Question

A beam of spin‐1/2 particles is prepared in the state:     20 2 20 4  i a) What are the possible results of a measurement of the spin component Sz and with what probabilities would they occur? b) What are the possible results of a measurement of the spin component SX and with what probabilities would they occur? c) Suppose that the SZ measurement yields the result SZ = ‐ħ/2. Subsequent to that result a second measurement is performed to measure the spin component SX. What are the possible results of that measurement and with what probabilities would they occur? d) Draw a schematic diagram depicting the successive measurements as described in part c. Use the Stern‐Gerlach diagrams as drawn in the book showing the possible numbers of events when measured.

Answer 1

(a) If we measure the component Sz of the intrinsic angular momentum of a silver atom in its ground state, we can find only one or the other of two values, corresponding to the deflections HN1 and HN2. We have to reject the classical picture and have to conclude that: Sz is a quantized physical quantity whose discrete spectrum includes only two eigenvalues, +! 2 and ?! 2.

Theoretical description

We associate an observable Sˆz with Sz. According to the experiment it has to satisfy the eigenvalue equations:

Sˆz |+! = + ! 2 |+!

Sˆz |?! = ?! 2 |?!

where |+! and |?! are normalized and orthogonal and thus form a basis of the twodimensional Hilbert space of one

spin 1/2 particle

|+!#+| + |?!#?| = 1

PROBABILITIES:

• In this case, we get Z up or down with equal probabilities! Measuring X has erased our original measurement of Z. Similarly, if we started with a definite state of X and measure Z, we erase the original value of X. By a more complicated arrangement, we can measure the component of spin in the direction Y . If we do, we find that measuring Y erases the value of either X or Z; measuring X erases Y or Z; and measuring Z erases X or Y .
• Suppose that a spin is up in the Z direction. If we measure the spin component along an axis at angle ? to the Z axis, we find the spin up along that axis with probability pup = cos 2 (?/2). This is not special to the Z direction. If the spin is initially up along a given axis, measuring along an axis at angle ? has outcomes up or down with probabilities cos 2 (?/2) and sin 2 (?/2). We need a mathematical framework to encompass these results.
• Suppose we multiply the state ? by a pure phase exp(i?), ? ? ei??, ? ? ei??. This doesn’t change the probabilities for a measurement along the Z axis. Nor does it change the probabilities for a measurement along any other axis. This means that multiplying by a pure phase has no observable physical consequences. We say that the global phase of a state is arbitrary.

(b) POSSIBLE RESULTS OF A MEASUREMENT OF THE SPIN COMPONENT SX:

Classical measurement: As mentioned in the introductory chapter, the concept of measurement in quantum mechanics differs significantly from its classical counterpart. Classically, measurements reveal intrinsic properties or observables of the system, particle are classical observables.

Classical observables: x, p, L, and E.

Classical measurement postulates: Two key postulates for classical measurements that fail for quantum measurements involve the implicit assumptions that 1. Independent reality: Measurements reveal “elements of physical reality” that exist independent of the observation. In other words, measurements reveal information about properties possessed by a particle, which existed prior to the measurement, and we are simply ignorant of the value of the observable before we make the measurement. For example, classically we would say that measurement of the position of an electron bound to a hydrogen nucleus just tells us where the electron was before the measurement. 2. No disturbance: Measurements can be performed that do not disturb the system. In classical physics, all that is needed to do this is to make our measurement interaction sufficiently weak so that there is no disturbance of the system.

In classical physics, and most experiences we have in the macroscopic world, we do not see any difficulties with these postulates. We have already discussed the concept of measurement disturbance in quantum mechanics in terms of the uncertainty principle and Heisenberg microscope. The idea that by trying to determine the position of a particle with ever increasing precision (smaller uncertainty ?x), at the cost of gaining momentum uncertainty of a high energy (and thus high momentum) photon seems plausible to our classical ways of thinking. However, the idea that the position of the moon does not exist unless we look seems absurd to our classical intuition. As we will see, this is precisely what quantum mechanics prescribes.

Quantum measurement: In quantum mechanics measurements are represented by operators that act on the wave function ?(x, t). For example the observables of position, momentum, angular momentum, and energy correspond to the following operators.The position operator acting on a wave function results in multiplication by the coordinate x inside the argument of the wave function. The momentum operator acting on a wave function takes a spatial gradient (multiplied by ?i~). The orbital angular momentum operator follows from these and the corresponding classical definition. The Hamiltonian operator, or energy operator, acting on a wave function takes a time derivative (multiplied by i~). As mentioned in the previous chapter, the outcome of a particular measurement cannot in general be predicted. One can only talk about the probability to obtain a particular measurement outcome, which is governed by the wave function. This is a fundamental feature of quantum mechanics. Let us take the example of the particle confined to an infinite potential well, and consider the situation in which the particle is prepared in the symmetric superposition state.

Quantum Measurement Postulates

1. Single-value measurement outcome: When a measurement corresponding to an operator Aˆ is made, the result is one of the operator eigenvalues,

2. Wave function collapse: As a result of a measurement yielding eigenvalue an, the wave function ‘collapses’ into the corresponding eigenstate of the measurement operator

?(x, t) ? ?n(x)

3. Outcome probability: The probability of a particular measurement outcome equals the squared modulus of the overlap between the wave functions before and after the measurement. For example, the probability to obtain measurement outcome an, corresponding to the eigenstate ?n(x) is given by

Pn(t) = |h?n|?(t)i|2 = Z ? ?? ? ? n (x)?(x, t)dx )^2 .

(c) We now have a mathematical description of that special relationship we saw in Chapter 1 between a physical observable, Sz say, the possible results ± ! 2 , and the kets ± corresponding to those results. This relationship is known as the eigenvalue equation and is depicted in Fig. 2.1 for the case of the spin up state in the z-direction. In the eigenvalue equation, the observable is represented by an operator, the eigenvalue is one of the possible measurement results of the observable, and the eigenvector is the ket corresponding to the chosen eigenvalue of the operator. The eigenvector appears on both sides of the equation because it is unchanged by the operator. This eigenvalue equations for the Sz operator in a spin-! system are:

Sz + = ! 2 +

Sz ! = ! ! 2 !

To determine the matrix representing the operator Sz, assume the most general form for a 2 * 2 matrix

(d) STERN-GERLACH DIAGRAMS

The projection postulate is at the heart of quantum measurement. This effect is often referred to as the collapse (or reduction or projection) of the quantum state vector. The projection postulate clearly states that quantum measurements cannot be made without disturbing the system (except in the case where the input state is the same as the output state), in sharp contrast to classical measurements. The collapse of the quantum state makes quantum mechanics irreversible, again in contrast to classical mechanics. We can use the projection postulate to make a model of quantum measurement, as shown in the revised depiction of a Stern-Gerlach measurement system in Fig. 2.6. The projection operators act on the input state to produce output states with probabilities given by the squares of the amplitudes that the projection operations yield. For example, the input state ! in is acted on the projection operator P+ = + + , producing an output ket ! out = + ( ) + ! in with probability !+ = + ! in 2 . The output ket ! out = + ( ) + ! in is really just a + ket that is not properly normalized, so we normalize it for use in any further calculations. We do not really know what is going on in the measurement process, so we cannot explain the mechanism of the collapse of the quantum state vector. This lack of understanding makes some people uncomfortable with this aspect of quantum mechanics, and has been the source of much controversy surrounding quantum mechanics. Trying to better understand the measurement process in quantum mechanics is an ongoing research problem. However, despite our lack of understanding, the theory for predicting the results of experiments has been proven with very high accuracy.