In Theme 1, the Stern—Gerlach experiment showed us the failure of the classical description of a magnetic moment, and led us to infer some elements of a new formalism: states represented by vectors, superpositions and complex amplitudes. We now have the postulates of Lesson 1 and the mathematical tools of Theme 2. It is time to proceed in the other direction: to start from the experimental results and the postulates, and deduce the mathematical model of the apparatus, that is, the state space and the observables representing measurements along the different axes.

We shall see that this model is almost entirely determined. The results of the simplest experiments, with one or two apparatuses, fix the dimension of the state space and the form of the eigenvectors, up to a few phase conventions. The spectral theorem then provides the matrices of the observables: these are the Pauli matrices. Once constructed, these matrices allow us to calculate the results of all the other experiments in series, in particular the $z$, $x$, $z$ sequence that had defeated classical physics, as well as measurements along an arbitrary axis. Along the way, we shall see why complex numbers are indispensable to quantum mechanics.

Throughout the lesson, we are concerned only with the internal state of the atom, responsible for its magnetic moment, leaving aside its motion in space. We also assume that this state does not evolve between two apparatuses.

1. What the experiment teaches us

Let us recall the results established in Theme 1. A Stern—Gerlach apparatus is characterised by the direction of its field gradient, which we call its axis: $\mathrm{SG}_z$ denotes an apparatus with axis $z$, $\mathrm{SG}_x$ an apparatus with axis $x$, and so on. At the output of an apparatus, the beam splits into two channels, denoted for example by $z+$ and $z-$. By blocking one of the channels, we prepare a beam in which all the atoms have given the same result; we shall denote by $\ket{z+}$ the state thus prepared in the $z+$ channel, and similarly for the other channels. The experiments have established the following facts.

These three facts will suffice to construct the whole model. The other results of Theme 1, such as the reappearance of the $z-$ channel in the $z$, $x$, $z$ sequence, or the dependence of the probabilities on the angle between two apparatuses, will then be predictions of the model, which we shall compare with experiment.

It is convenient to encode the two outcomes of an analysis by the dimensionless numbers $+1$ and $-1$, rather than by $\pm\mu_0$. The observables that we shall construct, denoted by $\sigma_z$, $\sigma_x$, $\sigma_y$, will therefore have eigenvalues $\pm1$. We shall return to physical units at the end of the lesson.

2. The state space

2.1. The observable associated with the z-axis

According to the second postulate, analysis along $z$ is represented by an observable, that is, a self-adjoint operator $\sigma_z$ acting on the state space $\H$, whose spectrum is the set of possible outcomes. According to (E1), this spectrum is $\{+1,-1\}$. Let $E_+$ and $E_-$ denote the two eigenspaces, and $P_+$ and $P_-$ the orthogonal projectors onto these subspaces. From Theme 2, two eigenvectors associated with distinct eigenvalues of a self-adjoint operator are orthogonal, and the eigenspaces of a self-adjoint operator in finite dimension span the whole space. We therefore have

The first relation also directly expresses the experimental fact that each atom emerges in one channel or the other: the sum of the probabilities of the two outcomes, $\bra\psi P_+\ket\psi+\bra\psi P_-\ket\psi=\braket\psi\psi$, is equal to one for every normalised state.

Fact (E2) is automatically compatible with the postulates: according to the reduction postulate, an atom that has given the result $+1$ is, immediately after the measurement, in a state in $E_+$, and a new measurement of $\sigma_z$ then gives $+1$ with probability $\bra\psi P_+\ket\psi=1$.

2.2. Dimension two

What is the dimension of the eigenspaces? Nothing in the postulates determines it. If $E_+$ had dimension greater than one, it would contain several different states all giving the result $+1$ with certainty; another observable, compatible with $\sigma_z$, would then be needed to distinguish them. Stern—Gerlach experiments reveal no additional internal degree of freedom of this kind: an atom selected in the $z+$ channel has fully determined behaviour in all subsequent experiments, as we shall verify. We therefore make the simplest modelling assumption: the two eigenvalues are non-degenerate. The subspaces $E_\pm$ are one-dimensional, spanned by two unit vectors $\ket{z+}$ and $\ket{z-}$, and the state space has dimension two.

The two components $\alpha$ and $\beta$ are not spatial coordinates, nor are they the components of a classical magnetic moment: they are probability amplitudes. According to the Born rule, an analysis along $z$ gives $+1$ with probability $|\braket{z+}{\psi}|^2=|\alpha|^2$ and $-1$ with probability $|\beta|^2$. Recall also that two vectors differing only by a global phase represent the same physical state.

Let us note a subtlety in passing: each of the vectors $\ket{z+}$ and $\ket{z-}$ is defined only up to a phase. If $\ket{z-}$ is replaced by $e^{i\gamma}\ket{z-}$, it remains a unit vector in the one-dimensional subspace $E_-$, and the projector $\ket{z-}\bra{z-}$ does not change. We shall use this freedom shortly.

2.3. The matrix of the observable along z

The spectral decomposition immediately gives the matrix of $\sigma_z$ in the basis $(\ket{z+},\ket{z-})$:

This is a real diagonal matrix: it is equal to its conjugate transpose and is therefore self-adjoint, as it should be. Its eigenvalues $\pm1$ can be read from the diagonal.

3. The observable along x

3.1. The eigenstates

Let us now turn to the $x$-axis. Analysis along $x$ is represented by an observable $\sigma_x$ acting on the same space $\C^2$, with eigenvalues $\pm1$ according to (E1), and whose eigenvectors $\ket{x+}$ and $\ket{x-}$ form another orthonormal basis. To construct it, we need only find these eigenvectors.

Write $\ket{x+}=a\ket{z+}+b\ket{z-}$. According to (E3), an atom prepared in the state $\ket{x+}$ and analysed along $z$ emerges in each channel with probability $1/2$. The Born rule therefore gives

The two coefficients have the same modulus and differ only in their phase. Since the global phase of $\ket{x+}$ is arbitrary, we may choose $a$ to be real and positive, obtaining

for some phase $\gamma$. This is where the freedom noted above comes into play: by replacing $\ket{z-}$ with $e^{i\gamma}\ket{z-}$, which changes nothing about the physics of the $z$-axis, we can always reduce to $\gamma=0$. We make this choice, which completely fixes the relative phase of the vectors $\ket{z+}$ and $\ket{z-}$:

The second eigenvector $\ket{x-}=c\ket{z+}+d\ket{z-}$ must likewise satisfy $|c|^2=|d|^2=\frac12$ according to (E3), and we may choose $c=\frac{1}{\sqrt2}$. It must also be orthogonal to $\ket{x+}$, since the two outcomes $x+$ and $x-$ are mutually exclusive, as expressed by fact (E2) applied to the $x$-axis. The condition

imposes $d=-c$, whence

We recover the superposition introduced in Theme 1: the state $\ket{x+}$ is not a mixture of atoms in the state $\ket{z+}$ and atoms in the state $\ket{z-}$, but a single state, an equal-weight superposition of the two.

3.2. The matrix

The spectral decomposition $\sigma_x=\ket{x+}\bra{x+}-\ket{x-}\bra{x-}$ gives the required matrix. The bra $\bra{x\pm}$ is the row vector of the conjugate components, which are real here, and the two projectors are

Their sum is readily seen to be the identity. Their difference gives

This matrix is real and symmetric, hence self-adjoint. Its eigenvectors can be checked directly: $\sigma_x(1,1)=(1,1)$ and $\sigma_x(1,-1)=-(1,-1)$. The operator $\sigma_x$ exchanges the two basis vectors: $\sigma_x\ket{z+}=\ket{z-}$ and $\sigma_x\ket{z-}=\ket{z+}$.

This result can also be read as a change of basis, in the sense of Theme 2. The change-of-basis matrix from the basis $(\ket{z\pm})$ to the basis $(\ket{x\pm})$ has as its columns the components of $\ket{x\pm}$:

which is unitary, like any change-of-basis matrix between orthonormal bases. In the basis $(\ket{x\pm})$, the matrix of the observable $\sigma_x$ is $\mathrm{diag}(1,-1)$, and we recover $\sigma_x=S\,\mathrm{diag}(1,-1)\,S^\dagger$ in the basis $(\ket{z\pm})$.

4. The observable along y and complex numbers

4.1. A phase imposed by experiment

Let us proceed in the same way for the $y$-axis. According to (E3) applied to the pair $(z,y)$, the two components of $\ket{y+}$ in the basis $(\ket{z\pm})$ have modulus $\frac{1}{\sqrt2}$, and, upon fixing the global phase, we may write

This time, we can no longer eliminate the phase $\delta$ by redefining $\ket{z-}$: that freedom has already been used for the $x$-axis. The phase $\delta$ therefore has physical content, and it is the pair $(x,y)$ that will determine it. According to (E3), an atom prepared in the state $\ket{y+}$ and analysed along $x$ emerges in the $x+$ channel with probability $1/2$. But

The condition $\frac{1+\cos\delta}{2}=\frac12$ imposes $\cos\delta=0$, that is, $\delta=\pm\frac\pi2$ and $e^{i\delta}=\pm i$.

It remains to choose between $\delta=+\frac\pi2$ and $\delta=-\frac\pi2$. These two choices are related by a mirror symmetry that reverses the direction of the $y$-axis: they describe a right-handed or left-handed frame $(x,y,z)$. Facts (E1) to (E3) cannot distinguish them, since they involve only pairs of axes. We choose $\delta=+\frac\pi2$, which corresponds, as we shall see, to a right-handed frame; this convention will be justified in Lesson 10 by the algebra of angular momentum. Thus

The second eigenvector must have components of modulus $\frac{1}{\sqrt2}$ and be orthogonal to $\ket{y+}$. Writing $\ket{y-}=\frac{1}{\sqrt2}(\ket{z+}+d\ket{z-})$, the inner product is $\frac12(1+(-i)d)$, since the bra $\bra{y+}$ has conjugate components $\frac{1}{\sqrt2}(1,-i)$. It vanishes for $d=-i$:

We can also check that $|\braket{x\pm}{y-}|^2=\frac12$, as required by (E3).

4.2. The matrix

The projectors are calculated as before, taking care with conjugation in the bras:

Their sum is the identity, and their difference gives

This matrix is not real. Let us check that it is self-adjoint by taking its transpose and then conjugating the coefficients:

This is indeed the case, as guaranteed by its construction as a real linear combination of orthogonal projectors. Finally, its eigenvectors can be checked: $\sigma_y(1,i)=(-i\cdot i,\,i\cdot1)=(1,i)$ and $\sigma_y(1,-i)=(-1,i)=-(1,-i)$.

5. The Pauli matrices

5.1. The result

We have constructed the three observables associated with the three axes. They are the matrices introduced by Pauli in 1927.

Let us summarise what comes from experiment and what comes from the postulates. Facts (E1) to (E3) fixed the eigenvalues and the moduli of the components of the eigenvectors, as well as the relative phase $\pm\pi/2$ between the components of $\ket{y+}$. The postulates provided the framework: a Hilbert space, self-adjoint observables and the Born rule for translating measured probabilities into inner products. Theme 2 provided the construction tool: the spectral decomposition $\hat A=\sum_aa\,P_a$. Three conventions remain: the phase of $\ket{z-}$, fixed by the choice $\ket{x+}=(\ket{z+}+\ket{z-})/\sqrt2$, the global phases of the eigenvectors, and the sign of $\delta$, which is related to the orientation of the frame. None of these conventions changes a measurable probability.

5.2. Algebraic properties

The three Pauli matrices have remarkable properties, which can be verified by direct calculation.

• They are self-adjoint: $\sigma_j^\dagger=\sigma_j$ for $j=x,y,z$, as we have checked.
• Their square is the identity: for example, $$\begin{aligned} \sigma_y^2=\begin{pmatrix}0&-i\\ i&0\end{pmatrix}\begin{pmatrix}0&-i\\ i&0\end{pmatrix}=\begin{pmatrix}(-i)(i)&0\\0&(i)(-i)\end{pmatrix}=\begin{pmatrix}1&0\\0&1\end{pmatrix}. \end{aligned}$$ This is the algebraic expression of the fact that their eigenvalues are $\pm1$: a self-adjoint operator whose square is the identity has eigenvalues $\lambda$ satisfying $\lambda^2=1$.
• They have zero trace and determinant $-1$, which again gives their eigenvalues as the roots of $\lambda^2-1=0$.
• The product of two different matrices is proportional to the third. For example, $$\begin{aligned} \sigma_x\sigma_y=\begin{pmatrix}0&1\\1&0\end{pmatrix}\begin{pmatrix}0&-i\\ i&0\end{pmatrix}=\begin{pmatrix}i&0\\0&-i\end{pmatrix}=i\,\sigma_z, \qquad \sigma_y\sigma_x=\begin{pmatrix}-i&0\\0&i\end{pmatrix}=-i\,\sigma_z . \end{aligned}$$ Similarly, $\sigma_y\sigma_z=i\sigma_x$ and $\sigma_z\sigma_x=i\sigma_y$, by cyclic permutation of $(x,y,z)$. With the choice $\delta=-\pi/2$, we would have obtained $\sigma_x\sigma_y=-i\sigma_z$: it is in this sense that our convention corresponds to the right-handed frame.

It follows from the last property that two different Pauli matrices anticommute, $\sigma_x\sigma_y=-\sigma_y\sigma_x$, and that their commutators are

None of these observables commutes with another. According to Theme 2, they therefore have no common eigenbasis; and one can check directly that they do not even have a single eigenvector in common, since the eigenvectors of each have components of modulus $\frac{1}{\sqrt2}$ in the eigenbasis of each of the others. This is the mathematical expression of the incompatibility observed in Theme 1: no state gives a certain outcome for two different axes.

5.3. Physical units

The observables $\sigma_j$ encode the outcomes as $\pm1$. For the electron in the silver atom, we shall see in Lesson 10 that the magnetic moment is associated with spin, an angular momentum whose components are

Relations (7) then become $[S_x,S_y]=i\hbar S_z$ and its cyclic permutations: these are the commutation relations of angular momentum, and this is the origin of our convention for the sign of $\delta$. The magnetic moment is proportional to the spin, $\boldsymbol\mu=\gamma\mathbf S$, with $\gamma\simeq-e/m_e$ for the electron; its components therefore take the values $\pm|\gamma|\hbar/2=\pm e\hbar/(2m_e)$, and we recover the Bohr magneton measured by Stern and Gerlach. Since the sign of $\gamma$ is negative, the channel of the apparatus in which the magnetic moment is positive corresponds to a negative spin: identifying the channels with the spin states requires a convention, which we shall assume has been fixed.

6. Experiments in series

6.1. The calculation rule

Now that the model has been constructed, we can calculate the result of any succession of apparatuses. We need only apply the postulates step by step. Consider an atom in the state $\ket\psi$, which passes through a first apparatus where the outcome $a$, with projector $P_a$, is selected, and then through a second where the outcome $b$, with projector $P_b$, is observed.

1. According to the Born rule, the atom emerges in channel $a$ with probability $p(a)=\bra\psi P_a\ket\psi=\norm{P_a\psi}^2$.
2. According to the reduction postulate, the state of the atoms selected in this channel is $\ket{\psi_a}=P_a\ket\psi/\norm{P_a\psi}$.
3. These atoms enter the second apparatus and emerge in channel $b$ with conditional probability $p(b\mid a)=\norm{P_b\psi_a}^2=\norm{P_bP_a\psi}^2/\norm{P_a\psi}^2$.

The probability of the complete path, relative to the beam entering the first apparatus, is the product of the two:

This formula generalises to any number of apparatuses. Conditional probabilities, relative to the beam entering a given apparatus, must be carefully distinguished from probabilities relative to the initial beam: the former are measured by counting the impacts at the output of an apparatus, the latter by taking into account all the atoms lost at the selection stages.

6.2. A single axis: ZZ

Let us prepare a beam in the state $\ket{z+}$ and analyse it along $z$. The amplitudes $\braket{z+}{z+}=1$ and $\braket{z-}{z+}=0$ give $p(z+)=1$ and $p(z-)=0$. We recover fact (E2), which was one of the inputs to the construction.

6.3. Two perpendicular axes: ZX

Let us now analyse the state $\ket{z+}$ along $x$. We can decompose $\ket{z+}$ in the eigenbasis of $\sigma_x$: adding the expressions for $\ket{x+}$ and $\ket{x-}$ gives

whence $p(x+\mid z+)=p(x-\mid z+)=\frac12$. Again we recover one of the inputs to the construction. Note that the expectation value of $\sigma_x$ in the state $\ket{z+}$ is zero: this can be seen from the probabilities, or directly, $\bra{z+}\sigma_x\ket{z+}=\bra{z+}\bigl(\sigma_x\ket{z+}\bigr)=\braket{z+}{z-}=0$.

6.4. Three apparatuses: ZXZ

Here is now a genuine prediction. Starting from the state $\ket{z+}$, let us select the $x+$ channel of a $\mathrm{SG}_x$ apparatus, and then analyse along $z$ again. After selection, the state is $\ket{x+}=\frac{1}{\sqrt2}(\ket{z+}+\ket{z-})$, and the last apparatus gives

The $z-$ channel, which was empty in the ZZ experiment, reappears: this is exactly what was observed in Theme 1, and what had defeated the picture of a simple classical sorting process. Relative to the initial beam, the probabilities of the two complete paths are, according to (8),

Each final output channel contains one quarter of the initial atoms, the remaining half having been stopped by the $x-$ filter.

The model accounts for this result without any additional assumption: the $x+$ filter does not merely sort atoms that already have a value of $\mu_x$ and a value of $\mu_z$; it prepares a new state, $\ket{x+}$, which retains no memory of the $z+$ outcome obtained previously. This is a consequence of the reduction postulate and the non-commutation of $\sigma_x$ and $\sigma_z$.

6.5. ZYZ: same probabilities, different states

Let us replace the intermediate apparatus with one whose axis is $y$, still selecting the positive channel. The last apparatus receives the state $\ket{y+}=\frac{1}{\sqrt2}(\ket{z+}+i\ket{z-})$, and

The ZXZ and ZYZ sequences therefore give exactly the same results. Yet the states $\ket{x+}$ and $\ket{y+}$ are different: an apparatus with axis $x$ gives $x+$ with certainty for the former, and $x+$ or $x-$ with probability $\frac12$ for the latter. Thus two states can have the same statistics for one axis and differ for another. What distinguishes them is the relative phase between their components along $\ket{z+}$ and $\ket{z-}$, which is $1$ in one case and $i$ in the other: a relative phase has physical consequences, whereas a global phase does not.

6.6. Selecting is not the same as doing nothing

In the ZXZ sequence, what would happen if neither of the two channels of the intermediate apparatus were blocked? If the atoms are detected, or if their trajectory makes it possible in principle to know which channel they passed through, the apparatus performs a measurement along $x$, and the last apparatus receives a mixture of atoms in the states $\ket{x+}$ and $\ket{x-}$, in equal proportions: each gives $z\pm$ with probability $\frac12$. But if the two beams emerging from the intermediate apparatus could be perfectly recombined, without any information about the path taken, the state $\ket{z+}=\frac{1}{\sqrt2}(\ket{x+}+\ket{x-})$ would be reconstituted, and the last apparatus would give $z+$ with certainty. A Stern—Gerlach apparatus therefore measures nothing until the path taken has been recorded in one way or another. Such recombination is extremely difficult to achieve with atoms; but this is exactly the situation in the interferometers studied in Theme 1, where the two paths are recombined coherently. We shall return to this connection between measurement and information when discussing composite systems.

7. An arbitrary axis

7.1. The observable

Finally, consider an apparatus whose axis has an arbitrary direction, specified by the unit vector

where $\theta$ is the angle with the $z$-axis and $\varphi$ the azimuth, measured from the $x$-axis in the $(x,y)$ plane. How do we construct the corresponding observable $\sigma_{\mathbf u}$? Classically, the component of a magnetic moment along $\mathbf u$ is the linear combination $\mathbf u\cdot\boldsymbol\mu=u_x\mu_x+u_y\mu_y+u_z\mu_z$. We assume that the same holds for quantum expectation values, $\langle\sigma_{\mathbf u}\rangle=u_x\langle\sigma_x\rangle+u_y\langle\sigma_y\rangle+u_z\langle\sigma_z\rangle$ in every state. Since two self-adjoint operators that have the same expectation value in every state are equal, this imposes

To obtain the matrix, we have combined $\sin\theta(\cos\varphi\,\sigma_x+\sin\varphi\,\sigma_y)$, whose upper-right coefficient is $\sin\theta(\cos\varphi-i\sin\varphi)=\sin\theta\,e^{-i\varphi}$. This matrix is self-adjoint, since it is a linear combination with real coefficients of self-adjoint matrices; this can also be checked directly, as its two off-diagonal coefficients are conjugates of one another.

7.2. A first check: quantisation along every axis

The preceding assumption is acceptable only if it respects fact (E1): an apparatus with an arbitrary axis must also give the two outcomes $\pm1$. This is not obvious a priori, since $\sigma_{\mathbf u}$ is a combination of the three matrices. Let us calculate its square. On expanding, the cross terms cancel in pairs owing to anticommutation, for example $u_xu_y(\sigma_x\sigma_y+\sigma_y\sigma_x)=0$, leaving

The eigenvalues of $\sigma_{\mathbf u}$ therefore satisfy $\lambda^2=1$, and since its trace is zero, they are $+1$ and $-1$. The model does indeed predict quantisation along every axis, in accordance with experiment. We can also calculate the characteristic polynomial of the matrix (9) directly: $(\cos\theta-\lambda)(-\cos\theta-\lambda)-\sin^2\theta=\lambda^2-1$.

7.3. The eigenstates

The eigenvectors of $\sigma_{\mathbf u}$ are

Let us check the first, setting $c=\cos\frac\theta2$ and $s=\sin\frac\theta2$, and using $\cos\theta=c^2-s^2$ and $\sin\theta=2sc$. The first component of $\sigma_{\mathbf u}\ket{u+}$ is

and the second is

Thus $\sigma_{\mathbf u}\ket{u+}=\ket{u+}$, and similarly one checks that $\sigma_{\mathbf u}\ket{u-}=-\ket{u-}$. We recover the special cases already constructed: $\theta=0$ gives $\ket{z+}$, $\theta=\frac\pi2$ and $\varphi=0$ give $\ket{x+}$, and $\theta=\frac\pi2$ and $\varphi=\frac\pi2$ give $\ket{y+}$. For an arbitrary axis in the $(x,y)$ plane, we obtain $\ket{u+}=\frac{1}{\sqrt2}(\ket{z+}+e^{i\varphi}\ket{z-})$, which is exactly the form inferred in Theme 1 to describe the continuous range of preparations $\ket{u+}$.

7.4. A prediction: the cosine-squared law

Let us prepare an atom in the state $\ket{z+}$ and analyse it along $\mathbf u$. The Born rule gives

The probability depends only on the angle $\theta$ between the two axes, and not on the azimuth, as required by rotational symmetry about the $z$-axis. It is $1$ for identical axes, $\frac12$ for perpendicular axes and $0$ for opposite axes: an atom prepared in the $z+$ channel emerges with certainty in the $-$ channel of a reversed apparatus, which is evident since reversing the apparatus amounts to exchanging its two channels. Between these values, the $\cos^2(\theta/2)$ law is a prediction of the model, confirmed experimentally. We deduce the expectation value $\langle\sigma_{\mathbf u}\rangle=\cos^2\frac\theta2-\sin^2\frac\theta2=\cos\theta$, which is the projection onto $\mathbf u$ of a unit vector directed along $z$: on average, everything behaves as for the projection of a classical magnetic moment, even though each individual measurement gives only $\pm1$.

For a $z$, $\mathbf u$, $z$ sequence with selection of the $u+$ channel, formula (8) gives the final probabilities, relative to the initial beam,

For $\theta=\frac\pi2$, we recover the values $\frac14$ of the ZXZ sequence.

7.5. The beam emerging from the oven

One final consequence deserves emphasis. After a choice of global phase that makes $\alpha$ real and positive, every normalised state $\alpha\ket{z+}+\beta\ket{z-}$ can be written in the form (10): it suffices to set $\alpha=\cos\frac\theta2$ and $\beta=e^{i\varphi}\sin\frac\theta2$, which is always possible since $|\alpha|^2+|\beta|^2=1$. In other words, every pure state is the state $\ket{u+}$ for some direction $\mathbf u$, and therefore gives a certain outcome for an apparatus oriented along that direction. The pure states of the system are thus in correspondence with directions in space, that is, with the points of a sphere: this is the Bloch sphere, which we shall study in Lesson 10.

Yet the beam emerging from the oven gives two spots of equal intensity whatever the orientation of the apparatus. From the preceding argument, no pure state can have this property. The atoms emerging from the oven are therefore not all in the same pure state: they form a statistical mixture of atoms in different, randomly oriented states. To describe such a beam, we shall have to generalise the notion of state and introduce mixed states, which we shall encounter in Theme 4.

8. Summary

Let us summarise the approach followed in this lesson, which is the one used to construct the quantum model of any system.

• The possible outcomes of a measurement determine the spectrum of the corresponding observable: here, two values for each axis.
• Reproducibility experiments show that the states prepared by selection are eigenstates and, together with a simplifying assumption, determine the dimension of the state space: here, two.
• The probabilities measured in experiments in series determine, through the Born rule, the moduli of the inner products between the eigenstates of the different observables, and ultimately the eigenvectors themselves, up to phase conventions. It was at this stage that complex numbers proved indispensable.
• The spectral decomposition then reconstructs each observable from its eigenvalues and its eigenprojectors: this gives the Pauli matrices.
• The model can then predict the results of other experiments: the reappearance of the $z-$ channel in the ZXZ sequence, quantisation along every axis, the $\cos^2(\theta/2)$ law, and the impossibility of describing the beam emerging from the oven by a pure state.

The following lessons will apply the same approach to richer systems whose state space is infinite-dimensional: a particle in space. For further discussion of two-level calculations and measurement, see the MIT notes, course 8.321, session 3^[1], as well as J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics^[2], Chapter 1, whose presentation of the Stern—Gerlach experiments inspired this lesson.

9. References