40-2 The Stern-Gerlach Experiment#

Prompts

Sketch the Stern–Gerlach experiment setup. What type of atom is used, and why? What did classical physics predict for the silver deposit when the magnet is on?
What was the actual result of the Stern–Gerlach experiment? Why is it surprising, and what does it imply about magnetic moments and angular momentum?
Why doesn’t the Lorentz force \(\vec{F} = q\vec{v}\times\vec{B}\) deflect the silver atoms? What type of force deflects them instead?
Derive the force on a magnetic dipole in a nonuniform field. How does the field gradient \(dB/dz\) affect the deflection?
For a Stern–Gerlach experiment with given \(dB/dz\), beam speed \(v\), and magnet length \(w\), how would you find the separation between the two subbeams?

Lecture Notes#

Overview#

The Stern–Gerlach experiment (1922) showed that the magnetic moment of silver atoms is quantized — the first direct evidence that atomic-scale magnetic moments and angular momentum take discrete values.
A magnetic dipole in a nonuniform magnetic field experiences a force; the force depends on the field gradient \(dB/dz\) and the component \(\mu_z\) of the dipole moment.
Classical physics predicts a continuous range of deflections (a vertical line); the experiment produced two distinct spots, proving quantization.
With modern quantum theory: the silver atom has one unpaired valence electron whose orbital moment is zero; its spin magnetic moment gives \(\mu_z = \pm\mu_B\), explaining the two beams.

The Experiment#

In 1922, Otto Stern and Walther Gerlach at the University of Hamburg demonstrated that magnetic moments of atoms are quantized.

Setup:

Silver is vaporized in an oven; atoms escape through a narrow slit.
A second slit (collimator) forms a narrow beam.
The beam passes between the poles of an electromagnet (with a strong vertical field gradient).
Atoms land on a glass detector plate.

Result:

Magnet off: a single narrow spot.
Magnet on:
- Classical expectation: \(\mu_z\) could take any value from \(-\mu\) to \(+\mu\) → continuous range of deflections → atoms land along a vertical line.
- Actual result: two distinct spots — one above and one below the undeflected position.

Why silver?

Silver (and other atoms like cesium) is used because it has a single unpaired valence electron. All other electrons pair up so their orbital and spin moments cancel. The net magnetic moment comes from that one electron, making the signal clear.

The Magnetic Deflecting Force#

The force that deflects the atoms is not the Lorentz force \(\vec{F} = q\vec{v}\times\vec{B}\) — a silver atom is electrically neutral (\(q=0\)), so that force is zero.

The force comes from the interaction between the magnetic dipole of the atom and the magnetic field. The potential energy of a dipole \(\vec{\mu}\) in a field \(\vec{B}\) is

(413)#\[ U = -\vec{\mu} \cdot \vec{B} = -\mu_z B \]

With the \(z\) axis along the field direction, the force on the dipole is

(414)#\[ F_z = -\frac{dU}{dz} = \mu_z \frac{dB}{dz} \]

Uniform field (\(dB/dz = 0\)): no force; the dipole may experience a torque but no net translation.
Nonuniform field (\(dB/dz \neq 0\)): the dipole experiences a force along the gradient. The Stern–Gerlach magnet poles are shaped to maximize \(dB/dz\) so deflections are measurable.

Classical vs. Quantum Prediction#

	Classical	Quantum (Stern–Gerlach)
\(\mu_z\)	Any value from \(-\mu\) to \(+\mu\)	Only discrete values
Deflection	Continuous range	Discrete spots
Deposit pattern	Vertical line	Two spots

The two-spot result proves that \(\mu_z\) is quantized — it can take only certain values. Because angular momentum is coupled to magnetic moment (Section 40-1), this also implies that angular momentum is quantized.

Modern Interpretation: Spin#

With quantum theory, we now understand the silver atom:

It has many electrons; most orbital and spin moments cancel.
One valence electron remains unpaired; its orbital angular momentum is zero (\(\ell=0\)).
The net magnetic moment is the spin magnetic moment of that electron.

From Section 40-1, \(\mu_{s,z} = -2m_s\mu_B\) with \(m_s = \pm\tfrac{1}{2}\):

(415)#\[ \mu_z = \pm\mu_B \]

So the force on the atom has only two values:

(416)#\[ F_z = \pm\mu_B \frac{dB}{dz} \]

— one for spin up, one for spin down — producing the two beams. The Stern–Gerlach experiment was the first experimental evidence of electron spin, even though spin was not understood until later (Uhlenbeck, Goudsmit, Dirac).

Calculating the Beam Separation#

The deflection depends on the force, the time the atom spends in the field, and the atom’s mass. The acceleration along \(z\) is \(a_z = F_z/M\). For constant \(F_z\) and \(dB/dz\) over the magnet length \(w\), with beam speed \(v\) along the beam direction:

(417)#\[ d = \frac{1}{2}a_z t^2 = \frac{\mu_B(dB/dz)}{M}\left(\frac{w}{v}\right)^2 \]

The separation between the two subbeams is \(2d\).

Worked example: Beam separation

A beam of silver atoms (\(M = 1.8\times 10^{-25}\) kg) passes through a Stern–Gerlach magnet with \(dB/dz = 1.4\) T/mm and length \(w = 3.5\) cm. The beam speed is \(v = 750\) m/s. Find the deflection \(d\) and the separation between the two subbeams.

Solution: Using \(\mu_B \approx 9.27\times 10^{-24}\) J/T and \(dB/dz = 1.4\times 10^3\) T/m:

\[ d = \frac{\mu_B(dB/dz)\,w^2}{2Mv^2} = \frac{(9.27\times 10^{-24})(1.4\times 10^3)(0.035)^2}{2(1.8\times 10^{-25})(750)^2} \approx 0.08\ \text{mm} \]

The separation is \(2d \approx 0.16\) mm — small but measurable.

Summary#

The Stern–Gerlach experiment demonstrated that atomic magnetic moments are quantized; the two-spot result contradicts the classical prediction of a continuous range.
A magnetic dipole in a nonuniform field experiences \(F_z = \mu_z\,dB/dz\); the gradient is essential for deflection.
For silver (and similar atoms), the net moment comes from one unpaired electron’s spin; \(\mu_z = \pm\mu_B\) gives two distinct beams.
The experiment provided the first experimental evidence of electron spin; it also shows that angular momentum and magnetic moment are coupled at the atomic level.