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SG ˆz
Sz = +ℏ/ 2
Sz = −ℏ/ 2
SG xˆ
Sx = +ℏ/ 2
Sx = −ℏ/ 2
SG zˆ
Sz = +ℏ/ 2
Sz = −ℏ/ 2
SG zˆ
Sz = +ℏ/ 2
Sz = −ℏ/ 2
Figure 1.9: Three successive Stern-Gerlach measurements for the circum- stance in which the particle emerges with Sz = +ℏ/2 after the first SG ˆz measurement.
Applying the rules of (1.6) one can deduce that the probability with which the particle emerges with Sz = +ℏ/2 after either of the latter SG zˆ measurements is 1/ 2. Similarly the probability with which the particle emerges with Sz = −ℏ/2 after either of the latter SG zˆ measurements is 1/ 2.
Exercise: Consider the arrangement of Fig. 1.9. Suppose that the particle emerges from the SG xˆ measurement with Sx = +ℏ/ 2. Show that the probability with which the particle emerges from the subsequent SG zˆ measurement with Sz = +ℏ/ 2 is 1/ 2. Repeat this for a particle that emerges with Sz = −ℏ/2 is 1/ 2. Re- peat this entire analysis for the case where the particle emerges from the SG xˆ measurement with Sx = −ℏ/ 2.
Exercise: Repeat the argument of the previous exercise for the case where the particle emerges with Sz = +ℏ/2 after the first SG zˆ measurement.
The implication is that it makes sense to speak of a particle as having a definite value for Sz in the context where the only subsequent operations are SG zˆ measurements. However whenever subsequent operations include SG nˆ measurements where nˆ is distinct from zˆ (or −zˆ), it is impossible to speak of a particle as having a definite value of Sz. Thus is quantum mechanics it is meaningful to speak of a particle as having a definite value for Sn only in certain very restricted contexts. One cannot universally ascribe a definite value of Sn to a particle.
1.3 Maximal description of physical states
It is possible to consider measuring more than one component of spin via successive Stern- Gerlach apparati as illustrated in Fig. 1.10.
SG xˆ
Sx = +ℏ/ 2
Sx = −ℏ/ 2
SG yˆ
Sy = +ℏ/ 2
Sy = −ℏ/ 2
SG yˆ
Sy = +ℏ/ 2
Sy = −ℏ/ 2
Figure 1.10: Successive Stern-Gerlach measurements in an attempt to mea- sure orthogonal spin components.
Any spin-1/2 particle to which these are applied will yield values for Sx and Sy. But this joint outcome can only be considered a measurement of a property of the particle if, when repeated again to a system that has undergone no extra measurements or interactions, it yields exactly the same value as it did previously.
Exercise: Consider the double SG experiment as illustrated in Fig. 1.10 and a spin-1/ particle that emerges from the uppermost output beam. Suppose that this is then reapplied to the same apparatus. Show that it does not emerge from the uppermost beam with certainty.
The preceding exercise indicates that two SG apparati cannot be combined to jointly measure two orthogonal components of spin in the sense that the measurement outcome is not repeatable. Thus, in the context of SG measurements it is not sensible to label the physical state of a particle via |+ mˆ, +nˆ〉 where mˆ and mˆ are not parallel; such a state would indicate that a combined SG mˆ and SG ˆn measurement would yield Sm = +ℏ/2 and Sn = +ℏ/2 with certainty. However, a general argument similar to that of the preceding exercise indicates that this is not possible. Thus, in terms of repeatable measurement outcomes, the only physically meaningful states are |+nˆ〉 and |−nˆ〉 where nˆ is any unit vector. The collection can be simplified by noting that, in terms of SG nˆ measurement outcomes, |−zˆ〉 is equivalent to |+(−nˆ)〉.
Exercise: Show that |+(−nˆ)〉 yields Sn = −ℏ/2 with certainty. To do so define mˆ = −ˆn and determine the probabilities with which |+ mˆ〉 yields Sn = ±ℏ/2.
Thus the collection of all possible physically distinct states of spin-1/2 particles is {|+nˆ〉} where nˆ ranges through all unit vectors.
It is important to note that the mean is an idealized quantity and that a given run of measurements will not necessarily yield the mean when averaged. Suppose that the outcomes of an experiment involving N measurements are s 1 , s 2 , s 3 ,... sN. This constitutes a sample of the probability distribution and the sample average is defined as:
m :=
N
∑^ N
i=
si. (1.11)
Typically for large N one expects that m ≈ 〈m〉.
Example: For an unbiased die, in one trial consisting of ten rolls, the following outcomes occurred: 2, 4 , 4 , 3 , 3 , 2 , 1 , 4 , 4 , 6. The sample average of these is 3.3 which differs from the mean, 3. 5.
Various theorems in probability theory quantify the extent to which a sample average approximates the mean. Generally as N increases the probability with which the sample average will be within a given range of the mean increases; the probability with which it is beyond the a certain range of the mean diminishes as 1/N. An important tool in quantifying such discrepancies and fluctuations away from the mean is the standard deviation or variance of a probability distribution. This is defined as:
∆m :=
∑n
i=
pi (mi − 〈m〉)^2 (1.12)
and this quantifies the extent to which measurement outcomes typically deviate from the mean. A general result that simplifies this calculation is:
∆m :=
〈m^2 〉 − 〈m〉^2 (1.13)
where 〈 m^2
∑^ n
i=
m^2 i pi.
These notions can be applied to physical systems subject to the laws of quantum me- chanics.
Example: Consider an ensemble of spin-1/2 particles each in the state | mˆ〉 where mˆ = √^1 2 xˆ^ +^ √^1 2 yˆ.^ Each particle is subjected to an SG^ xˆ^ measurement. Determine 〈Sx〉 and ∆Sx.
Answer: We can list the measurement outcomes and probabilities:
Outcome (Sz ) Probability
m 1 = +
p 1 =
m 2 = −
p 2 =
where we have used
Pr (Sx = +ℏ/2) =
(1 + mˆ · nˆ)
Pr (Sx = −ℏ/2) =
(1 − mˆ · nˆ)
etc,.... Thus
〈Sz 〉 =
∑^ n
i=
mipi
Then
∆Sz =
〈S z^2 〉 − 〈Sz〉^2.
Here
〈 S z^2
∑^ n
i=
m^2 i pi
ℏ^2
Thus
∆Sz =
ℏ^2
ℏ^2
