General Introduction and Foundational Experiments

1. Introduction

One has to start somewhere. Many introductions to quantum mechanics begin with Young's double-slit experiment, or with the historical study of black-body radiation and the hypothesis Planck introduced to account for it (Planck, 1900).

Planck introduced the idea that energy can only be exchanged in discrete amounts, the quanta from which the theory would take its name. But understanding his approach requires knowledge of statistical physics, which beginning students do not always have.

Young's double-slit experiment, for its part, is spectacular. In its quantum version, with particles sent one at a time, it reveals behaviour that is simply incomprehensible in classical terms. In the author's view, however, this experiment is much easier to grasp in hindsight, once the principles of quantum theory are known. In a sense, it alone encapsulates several of the theory's strangest aspects.

We shall therefore begin differently, with the Stern and Gerlach experiment (1922), which concerns an object that appears simple and well understood classically: a particle with a magnetic moment. In the historical experiment, this particle is a silver atom, and the only thing we must temporarily accept is that such an atom does indeed have a magnetic moment. When it is sent into a region containing an inhomogeneous magnetic field, the laws of mechanics and electromagnetism readily predict its trajectory.

The results are equally spectacular: they differ radically from classical predictions. By chaining together increasingly elaborate experiments, the classical description collapses and we successively encounter quantization (see section 4), quantum superposition (see section 7.2), and another purely quantum phenomenon that Young's slits do not directly reveal: the incompatibility of certain measurements (see section 6.1).

We shall begin by recalling what a magnetic moment is. We shall then describe the experiment and what we should observe if the atom behaved like an ordinary small magnet, before presenting the actual results and discussing their implications.

2. Review of the classical magnetic moment

2.1. The magnetic moment

A magnetic moment $\vec{\mu}$, or magnetic dipole, is a vector. A natural magnet has a magnetic moment, which by definition points from its South pole to its North pole. This is also the direction of the magnetic field lines inside the magnet; outside, these lines leave the North pole and return to the South pole, see Figure 1.

This moment is the fundamental quantity used to quantify a magnet's strength. For example, the repulsion between two magnets in a South-South orientation depends on the product of their magnetic moments and on their separation.

Natural magnets are not very convenient for continuing our discussion, however, because the origin of their magnetic field is not immediately obvious. Electromagnets, that is, current loops, are much more useful. For a planar loop carrying a current $I$ and enclosing an area $S$, one can show that the magnetic moment is $\vec{\mu}=I S\,\vec{n}$, where $\vec{n}$ is the unit vector normal to the loop, oriented according to the right-hand rule.

This is the only expression we shall need here. The theory of magnetic moments is, of course, more general. It defines the magnetic moment of an arbitrary current distribution by an integral over that distribution. The loop formula is a special case. We refer to standard textbooks for further details on this construction (for example, chapter 5 of ^[1]).

2.2. Coupling to the magnetic field

Classical mechanics derives three properties of a magnetic dipole immersed in an external magnetic field $\vec B$.

The third formula follows directly from the second through $\vec{F} = -\grad E_p$. It shows that a system carrying a magnetic moment in an inhomogeneous field experiences a net force that deflects its trajectory. The first two follow from an explicit calculation of the Laplace force on the current distribution (for the torque), and of the work done by this force (for the potential energy); we do not reproduce it here and refer in particular to sections 5.2 and 5.7 of ^[1] for the complete derivation¹.

Beyond the technical details, the main point is this. The potential energy in Eq. (2) is minimal when $\vec\mu$ is parallel to $\vec B$, and maximal when it is antiparallel: a magnetic dipole tends to align with the field through the torque in Eq. (1). This is how a compass works. Note, however, that without dissipation the compass needle would simply oscillate indefinitely about North, owing to the torque and the angular momentum theorem. It eventually comes to rest because friction at its rotation axis dissipates the energy of this oscillation.

2.3. Gyromagnetic ratio and precession

This evolution takes a particular form when the magnetic moment is proportional to an internal angular momentum associated with the dipole. In that case, we introduce the gyromagnetic ratio $\gamma$ through the proportionality:

A relation such as Eq. (4) arises, for example, for the magnetic moment of a classical charge in a circular orbit. Indeed, a charge $q$ of mass $m$ travelling around a circle of radius $R$ at speed $v$ is equivalent to a current $I=qv/(2\pi R)$. It therefore produces the magnetic moment

In this classical orbital model, $\gamma=q/(2m)$. The angular momentum theorem and Eq. (1) together give $\frac{d\vec L}{dt} = \vec{\tau} = \vec\mu\wedge\vec B$, and then, if $\gamma$ is constant, the precession equation:

In a uniform field, the solution of this equation shows that the vector $\vec\mu$ traces a cone around the field direction, i.e. precesses, at the Larmor angular frequency $\omega_L=|\gamma|B$, see Figure 2.

3. The Stern and Gerlach experiment

The principle of the experiment is summarized in Figure 3. A source sends neutral particles into an inhomogeneous magnetic field created by two facing magnets. A screen farther downstream records their impacts and allows any deflection to be measured.

3.1. The apparatus

The magnets.

The force in Eq. (3) vanishes identically if the external field is uniform. A dipole in a homogeneous field experiences only a torque, and its centre of mass continues straight ahead. The inhomogeneity of the field is therefore what makes the experiment interesting.

To generate this field, a Stern and Gerlach apparatus (commonly called a Stern-Gerlach, or SG) uses two shaped magnets, one with a pointed edge and the other with an inverted point or a slightly curved base, see Figure 3. This geometry concentrates the field lines towards the pointed edge of the upper magnet. As the field lines converge, conservation of magnetic flux $\mathrm{div} \vec B = 0$ within a flux tube tells us that the field increases towards this edge, here with $z$.

To first order, we might wish to write the field inside the apparatus as $\vec B\approx(B_0+kz)\,\vec e_z$ with $k>0$, but this would not satisfy the zero-divergence condition $\mathrm{div} \vec B = 0$. The field must therefore also have a transverse component. Neglecting edge effects and assuming invariance along $\vec e_x$, we find:

In order of magnitude, typical apparatuses have $B_0 \sim 0{,}1$ T, with $k\Delta z \ll B_0$ across the entire vertical gap between the magnets, typically a few millimetres. The magnets are a few centimetres long.

The source.

This is a small oven under vacuum containing solid silver. At high temperature, vaporized atoms leave through an aperture, and several narrow slits select those whose velocity is almost parallel to $\vec e_x$: this produces a collimated atomic beam with a typical speed of a few hundred metres per second and a spread inherited from the Maxwell-Boltzmann distribution.

Note that electrically neutral particles are used to probe only magnetic dipole effects. Charged particles would also experience the Lorentz force $q\,\vec v\wedge\vec B$, of magnitude $qvB_0$, which would overwhelmingly dominate the force $\grad(\vec\mu\cdot\vec B)$ of interest here.

The screen.

In the historical experiment, silver atoms gradually accumulated on the screen until they formed a visible trace. In a modern experiment, the flux can be reduced enough to record impacts one at a time. This possibility will matter later, and we shall return to it.

3.2. The prediction of classical mechanics

Let us send a neutral particle with magnetic moment $\vec\mu=\mu_x\vec e_x+\mu_y\vec e_y+\mu_z\vec e_z$ into the field region of Eq. (7). Equation 3 gives

The dipole is therefore deflected along both $z$ and $y$. Readers who have already encountered the Stern and Gerlach experiment elsewhere may be surprised by this, since many lecture notes confine themselves to what happens along the $z$ axis. In fact, the two terms in Eq. (8) initially play the same role.

The reason why we can neglect what happens along the $y$ axis lies in precession, and this is where we must make a crucial assumption that we shall justify afterwards. We shall assume that the atom's magnetic moment is proportional to an internal angular momentum $\vec L$ carried by the atom.

Under this assumption, and because the constant field $B_0\vec e_z$ overwhelmingly dominates the corrections $kz\,\vec e_z$ and $-ky\,\vec e_y$, the magnetic moment precesses to first order around $\vec e_z$ at angular frequency $\omega_p = |\gamma| B_0$, where $\gamma$ is its gyromagnetic ratio. During this precession, the component $\mu_z$ therefore remains constant, while $\mu_y$ oscillates around zero.

This precession is fast. The formulas above provide orders of magnitude that should be appropriate on the atomic scale. Thus $\gamma = \mathcal{O}(q/2m)$ becomes $\approx e/(2m_e)\approx 8{,}8\times10^{10} SI$ using the elementary charge and the electron mass². With $B_0\approx0{,}1$ T, we obtain $\omega_p\sim10^{10}\,\mathrm{rad}\,\mathrm{s}^{-1}$. Meanwhile, the flight time through the magnet is $T=\ell/v_x$, or about $10^{-4}$ s for $\ell$ of a few centimetres and $v_x\sim600\,\mathrm{m}\,\mathrm{s}^{-1}$. The magnetic moment thus makes a few hundred thousand revolutions during transit. The force along $y$ therefore averages to zero with excellent accuracy.

Within this model, the only relevant average force is therefore $\langle F_z\rangle=k\mu_z$. This immediately implies that the final deflection on the screen must be proportional to $\mu_z$. We need only consider the initial conditions. The atoms emerge from an oven, that is, a thermal bath, where no spatial direction is preferred. At the entrance to the SG apparatus, the magnetic moments must therefore be randomly oriented, and their vertical component $\mu_z$ must continuously take all values between $-\mu$ and $+\mu$, with a uniform distribution.

The classical prediction is then unambiguous: we should observe a continuous band of impacts along the $z$ axis (although not necessarily a uniform one, because of the velocity spread).

4. The experimental results

This is not what is observed. The deflection does occur only along $z$, consistently with the force along $y$ averaging to zero, but the screen does not display a continuous band: it shows two spots, symmetric about the initial axis and separated by an empty region.

4.1. Measuring $\mu_0$

The precise positions of the two spots allow us to estimate $\mu_0$. Assuming that the gradient $k=\partial B_z/\partial z$ and the longitudinal velocity $v_x$ remain approximately constant inside the magnet, the atom experiences the transverse acceleration

for a duration $T=\ell/v_x$. We integrate this by elementary means and then extend the trajectory in a straight line to the screen. Taking the geometry of the experiment into account, Otto Stern and Walther Gerlach could thus measure its value for silver atoms and found the value of the Bohr magneton

This measurement allows us to revisit assumption 4. However, it does not give $\gamma$ directly, since it measures the magnetic moment rather than the angular momentum $L$ that would be associated with it. We can only check an order of magnitude: the natural scale of atomic angular momentum is $\hbar$, and if we write $L\sim\hbar$, then $|\gamma|\sim\mu_0/L\sim e/(2m_e)$. We thus recover the order of magnitude assumed above and a precession angular frequency of order $10^{10}\,\mathrm{rad}\,\mathrm{s}^{-1}$, consistently with the observed absence of deflection along $y$.

4.2. The problem for classical mechanics

The experiment reveals a phenomenon foreign to classical mechanics: the component of a magnetic moment along a given axis does not vary continuously, but is quantized.

Even accepting that a silver atom has a magnetic moment, we still have to explain how arbitrary initial orientations, emerging from a thermal bath, lead to exactly two beams. Let us examine two classical mechanisms that might explain this result.

Could the field inside the magnet align the moments? Without dissipation, we have seen that the magnetic torque makes the dipole precess while preserving its angle with the field. It therefore aligns it with neither $+\vec B$ nor $-\vec B$. But can all dissipation be neglected? Electromagnetic theory does predict that a precessing moment radiates³. This dissipation is nevertheless negligible: shortly after the results were published, Einstein calculated it and estimated that relaxation would take more than a century, compared with a transit time of about $10^{-4}$ s through the magnet ^[3]. Even if it were fast enough, it would drive the moments towards the lowest-energy orientation, parallel to the field: we would then observe one spot, not two.

Could fringe fields explain the two spots? At the entrance and exit of the magnet, the field's direction and strength vary and may complicate the precession. These effects do, however, depend on the precise geometry of the magnets and of the experiment in general, and should therefore vary between experiments. Yet the separation into two beams does not depend on them. Moreover, in this classical model, evolution under the laws of mechanics and Maxwell's equations is a continuous function of the initial orientation: it cannot transform the continuum of initial orientations into just two output values.

4.3. The emergence of randomness

In a modern version of the experiment, the flux can be reduced until the atoms pass through the apparatus one by one. Each atom produces a single impact in one or the other of the two spots. These gradually take shape as the impacts accumulate.

This randomness is not yet enough to establish the existence of fundamental randomness: the atoms emerge from an oven, and their initial conditions vary from one atom to another. The unpredictability could therefore still be attributed to our ignorance of these conditions. The question will become more pressing once we have better control over the preparation of the beam.

5. Several Stern-Gerlach apparatuses in series

Experiments combining several SG apparatuses in series will clarify the role of preparation and probabilities, and then reveal the incompatibility of certain measurements.

5.1. Preparing a beam

At the exit of an $\mathrm{SG}_z$ apparatus, the beam separates into two channels. We shall denote the channel corresponding to the positive result by $z+$, and the one corresponding to the negative result by $z-$.

We can place an obstacle in one channel and keep only the other. If we block $z-$, all the atoms that continue the experiment gave the result $z+$ in the first apparatus. The device therefore no longer serves only to measure: it serves to prepare a well-defined beam for the next experiment.

We shall denote the resulting preparation by $\ket{z+}$. At this stage, this notation simply means “an atom that has just emerged through the $z+$ channel of an $\mathrm{SG}_z$ analyser”. We shall gradually discover the mathematical structure associated with this symbol.

5.2. Reproducibility along the same axis

Let us place a second $\mathrm{SG}_z$ after the first, keeping only the $z+$ channel from the first apparatus, see Figure 5. Experimentally, the second apparatus then produces only one spot, in the $z+$ channel. The $z-$ channel is empty.

The classical picture poses no problem here. If the first apparatus selected a value of $\mu_z$ or aligned the moments along $z$, and if nothing subsequently changed their $\mu_z$ component, the second apparatus should give the same deflection for all the selected atoms. This result is therefore compatible with a picture of simple sorting.

5.3. Two different axes

Now replace the second apparatus with an $\mathrm{SG}_x$, whose analysis axis is perpendicular to that of the first, see Figure 6. Two spots are again observed, in equal proportions.

Two classical interpretations are possible. If the first apparatus has completely oriented $\vec\mu$ along $z$, then $\mu_x=0$: the second apparatus should produce a central spot. If it has merely selected positive values of $\mu_z$, without fixing the other components, then $\mu_x$ remains continuously distributed: we should observe a continuous band. Neither interpretation predicts the two observed spots.

Let us return to atoms sent one by one. Each atom produces an impact in the $x+$ or $x-$ channel; over a large number of atoms, the two results occur in equal proportions. Yet the beam entering $\mathrm{SG}_x$ has a better-controlled preparation than the beam emerging directly from the oven: all its atoms were selected in the $z+$ channel of the first apparatus. As we have just seen, their preparation $\ket{z+}$ even guarantees a certain result in a further measurement along $z$.

Nevertheless, this preparation does not allow us to predict which channel will register the impact in a measurement along $x$. An appeal to mere disorder at the oven's exit is therefore no longer enough: the same preparation makes one measurement certain while leaving the other random. The role of probabilities in quantum mechanics becomes clearer.

6. Three SG apparatuses in series

6.1. ZXZ: filtering changes the preparation

Now consider the sequence $\mathrm{SG}_z\to\mathrm{SG}_x\to\mathrm{SG}_z$, see Figure 7. After the first apparatus, we retain only $z+$. After the second, we retain only $x+$. Finally, we analyse this new beam along $z$.

Classically, we have already ruled out complete alignment of the magnetic moment along the SG axis, since this would predict a central spot in the second apparatus, leaving no $x+$ channel to select. If we retain the picture of simple sorting, the first apparatus selects a positive value of $\mu_z$, and the second sorts the atoms by $\mu_x$ without changing $\mu_z$. The third apparatus should then produce only the $z+$ channel. Yet the $z-$ channel reappears.

Filtering along $x$ has therefore changed the preparation. After selecting $x+$, the earlier result $z+$ no longer predicts the new measurement of $z$ with certainty. It seems that the state itself has been modified by the second apparatus.

This is sometimes summarized by saying that “measuring $x$ destroys the information about $z$”. The phrase is useful, provided we remember that nothing here depends on what the experimenter knows or looks at. The $x+$ filter has physically prepared a new state, $\ket{x+}$, which gives the two results $z+$ and $z-$ with equal probabilities.

6.2. ZYZ: same probabilities, different preparation

Finally, replace the intermediate analyser along $x$ with an analyser along $y$, still retaining only its positive channel. The sequence $\mathrm{SG}_z\to\mathrm{SG}_y\to\mathrm{SG}_z$ again gives two equally probable results in the final measurement. This generalizes to any intermediate axis perpendicular to $z$: for a direction $\vec u=\cos\varphi\,\vec e_x+\sin\varphi\,\vec e_y$ in the $(x,y)$ plane, selecting the $u+$ channel again leads to two equally probable results in the final measurement along $z$.

This does not mean that the preparations $\ket{x+}$, $\ket{y+}$ and $\ket{u+}$ are all identical. An analysis along $x$ would distinguish them: it would certainly give $x+$ for the first, but $x+$ or $x-$ in equal proportions for the second, and experiments show that the probabilities of the two results depend on $\varphi$ for the third. Two preparations can therefore give the same probabilities for one measurement while being physically different.

7. The formalism taking shape

We shall not claim to deduce the entire quantum formalism from these experiments alone. Nevertheless, the failure of the classical description forces us to reconsider our vocabulary and suggests a new way to represent states.

7.1. A state is not a list of values

The preparation $\ket{z+}$ cannot reduce to the specification “$\mu_z=+\mu_0$, with unknown but already fixed values of $\mu_x$ and $\mu_y$” that the apparatuses merely reveal. In this picture of simple sorting, the sequence $Z\!X\!Z$ would again give $100\,\%$ of $z+$, which it does not. How, then, can we represent a state, if not by a list of pre-existing values?

7.2. Towards quantum superposition

The major novelty of the quantum formalism is that the state itself contains the information needed to calculate the probabilities of measurement results.

The preparation $\ket{x+}$ gives a certain result along $x$, but two equally probable results along $z$. To describe this situation, the quantum formalism represents $\ket{x+}$ as a superposition of the two states $\ket{z+}$ and $\ket{z-}$:

The coefficients (here $1/\sqrt{2}$ before each term) are called amplitudes: their squared moduli give the probabilities of the results along $z$, here $1/2$ for each. All atoms prepared in $\ket{x+}$ have the same state, represented by this linear combination. This is not a statistical mixture in which some atoms would be in $\ket{z+}$ and the others in $\ket{z-}$.

We cannot fully deduce this structure from the results presented here alone. The interference experiments in the next lesson will provide further justification. Ultimately, we shall have to postulate it.

7.3. A complex vector space

The ZYZ and, more generally, ZUZ variants raise the following question: the preparations $\ket{y+}$ or $\ket{u+}$ also give $50/50$ along $z$, while differing from $\ket{x+}$ when $\varphi\not\equiv0\pmod{2\pi}$. Their two amplitudes in the basis $\ket{z+},\ket{z-}$ must therefore have modulus $1/\sqrt2$. If the amplitudes were real, only the four choices of signs $\pm1/\sqrt2$ would remain: this finite number of possibilities would not suffice to represent the continuous variety of preparations $\ket{u+}$.

The quantum formalism therefore uses complex amplitudes, whose phases can vary without changing their moduli. For example, we shall be led to write:

and more generally $\ket{u+}=\frac{1}{\sqrt2}\left(\ket{z+}+e^{i\varphi}\ket{z-}\right)$ for any angle $\varphi$. The states are then distinct, but the probabilities along $z$ remain $1/2$. Taking these linear combinations seriously leads us to represent states by vectors in an abstract complex vector space. The word “abstract” reminds us that their coefficients are probability amplitudes, rather than the spatial components of a classical magnetic moment. More generally, a state preparation can be written as

To obtain the amplitudes and then their squared moduli, we shall need to project the state onto the states associated with the possible results. The simplest way to recover these components will be to use an inner product. This will also define a norm, with, here,

The squared norm thus corresponds to the sum of the probabilities of the possible results: this sum must be one, so the state will be represented by a vector of norm one.

In the emerging formalism, states will therefore be vectors in a complex Hilbert space, that is, a space equipped with an inner product for calculating probabilities. Its linear structure allows amplitudes to be added and expresses the principle of quantum superposition. All this will be studied in detail in theme 2.

8. Going further

A century after these experiments, we can specify what they actually measure. Following Stern and Gerlach, it was gradually understood that the magnetic moment measured in the silver atom is essentially associated with the spin of an unpaired electron. Spin is an intrinsic angular momentum, distinct from the electron's orbital angular momentum. The electron is a spin-$1/2$ particle: measuring its spin projection along an axis gives two possible values, $S_z=\pm\hbar/2$.

This spin angular momentum has an associated magnetic moment, proportional to the spin:

where $e$ denotes the positive value of the elementary charge, and $\gamma_s$ is the electron's gyromagnetic ratio. The minus sign is associated with the electron's negative charge: its magnetic moment points opposite to its spin. The two measurable values of its component are therefore $\mu_{s,z}=\pm\mu_B$, with $\mu_B=e\hbar/(2m_e)$. Spin and magnetic moment are two distinct quantities, proportional to one another.

In an atom, we must also account for the electron's orbital angular momentum, which has another associated magnetic moment. Hydrogen provides a simple example: its single electron has both a spin and an orbital state. In the $1s$ ground state, the orbital angular momentum is zero, so the electronic contribution to the atom's magnetic moment comes from spin. Phipps and Taylor performed the Stern-Gerlach experiment with hydrogen in 1927.

The silver atom contains more electrons, but its electronic configuration, $[\mathrm{Kr}]\,4d^{10}\,5s^1$, reduces the situation to the same principle. One can show that the orbital and spin contributions cancel in filled shells. This leaves the $5s$ electron, whose orbital angular momentum happens to vanish in the ground state and whose spin is $1/2$. It is therefore its spin magnetic moment that the historical experiment revealed.

9. References