Bucknell Physics 212 Supplement

Section 5.4 Spin

We would now like to introduce a new property of particles that lends itself nicely to being described using the new mathematical formalism that we have just introduced. In addition to the angular momentum associated with a particle's motion about some origin, many types of elementary particles have an intrinsic angular momentum, as though the particle were a tiny sphere rotating about an internal axis. This internal angular momentum is given the name spin and is represented by the vector symbol \(\vec{S}\text{.}\) Every particle is characterized by a spin quantum number, \(s\text{,}\) and the magnitude and \(z\)-component of spin angular momentum are given by

An important issue for spin angular momentum is the allowed values associated with the quantum numbers \(s\text{.}\) These values can be determined from a complicated mathematical analysis whose details are beyond the level of this class. However, the interesting result is that the possible values for spin quantum numbers are integer multiples of \(1/2\text{,}\) i.e., \(s = 0\text{,}\) 1/2, 1, 3/2, 2, 5/2, …, which differ from the allowed values of \(l\) because the list now includes half-integer values. Electrons, protons, neutrons, and \(^3\)He nuclei all have the same spin quantum number \(s=1/2\) (sometimes referred to as “spin-\(\frac{1}{2}\)” ); photons have \(s=1\) (or “spin-\(1\)” ); He\(^4\) nuclei have \(s=0\) (or “spin-\(0\)”).

Every kind of particle has its particular value of \(s\) — for example electrons have \(s=1/2\text{.}\) This means that every electron in the universe has exactly the same intrinsic angular momentum magnitude: \(\vert \vec{S}\vert = (\sqrt{3}/2)\hbar\text{.}\) For classical, macroscopic-sized objects, we can change the spin angular momentum by changing the rotation rate of the object. However, in quantum mechanics we are stuck with a constant magnitude of spin angular momentum for elementary particles.

All elementary particles are classified according to their spin quantum numbers \(s\text{.}\) All particles with integer spin quantum numbers (\(s\) = 0, 1, 2, … ) are called bosons, and all particles with half-odd integer spin quantum numbers (\(s\) = 1/2, 3/2, 5/2, … ) are called fermions. This distinction is crucial in determining the behavior of systems consisting of many of these particles, as will be discussed in later chapters.

Equation (5.12) gives the possible values one can measure for the component of spin in the \(z\)-direction, \(S_z\text{.}\) For spin quantum number \(s\text{,}\) the possible values of \(S_z\) are given by \(m_s\hbar\text{,}\) where he quantum number \(m_s\) can assume values

For a spin–1/2 particle, for instance, \(S_z\) can be \(-\hbar/2\) or \(+\hbar/2\text{.}\) For a spin-1 particle, \(S_z\) can be \(-\hbar\text{,}\) 0 or \(+\hbar\text{.}\) For a spin-3/2 particle, \(S_z\) can be \(-3\hbar/2\text{,}\) \(-\hbar/2\text{,}\) \(+\hbar/2\text{,}\) or \(+3\hbar/2\text{.}\)

The points raised in the previous paragraphs should make your head spin (no pun intended)¹. There are a few issues here that are inherently quantum in nature:

• (1).

  The word spin glosses over just how strange this phenomena actually is. It is easy to envision, say, a basketball that is spinning on its axis, resulting in the ball having angular momentum. However, elementary particles can have spin angular momentum even if the particle has no spatial extent! The electron, for instance, is a point-particle (as far as our best measurements can determine), as are quarks, and yet these particles carry angular momentum. A photon doesn't even have any mass and yet it, too, carries angular momentum.

• (2).

  A spinning basketball could always be stopped, such that its angular momentum becomes zero. This is not the case for an elementary particle. An electron can't be stopped from “spinning” — the spin angular momentum is always there.

• (3).

  Although spin angular momentum is a vector with both a magnitude and direction, measuring components of that vector along different directions produces results that simply can't be explained classically. For instance, if a basketball were spinning, you could define a direction for the angular momentum \(\vec{S}\text{.}\) Measurement of the component of the angular momentum perpendicular to \(\vec{S}\) would produce a value of zero. This is often not the case for spin angular momentum for elementary particles. For instance, for an electron it is impossible to obtain a value of zero from any measurement of any component of spin angular momentum.

To illustrate some of these issues, we will introduce a device capable of measuring the \(z\)-component of spin, \(S_z\text{,}\) for a particle. This device is called a “Stern-Gerlach” (SG) device, named after the two physicists who first performed experiments like those we describe. Although the technical details of the device are not important to our discussion, the significant thing is that this device is capable of separating a beam of particles according to the \(z\)-component of their spins, where the \(z\)-axis is determined by the orientation of the SG device.

If we take a source of electrons with random spin component (say from a heated filament) and direct a beam of these electrons through a SG device to measure \(S_z\text{,}\) as shown in Figure 5.4, we find that the SG device splits the beam into two beams: one composed of electrons with \(S_z = +\hbar/2\) (called “spin-up” ) and the other with \(S_z = -\hbar/2\) (called “spin-down”). Approximately equal numbers of electrons appear in the two beams.

If we now direct the spin-up beam from the SG device to a second SG device, as shown in Figure 5.5, the second SG device outputs only a spin-up beam (\(S_z = +\hbar/2\)) and no spin-down beam. What does this mean? Obviously if you measure \(S_z = +\hbar/2\) in the first SG device, then if you measure \(S_z\) again you will always get the same result. This means that all of the electrons entering the second SG device are spin-up electrons, or are in the spin-up “state.”

Now imagine that we take the spin-up beam from the first SG device and direct it to a second SG device which measures the \(x\)-component of spin \(S_x\text{,}\) as in Figure 5.6. This can be accomplished by orienting a SG device along the \(x\)-axis as shown. What will be the result of the measurement made by this second SG device?

The measurement of \(S_x\) results in two beams with approximately equal count rates, one of \(S_x = +\hbar/2\) (spin-up along \(x\)) and the other of \(S_x = -\hbar/2\) (spin-down along \(x\)). This result is true regardless of whether we had used the spin-up or spin-down beam from the first SG device. Also if we send the electrons into the second SG device one at a time, what we observe from the output is an electron randomly coming out of the second SG device as either \(S_x = +\hbar/2\) or \(S_x = -\hbar/2\text{.}\) So the result of this experiment is that we have measured \(S_z\) to be \(+\hbar/2\) for an electron in the first SG device and \(S_x\) to be either \(+\hbar/2\) or \(-\hbar/2\) for the same electron in the second SG device.

Now we perform a very interesting experiment. As shown in Figure 5.7, we select electrons in the state spin-up along \(z\) in the first SG device and then select electrons in the state spin-down along \(x\) in the second SG device. If we now pass these electrons through a third SG device to measure \(S_z\) once again, what will be the result? Shouldn't we get just all electrons coming out in the spin-up (\(S_z = +\hbar/2\)) beam, since that is what we measured in the first SG device?

The answer is — “NO!” The measurement of \(S_z\) in the third SG device shows electrons coming out as either spin-up (\(S_z = +\hbar/2\)) or spin-down (\(S_z = -\hbar/2\)) with equal count rate. This certainly is not the classically expected result! It is as if the electrons lost all memory of being in the spin-up state from the first SG device.

This stuff should really bother you! There is simply no classical way to explain the observations (and these results are from experimental observations) about spin and its components. But that's the way in which sub-atomic particles work.