Case Study 1: The Stern-Gerlach Experiment — From Silver Atoms to Qubits

Overview

The Stern-Gerlach experiment (1922) is one of the most important experiments in the history of physics. It provided the first direct evidence that angular momentum is quantized and, serendipitously, the first observation of electron spin — a property that would not be theoretically understood for another three years. Today, the two-state quantum system that the Stern-Gerlach experiment revealed is the foundation of quantum information science and quantum computing.

This case study traces the experiment from its original execution, through the surprising physics it uncovered, to its modern incarnation as the conceptual prototype for the qubit.

Part 1: The Original Experiment

Historical Context

By 1921, Bohr's model of the atom had introduced the idea that certain atomic properties are quantized — they take only discrete values. The orbital angular momentum of the electron, according to Bohr and Sommerfeld, was quantized as $L = n\hbar$, and its component along any chosen axis was expected to be quantized as $L_z = m\hbar$ where $m$ ranges from $-l$ to $+l$ in integer steps.

But was this quantization real, or just a convenient mathematical fiction that happened to give correct spectral predictions? Otto Stern, a theoretical physicist working in Max Born's laboratory in Frankfurt, proposed an experiment to settle the question.

The Setup

Stern's idea was elegant. Silver atoms are evaporated from an oven, collimated into a narrow beam, and passed through a region of strongly inhomogeneous magnetic field — a field whose magnitude varies sharply in the vertical ($z$) direction.

A magnetic dipole with moment $\boldsymbol{\mu}$ in a non-uniform field $\mathbf{B}$ experiences a force:

$$F_z = \mu_z \frac{\partial B}{\partial z}$$

If $\mu_z$ can take any value (as classical physics predicts), the beam should spread out into a continuous smear on a detector plate. If $\mu_z$ is quantized, the beam should split into discrete spots.

The magnetic moment of the silver atom comes primarily from its single unpaired electron in the 5s orbital. At the time, Stern and Gerlach expected to see quantization into an odd number of spots (consistent with integer angular momentum $l = 1$, giving $m = -1, 0, +1$ — three spots).

The Result

Walther Gerlach, the experimentalist of the pair, performed the experiment in early 1922 in Frankfurt. After hours of running the apparatus, he developed the detector plate (a glass slide coated with sulfur, which reacted with the deposited silver to form visible silver sulfide).

The beam had split into exactly two spots. Not three. Not a continuous smear. Two.

This was profoundly puzzling. An angular momentum of $l = 1$ should give three spots. An angular momentum of $l = 0$ should give one spot (no deflection). No integer value of $l$ gives two spots.

The Resolution: Electron Spin

The puzzle was resolved in 1925 by George Uhlenbeck and Samuel Goudsmit, two graduate students at the University of Leiden. They proposed that the electron possesses an intrinsic angular momentum — spin — with quantum number $s = 1/2$. The component of spin along any axis can take only two values:

$$S_z = m_s \hbar, \quad m_s = +\frac{1}{2} \text{ or } -\frac{1}{2}$$

This gives exactly two spots in the Stern-Gerlach experiment, corresponding to "spin up" ($m_s = +1/2$) and "spin down" ($m_s = -1/2$).

The magnetic moment associated with electron spin is:

$$\mu_z = -g_s \mu_B m_s$$

where $g_s \approx 2$ is the electron spin g-factor and $\mu_B = e\hbar/(2m_e) = 9.274 \times 10^{-24}$ J/T is the Bohr magneton.

Part 2: Sequential Stern-Gerlach Experiments

The Stern-Gerlach apparatus becomes a powerful tool for exploring quantum mechanics when multiple devices are chained together. These sequential experiments reveal the deepest features of quantum measurement.

Experiment A: Same Axis Twice

Setup: Pass a beam through a $z$-oriented SG apparatus. Block the spin-down beam. Pass the remaining spin-up-in-$z$ atoms through a second $z$-oriented SG apparatus.

Result: 100% of atoms emerge from the spin-up port. Zero from spin-down.

Interpretation: If a system has been prepared in a definite state (spin-up along $z$), repeated measurement of the same observable ($S_z$) gives the same result with certainty. The state is not disturbed by measuring the same quantity again. In the language of quantum mechanics, the system is in an eigenstate of the measured observable.

Experiment B: Perpendicular Axes

Setup: Prepare spin-up-in-$z$ atoms (using the first SG + blocking as above). Pass them through an SG apparatus oriented along the $x$-axis.

Result: The beam splits 50-50 into spin-up-in-$x$ and spin-down-in-$x$.

Interpretation: A state that is definite in $S_z$ is completely indefinite in $S_x$. The probability of measuring $S_x = +\hbar/2$ or $S_x = -\hbar/2$ is each exactly $1/2$. In the formalism of quantum mechanics (Chapter 13), a spin-up-in-$z$ state is a superposition of spin-up-in-$x$ and spin-down-in-$x$:

$$|+z\rangle = \frac{1}{\sqrt{2}}|+x\rangle + \frac{1}{\sqrt{2}}|-x\rangle$$

Experiment C: The Eraser

Setup: Prepare spin-up-in-$z$. Filter through $x$ (keeping only spin-up-in-$x$). Pass through a third SG apparatus oriented along $z$.

Result: The beam splits 50-50 into spin-up-in-$z$ and spin-down-in-$z$.

Interpretation: This is the most startling result. The atoms were prepared as spin-up-in-$z$. They should "remember" this. But the intermediate $x$-measurement has completely erased the $z$-information. After the $x$-measurement, the state is spin-up-in-$x$, which is:

$$|+x\rangle = \frac{1}{\sqrt{2}}|+z\rangle + \frac{1}{\sqrt{2}}|-z\rangle$$

So a subsequent $z$-measurement gives 50-50. The $x$-measurement did not merely reveal a pre-existing property; it changed the state of the system.

The Deep Lesson

These sequential experiments demonstrate three foundational principles of quantum mechanics:

1. Superposition: A spin state can be a superposition of eigenstates of a different spin component.
2. Incompatible observables: $S_z$ and $S_x$ cannot simultaneously have definite values. Measuring one randomizes the other.
3. Measurement disturbance: Quantum measurement is not passive observation — it actively transforms the quantum state.

These principles will be formalized as postulates in Chapter 3 and explored mathematically in Chapter 13.

Part 3: From Silver Atoms to Qubits

The Two-State System

The Stern-Gerlach experiment isolates the simplest possible quantum system: a system with exactly two distinguishable states. We call these states $|0\rangle$ and $|1\rangle$ (or $|\uparrow\rangle$ and $|\downarrow\rangle$, or $|+\rangle$ and $|-\rangle$ — the notation varies, but the physics is the same).

A general state of this two-state system is a superposition:

$$|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$$

where $\alpha$ and $\beta$ are complex numbers satisfying $|\alpha|^2 + |\beta|^2 = 1$. The probability of measuring $|0\rangle$ is $|\alpha|^2$ and the probability of measuring $|1\rangle$ is $|\beta|^2$.

This is a qubit — the quantum analog of a classical bit.

Classical Bit vs. Qubit

A classical bit is either 0 or 1. Nothing in between, nothing else. It is a switch with two positions.

A qubit can be in state $|0\rangle$, state $|1\rangle$, or any superposition of the two. The state $\frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)$ is neither 0 nor 1 — it is something that has no classical analog. When measured, it collapses to $|0\rangle$ or $|1\rangle$ with equal probability. But before measurement, it is genuinely in both states simultaneously.

The space of all possible qubit states can be visualized as the Bloch sphere — a unit sphere where the north pole represents $|0\rangle$, the south pole represents $|1\rangle$, and every point on the surface represents a valid superposition state. (This will be developed fully in Chapter 25.)

Physical Implementations

The abstract qubit — the two-state quantum system revealed by Stern and Gerlach — can be physically realized in many ways:

Each of these is a physical realization of the same abstract quantum mechanics that Stern and Gerlach first observed with silver atoms.

Quantum Gates as Generalized Stern-Gerlach Rotations

In quantum computing, operations on qubits are called quantum gates. A quantum gate transforms the state $|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$ into a new state $|\psi'\rangle = \alpha'|0\rangle + \beta'|1\rangle$. Mathematically, this is a rotation on the Bloch sphere.

The sequential Stern-Gerlach experiments are, in essence, sequences of quantum gates:

• Preparing spin-up-in-$z$ is the state preparation $|0\rangle$.
• Measuring spin along $x$ is equivalent to applying a rotation gate (the Hadamard gate, in quantum computing language) and then measuring.
• The "eraser" experiment ($z \to x \to z$) is a sequence of gate + measurement + gate + measurement.

The entire apparatus of quantum computing — superposition, measurement, incompatible bases, and state transformation — was present in embryonic form in Stern and Gerlach's 1922 laboratory.

Discussion Questions

1. Stern and Gerlach expected to see three spots (from $l = 1$ quantization) but saw two (from $s = 1/2$ spin). Discuss the role of "wrong" predictions in scientific discovery. How did the unexpected result lead to deeper understanding?

2. In Experiment C (the "eraser"), the $x$-measurement destroys information about $z$-spin. A classical analogy might be: measuring whether a coin is heads-or-tails destroys information about whether it is scratched-or-smooth (if flipping is required for measurement). Why is this analogy ultimately inadequate? What makes quantum measurement fundamentally different from any classical analog?

3. The qubit exploits the fact that a quantum system can be in a superposition of $|0\rangle$ and $|1\rangle$. A classical bit cannot. Does this mean a qubit stores "more information" than a classical bit? (Careful — this question is subtler than it appears. Consider what happens when you measure the qubit.)

4. If you were advising a student in 1921 about whether the Stern-Gerlach experiment was worth performing, what would you say? How does the experiment illustrate the value of testing even "obvious" predictions?

Further Investigation

• Read the first chapter of J.J. Sakurai's Modern Quantum Mechanics, which uses the Stern-Gerlach experiment as the foundation for the entire theory.
• Research the history of the Stern-Gerlach experiment: the cigar smoke story (Gerlach's cigar supposedly helped develop the detector plate), and the fact that Stern and Gerlach initially misinterpreted their own result.
• Explore the IBM Quantum Experience (online) to manipulate virtual qubits — the conceptual descendants of Stern-Gerlach's silver atoms.