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5.2.1 The Stern-Gerlach experiment

Stern-Gerlach (SG) experiments in 1921-1922 _{) abandonment of Classical Mechanics} concepts.

• Description of the experiment (see Fig. 5.1)

Use of47Ag: massive _{) can use trajectory, force, and so forth.}

Non-uniform field along the z-axis ) measurement of z component of magnetic moment µ_{! Since}

µ = e mec

~

S (5.2.1)

where e < 0 is the electron charge, me is its mass, c is the speed of light, and ~S

is the physical quantity denoted spin, or intrinsic angular momentum; hence the spin is a measure of the particle’s magnetic moment (more on this throughout this course). _{! denote this by ‘SGˆz experiment’.}

The particles in the beam hit the detector: this provides distribution of final positions, according to the projections of ~_{S along the SG axis.}

Figure 5.1: The Stern-Gerlach experi- ment. [From Sakurai, MQM, Fig. 1.1.]

Figure 5.2: Beams from the SG exper- iment (schematic): (a) classical expec- tation; (b) actual observation. [From Sakurai, MQM, Fig. 1.2.]

• Expectation from classical point of view:

Random orientation of atoms ) no preferential direction of ~S ) outcome should be a continuous distribution of hits around zero, corresponding to all possible values between _{|µ| and +|µ|; see Fig. 5.2(a).}

• Actual observation:

Two spots are detected, centred around positions corresponding to _Sz =±~/2.

Note that ˆz is not special: any direction the SG apparatus pointed to (e.g., ˆx), would produce the same result, namely _Sx=±~/2.

Let us introduce a spin operator S associated with the physical quantity ~S; when no possibility of ambiguity arises, from now on we will indistinctly use S as meaning both the physical quantity and the operator.

5.2.2 Sequential Stern-Gerlach apparatus

Let us now discuss the e↵ects of directing a bean through more than one SG apparatus, as in Fig. 5.3:

(a) Two SGˆz in sequence: after going through the first apparatus, the beam with say, Sz = ~/2 is blocked; the beam with Sz = +~/2 then passes through the second

SGˆz, from which only the beam with Sz = +~/2 emerges.

(b) SGˆz and SGˆx in sequence: after going through the first apparatus, the beam with say, Sz = ~/2 is blocked; the beam with Sz = +~/2 then passes through the SGˆx

apparatus, which splits the beam between Sx=±~/2 with equal intensities.

(c) SGˆz, SGˆx, and SGˆz in sequence: we follow the steps of (b), but after emerging from the second (SGˆx) apparatus, the Sx = ~/2 is blocked; the beam with Sx = +~/2

then passes through the second SGˆz apparatus, which splits the beam between Sz=±~/2 with equal intensities.

5.2. STERN-GERLACH EXPERIMENTS 89

Given that particles with Sz = ~/2 had been blocked by the first SGˆz apparatus,

and reappear after the second SGˆz, we are led to conclude that in quantum me- chanics, one cannot determine both Sx and Sz simultaneously; or, equivalently, the

selection of particles with Sx= +~/2 by the second apparatus, completely destroys

any previous information about Sz.

Figure 5.3: Sequential SG experiments; see text. [From Sakurai, MQM, Fig. 1.3.]

5.2.3 Construction of spin states and operators

In view of the outcomes of a single SGˆz experiment, and of spectral decomposition, we may write for the Sz-operator,

Sz = ~

2|+ih+| ~

2| ih |, (5.2.2) where we adopt the notation _{|±i for the spin states with eigenvalues ±~/2 of S}z. We

see that Sz is, by itself, a CSCO, and on the|±i basis can be represented by the matrix

Sz= ~ 2 ✓ 1 0 0 1 ◆ ⌘ ~ 2 z, (5.2.3)

where z is one of the Pauli matrices (see below).

In order to build Sx, we recall that the +~/2 beam emerging after the SGˆx apparatus

in Fig.5.3(c) is split into two equal-intensity beams with Sz eigenvalues ±~/2; therefore,

the probability of|+ix being found in |±i is 50% each,1

|h+|+ix| = |h |+ix| =

1 p

2. (5.2.4)

1_{Eigenkets of S}

We may therefore write |+ix= 1 p 2[|+i + e i 1_{| i], (5.2.5)} and, by orthogonality, | ix= 1 p 2[|+i e i 1_{| i], (5.2.6)}

where 1 is to be determined. The spectral decomposition of Sx then yields

Sx= ~

2[|+ix xh+| | ix xh |] = ~

2[e

i 1_{|+ih | + e}i 1_{| ih+|]. (5.2.7)}

A similar analysis for Sy leads to

|±iy = p1 2[|+i ± e i 2_{| i], (5.2.8)} and Sy = ~ 2[e i 2_{|+ih | + e}i 2_{| ih+|]. (5.2.9)}

In order to determine 1 and 2, we make the beam go through SGˆx followed by a

SGˆy. Analogously to what we concluded before, we must have

|yh±|+ix| = |yh±| ix| =

1 p

2. (5.2.10)

Taking Eqs. (5.2.5),(5.2.6), and (5.2.8) into (5.2.10), leads to

1 2|1 ± e i( 1 2)_{| = p}1 2 ) 2 1 =± ⇡ 2. (5.2.11)

Therefore, the matrix elements of Sx and Sy cannot be all real; if we take those of

Sx as real, then the matrix elements of Sy will be complex. Accordingly, let us choose 1 = 0, which in turn imposes 2 = +⇡/2 (the choice 2 = ⇡/2 would lead to a

left-handed system of coordinates). In summary, we obtain |±ix = p1 2[|+i ± | i] (5.2.12) |±iy = 1 p 2[|+i ± i| i], (5.2.13) and Sx = ~ 2 ✓ 0 1 1 0 ◆ ⌘ ~ 2 x, and Sy = ~ 2 ✓ 0 i i 0 ◆ ⌘ ~ 2 y, (5.2.14) thus defining the other Pauli matrices, x and y.

5.2. STERN-GERLACH EXPERIMENTS 91

Figure 5.4: Definition of the polar angles used to define ˆu. [From CT, Fig. IV.I.]

The spin operators are hermitian, and their eigenkets form a basis in this two- dimensional state space; it is therefore an observable, as expected. Using their matrix representations, we can also note that the spin operators satisfy the following commu- tation relations

[Sx, Sy] = i~Sz, [Sy, Sz] = i~Sx, and [Sz, Sx] = i~Sy, (5.2.15)

which can be regarded as circular permutations (i.e., x_{! y ! z ! x) of each other.} Having discussed the projections of the spin ~_{S along the x, y, and z directions, let} us now generalise to an arbitrary direction, characterized by the unit vector ˆu; this situation is equivalent to pointing the SG apparatus along ˆu. From Fig. 5.4, we see that

ˆ

u = sin ✓ cos ' ˆx + sin ✓ sin ' ˆy + cos ✓ ˆz, (5.2.16)

so that

Su = ~S · ˆu = Sxsin ✓ cos ' +Sysin ✓ sin ' +Szcos ✓. (5.2.17)

The replacement ~_{S ! S allows one to write the corresponding expression for the arbi-} trary u-component of the spin operator,

Su = S· ˆu = Sxsin ✓ cos ' + Sysin ✓ sin ' + Szcos ✓, (5.2.18)

whose matrix representation (in the basis of Sz eigenstates) is

Su= ~ 2 ✓ cos ✓ e i'sin ✓ ei'_{sin ✓ cos ✓} ◆ . (5.2.19)

The eigenvalues are, certainly,_{±~/2, and the corresponding eigenvectors are}

|+iu= e i'/2cos ✓/2|+i + ei'/2sin ✓/2| i (5.2.20)