Lecture 2: The Classical Stern-Gerlach experiment

Introduction: Separating Current Loops by Their projection

We have a collection of very small objects that act like current loops. We want to separate them according to the projection of the effective magnetic needle (representing the current loop) onto the axis of an increasing magnetic field. This separation allows us to determine the different values the projection can take and helps us understand the properties of these particles.

The Prism Analogy

This type of separation experiment is common in physics. The simplest example: shining white light into a prism and separating it into the spectrum of the rainbow.

• Separating light by color = separating by energy = separating by frequency
• Our experiment with current loops is analogous, but selecting by projection of the effective needle on the magnetic field axis

The Experimental Setup

• The “current loops” are actually atoms — extremely small
• We shoot atoms perpendicular to the direction of increasing magnetic field
• They feel a push or pull for a fixed time near the magnet
• The deflection of the path is proportional to the magnitude and sign of the projection
• Collecting far enough away amplifies the differences (acts like a microscope)
• Final experiment: beam of atoms → magnet → screen

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Designing the Classical Stern-Gerlach Experiment

The Stern-Gerlach apparatus

• Otto Stern (Nobel Prize 1943) and Walter Gerlach co-invented the experiment
• Gerlach was excluded from the Nobel because of his association with the Nazis
• Silver atoms are injected at a fixed velocity through a region with a non-uniform magnetic field
• Current loops precess in the field but keep their projection on the axis of increasing field constant
• Result: they feel a push (up) or pull (down) depending on the projection

Patterns on the Screen

Three possible scenarios for what the screen might show:

1. Random projection of effective needle — a continuous smear on the screen
2. Only one value — a single spot
3. Only two values — two discrete spots

(Video demonstration: sge_exp_1-01_sg_current_loops.html)

The quantum case: the atom always shows two projections with the same splitting amount, regardless of field orientation — a result that cannot be reconciled with any classical picture.

(Video demonstration: sge_exp_1-02_sg_atoms.html)

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Our Philosophy: Embracing the Quantum

Two key realizations emerge:

1. Quantum mechanics is fundamentally different from classical physics. Something unexpected has happened. We must incorporate this different behavior into a new model — we cannot keep forcing classical explanations.
2. Quantum mechanics is random (stochastic). We cannot say what will happen to any particular particle; we can only calculate the probability of outcomes over many trials.

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The Stern-Gerlach analyzer

To study quantum behavior systematically, we package the Stern-Gerlach experiment into a compact device called a Stern-Gerlach analyzer.

(Video demonstration: sge_exp_1-03_analyzer_intro.html)

Repeated Measurements: Testing Reproducibility

Setup: Two analyzers (A and B) placed side-by-side. Atoms from A’s + output enter B.

Key Finding: Once an atom is measured as +z, it always measures as +z again.

(Video demonstration: sge_exp_1-04_repeated_measurements.html)

• Atoms with a -z measurement always measure as -z again
• The measurement defines the state; subsequent identical measurements are fully reproducible

Measurements and Quantum States

Key insight: Measuring the direction of an atom’s effective magnetic arrow defines that direction. We represent this knowledge using blue and red cones — because the needle precesses around a cone, any direction on the cone gives the same projection.

State representation:

• Atoms from source: No cone — we have no knowledge of the orientation (any direction equally likely)
• Atoms leaving + output: Red cone points up (along +z)
• Atoms leaving - output: Red cone points down (along -z)
• Direction is relative to the magnets inside the analyzer

Flipping analyzer B upside-down reverses the roles of + and - outputs, but the deflection and cones remain unchanged. This means we must carefully track what we are measuring and how.

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Perpendicular Stern-Gerlach Analyzers

The “Conundrum of Projections”

• Only two results are possible for any measurement, regardless of direction
• The magnitude of the effective arrow is always the same — completely at odds with the classical notion of a definite pre-existing direction
• The act of measuring defines what the direction is

Experiment: Rotating Analyzer B by 90°

Setup: Analyzer A measures along z. Analyzer B is rotated to measure along x.

Classical expectation: Since x and z are perpendicular, the projection should be zero. But that’s not what we see.

(Video demonstration: sge_exp_1-05_perpendicular_analyzers.html)

Result: Atoms emerge from both outputs of B, with ~50% probability each.

An atom with a known state in the z-direction does not have a definite state in the x-direction. The measurement in a new direction destroys the previous state’s definiteness.

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Three Perpendicular Stern-Gerlach Analyzers

(Video demonstration: sge_exp_1-06_three_analyzers.html)

“Atoms Are Stupid”

The Rule: Atoms only remember the last axis their effective magnetic needle was measured on. They suffer from permanent memory loss.

• When measured on a particular axis, an atom has a projection on that axis
• Change the axis and measure again: the atom now has a projection on the new axis
• It no longer has any definite projection on the previous axis (with two special exceptions we’ll encounter later)
• The atom can have a definite state in only one direction at a time

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Analyzers at an Angle

When the second analyzer is at an arbitrary angle θ (not 0°, 90°, or 180°):

(Video demonstration: sge_exp_1-07_arbitrary_angles.html)

The probability follows:

$$P(+,\theta )={\cos }^2\left(\frac{\theta }{2}\right)$$ $$P(-,\theta )={\sin }^2\left(\frac{\theta }{2}\right)$$

Three key angles to memorize:

Angle θ	$P(+,\theta )$	$P(-,\theta )$
0°	1	0
90°	1/2	1/2
180°	0	1

Statistical fluctuations mean experimental points won’t lie exactly on the curve — especially with only 100 measurements. Larger samples approach the theoretical curve more closely.

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Why We Must Use Probability

Classical vs. Quantum Randomness

Classical	Quantum
Appears random due to unknown initial state	Truly random
In principle, path could be followed precisely	Cannot follow the path
deterministic underneath	Intrinsically probabilistic

In the quantum case, we cannot detect atoms until they hit the screen, so we don’t know their precise path. The overwhelming consensus: quantum results are truly random, and we can only predict probabilities — but we can predict those probabilities precisely.

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Accuracy of Quantum Measurements

⚠️ Don’t confuse randomness with imprecision or irreproducibility.

Quantum mechanics is one of the most accurately tested theories in all of science:

• Energy levels can be measured to more than 10 digits of accuracy
• The NIST Yb lattice clock is accurate to one part in $10^{18}$
• This is like measuring the distance from Washington, DC to Los Angeles to within the width of a single atom

The most accurate quantum measurements typically involve:

• Time between two events
• Frequency/color of photons

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The End of Determinism

Quantum phenomena have no classical description. Quantum events have unavoidable randomness.

These two facts undermine the philosophy of determinism. The quantum world is manifestly not deterministic — randomness is inherent whenever a measurement is performed.

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Key Takeaways

1. The Stern-Gerlach experiment reveals quantization: Silver atoms always produce exactly two discrete spots on the screen, corresponding to two possible projections — never a continuous distribution.

2. Measurements define quantum states: The very act of measuring a projection along a given axis forces the atom into a definite state along that axis. Repeated measurements along the same axis are fully reproducible.

3. “Atoms are stupid”: An atom only remembers the last axis it was measured on. Measuring along a new axis destroys any definite projection along the previous axis.

4. Quantum probability follows ${\cos }^2(\theta /2)$: The probability for obtaining a positive result after rotation by angle $\theta$ is precisely ${\cos }^2(\theta /2)$. The three special cases (0°, 90°, 180°) give probabilities of 1, 1/2, and 0.

5. Quantum randomness does not mean imprecision: Quantum mechanics is the most accurately tested theory in science — quantities like energy levels and atomic clock frequencies can be measured to over 10 digits of precision.