
xjlow < x < xjhigh
j ∑high
j=jlow
Ψ(xj )∗Ψ(xj )
j ∑high
j=jlow
〈xj |Ψ〉∗^ 〈xj |Ψ〉
j ∑high
j=jlow
〈Ψ|xj 〉 〈xj |Ψ〉
j ∑high
j=jlow
|xj 〉 〈xj |
|Ψ〉 (IV.1.27)
and in this sense the operator j ∑high
j=jlow
|xj 〉 〈xj |
is associated with position measurements in the entire range xjlow < x < xjhigh.
1.2 Position Measurements: Continuous approach
The resolution of position measurements can always be improved; this amounts to decreas- ing the size of the bins, ∆x, in the discretized version. The mechanisms described for the discretized quantum description above, apply whenever ∆x 6 = 0. Is there an underlying description when ∆x = 0? In a thorough mathematical treatment there is such a descrip- tion but the mathematical tools used in the discretized version of position measurements have to be modified substantially. Nevertheless, the basic rules for calculation have many features similar to those of the discretized version. The set of all position states is { |x〉 | all real x
Note that although the labels inside the kets are real numbers, they are still just labels, and they are not explicitly involved in typical real number calculations. Thus
| 2. 5 〉 + | 3. 6 〉 6 = | 6. 1 〉.
Additionally there is an important conceptual caution: it is impossible to prepare a particle in any of these states |x〉. Thus they do not have any reasonable physical interpretation in terms of the outcomes to position measurements. The role of these states is to support superpositions, from which quantities pertaining to measurement outcomes can be calculated. In superpositions over states with continuous labels, the sums must be replaced by integrals. Thus the general state will have the form
−∞
Ψ(x) |x〉 dx (IV.1.28)
