Theoretical Physics : The Stern-Gerlach experiment and the qubit model of the spin

The Stern-Gerlach experiment and the qubit model of the spin

The Stern-Gerlach experiment

The Stern-Gerlach experiment is one of the greatest experiments in physics, atomic physics and quantum mechanics.

This experiment plays also a major role in the foundation and philosophy of quantum mechanics. The Stern-Gerlach apparatus can be used to prepare the initial state of the system, i.e. it allows us to select any desired polarization for the initial state. More importantly, the Stern-Gerlach experiment provides an almost idealized conceptual model for the process of quantum measurement as we will discuss below.

It was designed originally to decide between Larmor's classical theory and Sommerfeld's old quantum theory describing the motion of charged particles in magnetic fields. As it turns out both theories are in fact wrong and the correct description is given by quantum mechanics.

It was precisely designed to test the so-called "space quantization", i.e. the quantization of the angular momentum suggested originally by Bohr.

It led immediately to the discovery of the "spin" which is a purely quantum property of the electron.

In the Bohr-Sommerfeld model we imagine the atom as a positively charged dense nucleus with negatively charged electrons moving in orbits around the center.

In order for the electrons to not spiral towards the center, as they radiate energy in the electric field of the nucleus, Bohr restricted their motion to specific orbits, called orbitals, determined by some integer (the principal quantum number $n$). This is called orbital quantization and it allowed Bohr to explain why atoms emit and absorb radiation only under a discrete form, i.e. at a discrete set of electromagnetic frequencies. In fact, Bohr was also successful in deriving the celebrated Rydberg formula for these frequencies.

Bohr's model relied on the fundamental assumption that the angular moment of the electron must be quantized. We write this in the form

\[\oint p_{\varphi}d\varphi=n_{\varphi} h.\]

Thus, in the ground state of the atom the electron must have only two values of the angular momentum along any direction in space corresponding to $n_{\varphi}=\pm 1$.

This prediction was called space quantization and Stern devised his experiment to test it directly. In fact, Stern was very sceptical about this prediction and he went in his scepticism as far as to raise a wager to quite physics if Bohr's quantization turns out to be correct. Stern was shortly joined by Gerlach who was instrumental in the success of the experiment. Yet, it is Stern who won the Nobel prize in physics on this work (called the molecular ray method) in 1943.

As we will see this one assumption, which seemed to enrage Stern, turns out to be wrong yet the beams of atoms prepared in the ground states (such as the silver and the hydrogen atoms used in the experiments) are fund to suffer a splitting into two streams in a non-uniform magnetic field as predicted by Bohr (but for the wrong reason). The orbital angular momentum is indeed quantized but not according to the above rule and the Stern-Gerlach experiment which sought to discover this space quantization ended up discovering the spin quantum number and vindicating the new theory of quantum mechanics.

We can also add to the above quantization of the angular momentum, following Sommerfeld, the following radial quantization

\[\oint p_r dr=n_rh.\]

This will extend Bohr's original circular orbits to elliptical orbits. These quantization conditions govern the so-called Bohr's correspondence principle which states that in the limit of large quantum numbers $n_{\varphi}$ and $n_{r}$ the predictions of the old quantum theory should reduce to those of classical physics.

In the Stern-Gerlach experiment (1922) a beam of hot silver atoms (characterized by a single unpaired electron) is sent through a non-uniform magnetic field. After going through the magnets the beam reaches a detector plate. The atoms, as we will see, are characterized by a dipole magnetic moment and thus as they move in the non-uniform magnetic field they will experience a torque which acts differently on the two ends of the dipoles leading to a net force on the atoms. This causes the atoms to deflect differently, in the non-uniform magnetic field, according to the magnitude of their magnetic moments.

This experiment was repeated with hydrogen atoms (which are characterized by a single electron) by Phipps and Taylor in 1927.

Both silver and hydrogen are neutral atoms found in the ground state $l=0$ (as opposed to the assumption of the Bohr-Sommerfeld old quantum mechanics which predicts $l=1$). Yet, in both cases the Stern-Gerlach experiment shows that the beam of atoms splits in fact into two beams, as originally thought by Bohr and Sommerfeld but not for the right reasons, signaling therefore the existence of another quantum property (the so-called spin angular momentum with value $s=1/2$ which was first proposed first by two graduate students Goudsmit and Uhlenbeck and then confirmed later by Dirac in his quantum relativistic theory of the electron). In the case of the hydrogen atom the extra spin angular momentum is associated with its single electron whereas in the case of the silver atom the extra spin quantum number is associated with its unpaired outermost electron.

Thus, the Stern-Gerlach experiment invalidates both classical mechanics and the old quantum theory. The Stern-Gerlach experiment is a major vindication of the new quantum theory of Heisenberg, Schrodinger, Dirac, Pauli and Bohr.

In general if a beam of atoms with total angular momentum $\vec{J}=\vec{L}+\vec{S}$ is prepared in the orbital ground state $l=0$ then allowed to go through the Stern-Gerlach appartus it will be observed to split into $2s+1$ beams corresponding to the eiegnvalues $m_s$ of the spin angular momentum ${S}_z$ which are given by $m_s=s, s-1,...,-s$. In other words, if we measure the angle between the spin angular momentum and the $z-$axis there can only be $2s+1$ possible values.

Every atom acts as an electromagnet, i.e. it is characterized by a magnetic dipole moment $\vec{\mu}$. Thus, it will experience a torque $\vec{\mu}\times \vec{B}$ in a magnetic field $\vec{B}$ which makes the dipole moment $\vec{\mu}$ to want to align with the direction of the magnetic field $\vec{B}$ giving therefore a minimum energy. The corresponding potential energy is given by

\[H=-\vec{\mu}.\vec{B}.\]

In the simplest case of the hydrogen atom we have a single electron moving around its orbit and thus creating a tiny loop of electric current $I$. The magnetic moment of this current is $\mu_L=I.A$ where $A$ is the area of the loop. If $v$ is the speed of the electron and $r$ is the radius of the orbit then $2\pi r$ is the distance traveled by the electron in a single period, i.e. $v/2\pi r$ is the inverse period and $ev/2\pi r$ is precisely the current, viz $I=ev/2\pi r$ where $e$ is the charge of the electron. The angular momentum of the electron is $L=mrv$. From all these considerations we have

\[\mu_L=\frac{e}{2m} L.\]

Similarly, the magnetic dipole moment of a spinning charge is given in terms of the spin angular momentum $\vec{S}$ by

\[\vec{\mu}=\gamma \vec{S}\Rightarrow H=-\gamma \vec{S}.\vec{B}.\]

$\gamma$ is the gyromagnetic ratio.

In a non-uniform magnetic field there exists, in addition to the above torque, a force on the magnetic dipole given by

\[\vec{F}=\vec{\nabla}(\vec{\mu}.\vec{B}).\]

Let us imagine that the beam is directed in the direction of the $y-$axis and that the non-uniform magnetic field is given by

\[\vec{B}=-\alpha x\hat{i}+(B_0+\alpha z)\hat{k}~,~\vec{\nabla}\vec{B}=0.\]

The parameter $\alpha$ provides the non-uniformity. $B_0$ represents a uniform magnetic field giving rise to the so-called Larmor precession. This effect is given by the following expectation values of the spin quantum number

\[\langle S_x\rangle=\frac{\hbar}{2}\sin\alpha\cos\gamma B_0 t~,~\langle S_y\rangle=-\frac{\hbar}{2}\cos\alpha\sin \gamma B_0 t~,~\langle S_z\rangle=\frac{\hbar}{2}\cos\alpha.\]

Thus $\langle \vec{S}\rangle$ precesses around the $z-$axis at a constant angle $\alpha$ with a frequency $\omega=B_0 t$ (Larmor frequency). The above result is an instance of Ehrenfest's theorem, viz

\[\frac{d}{dt}\langle \vec{S}\rangle=\langle \vec{\mu}\times\vec{B}\rangle.\]

Now we return to the non-uniform magnetic field and we compute the corresponding force. We find the result

\[\vec{F}=\gamma \alpha(-S_x\hat{i}+S_z\hat{k}).\]

But we know from Larmor precession that $S_x$ oscillates rapidly and averages to zero leaving only the $z-$component of the force, viz

\[\vec{F}=\gamma \alpha S_z\hat{k}.\]

For a spin $1/2$ particle the beam is either deflected up for the eigenvalue $m=+\hbar/2$ of $S_z$ or down for the eigenvalue $m=-\hbar/2$. In general the beam is split into $2s+1$ components corresponding to the $2s+1$ values of the force

\[\vec{F}=\gamma \alpha\hbar m\hat{k}~,~m=s,s-1,...,-s.\]

This is the basic physics behind the Stern-Gerlach experiment. The experiment showed first that atoms enjoy a magnetic dipole moment. Then the splitting into two beams was observed by Gerlach for the first time in February 1922. As we understand today the correct explanation is not the Bohr-Sommerfeld old quantum theory which suggested that the ground state of the silver atom is characterized by a quantized angular momentum equal $l=1$. We know today that this ground state is in fact characterized by $l=0$ and thus does not lead to any splitting (and even if it was characterized by $l=1$ it would then give a splitting into three beames corresponding to $l=+1,0,-1$ and not two beams corresponding to $l=+1,-1$). The correct explanation is given by the spin angular momentum of the outermost unpaired electron. The total angular momentum $\vec{J}=\vec{L}+\vec{S}$ with $l=0$ and $s=1/2$ leads to $j=1/2$ and thus a splitting into two beams.

Thus, the Stern-Gerlach experiment is an experimental proof for the spin quantum number and not for space quantization (although we also know today that orbital angular momentum is indeed quantized). More importantly, Stern-Gerlach experiment is an experimental proof for the new theory of quantum mechanics.

The qubit model of the spin

In summary, the Stern-Gerlach experiment comes with two main results:

-First: The magnetic dipole moment of the atoms is quantized not continuous (discrete set of angles instead of a continuous distribution).

-Second: In the ground state of atoms we have zero orbital angular momentum $l=0$ which corresponds to zero magnetic dipole moment $\mu=0$ giving rise therefore to no deflection at all. Yet, we observe two peaks.

The conclusion is that there must exist a new physical quantity making an extra contribution to the magnetic dipole moment (spin). This contribution has no relation with the rotational motion of the electron.

The spin can be characterized by a single bit or more precisley by a single quantum bit or qubit as follows. The output of the Stern-Georlach apparatus consists of two beams $|+Z\rangle$ (up) and $|-Z\rangle$ (down). We call this the $\hat{z}-$Stern-Gerlach apparatus (since it measures the spin or the qubit in the $\hat{z}-$direction as it is oriented in the $\hat{z}-$direction).

We place now an $\hat{x}-$Stern-Gerlach apparatus oriented in the $\hat{x}-$direction (and thus measures the spin or qubit in the $\hat{x}-$direction) in series with the original $\hat{z}-$Stern-Gerlach apparatus. And we will block the beam $|Z-\rangle$.

Thus, all transmitted atoms have magnetic dipole moment which is up. And classically since the magnetic dipole moment is oriented in the $\hat{z}-$direction there will be no deflection in a magnetic field oriented in the $\hat{x}-$direction. In other words, one should have one central peak. However, again we observe two peaks labelled $|+X\rangle$, $|-X\rangle$.

Let us again block the beam $|-X\rangle$ and place a third $\hat{z}-$Stern-Gerlach apparatus in the way of the beam $|+X\rangle$. We expect to see one peak corresponding to the fact that the atoms in the beam $|+X\rangle$ retained their $|+Z\rangle$ orientation. But again we observe a splitting into two beams $|+Z\rangle$ and $|-Z\rangle$.

In terms of the qubit computational states $|0\rangle$ and $|1\rangle$ we have then the following correspondence :

\[|+Z\rangle\equiv |0\rangle~,~|-Z\rangle\equiv 1\rangle.\]

\[|+X\rangle\equiv |+\rangle=\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)~,~ |-X\rangle\equiv |-\rangle \frac{1}{\sqrt{2}}(|0\rangle-|1\rangle).\]

Thus, the $\hat{z}-$Stern-Gerlach apparatus measures the spin quantum number or the qubit in the computational basis $\{|0\rangle, |1\rangle\}$. Similarly, the $\hat{x}-$Stern-Gerlach apparatus measures the spin quantum number or the qubit in the computational basis $\{|+\rangle,|-\rangle\}$.

The spin can thus be completely captured by the qubit model.

Exercises:

Exercise 1:

Derive Bohr's energy levels from Bohr's quantization condition of the angular momentum. Of course, assume circular orbits and the Coulomb potential $V=-ke^2/r$. Derive Rydberg formula for atomic transitions or quantum jumps.

Exercise 2:

Derive Larmor precession and the corresponding Ehrenfest's theorem.

Exercise 3:

Show in the context of the $\hat{z}-\hat{x}-\hat{z}$ experiment using three cascaded Stern-Gerlach appartuses that the spin quantum number is captured by the qubit model. In particular, calculate the probabilities for obtaining the two beams after exiting each Stern-Gerlach apparatus.