3.3Observables

So far the outcomes of a test have had names: plus and minus, up and down, horizontal and vertical. Physicists like numbers. This lesson writes a number on each outcome, and finds that the test and its numbers together are captured by a single mathematical object, a matrix, whose properties answer the questions an experimenter would ask.

Numbers on outcomes

Take a complete test and attach a real number to each of its $N$ outcomes: $a_1, a_2, \dots, a_N$. Sometimes the numbers suggest themselves. A Stern–Gerlach magnet deflects atoms by amounts that correspond to $+\mu$, $0$ and $-\mu$ of magnetic moment, so those are the natural labels. Sometimes they are arbitrary: for the two exits of a polarising crystal we might as well write $+1$ and $-1$.

Once that is done, each run of the test yields a number. We then say, loosely, that we have measured something, and call the something an observable, $A$. Remember from lesson 1.5 what this does and does not mean. There was no value of $A$ sitting in the system beforehand. A quantum measurement is nothing more than a test whose outcomes carry numerical labels, and the only values $A$ can ever show are the labels $a_1, \dots, a_N$.

Do the measurement many times on systems all prepared in the state $\mathsf v$. If $\mathsf e_r$ is the vector for the $r$th outcome, rule H says that outcome occurs with probability $|\ip{\mathsf e_r}{\mathsf v}|^2$. The average of the numbers obtained, called the mean or expectation value and written $\langle A\rangle$, is each label weighted by how often it turns up:

The same thing as a matrix

That formula for the mean mixes together two quite separate things: the state $\mathsf v$, which belongs to the preparation, and the axes $\mathsf e_r$ with their labels $a_r$, which belong to the test. A little algebra pulls them apart. Write out the inner product in components, $\ip{\mathsf e_r}{\mathsf v} = \sum_n \overline{(e_r)_n}\,v_n$, substitute, and collect everything that does not involve $\mathsf v$:

The numbers $A_{mn}$, arranged in a square with $m$ labelling rows and $n$ columns, form a matrix. It contains everything about the test and its labels, and nothing about the state. The mean is then written compactly as

Because the labels $a_r$ are real, the matrix has a special symmetry: flipping it across its diagonal and conjugating every entry gives it back, $A_{mn} = \overline{A_{nm}}$. Such a matrix is called Hermitian. Every observable is represented by a Hermitian matrix. The converse is true as well, as we shall see: every Hermitian matrix represents an observable.

Three examples worth knowing by heart

Label the exits of the three standard polarisation tests $+1$ and $-1$. Working out $A_{mn}$ from the formula, with components taken along the horizontal and vertical axes, gives

These three are known as the Pauli matrices. Their means in a state are the three numbers $s_1$, $s_2$, $s_3$ of the last lesson: the coordinates of the state on the sphere. (They are numbered here in the order of the tests. Most books call the same three $\sigma_z$, $\sigma_x$ and $\sigma_y$, after the directions of a magnet rather than the settings of a crystal.)

Yes-or-no questions

The simplest observables are those whose labels are all 0 or 1: yes or no. For a single state $\mathsf v$, the question “does the system pass the test for $\mathsf v$?” is represented by the matrix

It takes any vector $\mathsf u$ and returns its shadow along $\mathsf v$, which is why it is called a projector. Casting a shadow twice is the same as casting it once, so $P^2 = P$. And any observable is a sum of projectors on its outcomes, each weighted by its label:

Eigenvectors: the states with a sure answer

Apply the matrix $A$ to one of the vectors that define it. Using the formula above, $P_r$ leaves $\mathsf e_r$ alone and every other projector kills it, so

The matrix does not turn this vector at all; it only stretches it, by the factor $a_r$. A vector that a matrix merely stretches is called an eigenvector of the matrix, and the stretch factor is the corresponding eigenvalue. (Eigen is German for “own”: these are the matrix’s own special directions.)

So the dictionary between the laboratory and the algebra reads:

• the possible results of a measurement are the eigenvalues of its matrix;
• the states for which the result is certain are its eigenvectors. Prepare the system in $\mathsf e_r$ and the measurement gives $a_r$ every time;
• conversely, if a state gives the same result every time, it is an eigenvector. The spread of the results about their mean, $\Delta A$, defined by $(\Delta A)^2 = \langle A^2\rangle - \langle A\rangle^2$, is zero exactly for eigenvectors.

Now run the dictionary backwards. Suppose someone hands you a Hermitian matrix, with no test attached. It is a theorem of algebra that its eigenvalues are all real, and that its eigenvectors can be chosen to form an orthonormal basis. By rule G* that basis is a test that can be carried out, and the eigenvalues are real numbers to write on its outcomes. So the matrix defines an observable after all. To find out what a measurement of it can give, you find its eigenvalues.

For a two-by-two matrix this is a short calculation. The eigenvalues $a$ are the numbers for which $A - a\cdot\mathbb 1$ squashes some vector to zero, and that happens when its determinant vanishes. For $\sigma_2$:

The results are $\pm1$, and the sure states are diagonal and anti-diagonal, as they should be.

Tests that do not share axes

Two observables defined on the same test, with different labels on its outcomes, are both diagonal matrices when written along that test’s axes, and diagonal matrices can be multiplied in either order: $AB = BA$. They are said to commute. The converse holds too. If two Hermitian matrices commute, there is a single set of axes that are eigenvectors of both, and so a single test that measures both at once.

If they do not commute, there is no such test. Try $\sigma_1$ and $\sigma_2$:

The difference $AB - BA$ is called the commutator and written $[A, B]$. It is not zero here, and that is the algebraic face of the trade-off in the last lesson: the horizontal-against-vertical test and the diagonal test have no eigenvectors in common, so no state is sure for both.

A sum is a new test

Add two Hermitian matrices and you get another, so $A + B$ is an observable. It is natural to assume you would measure it by measuring $A$, measuring $B$, and adding the results. If $A$ and $B$ do not commute, that is wrong. $A + B$ has its own eigenvectors, different from those of either, and so its own test; and its possible results are its eigenvalues, which need not be sums of theirs. For instance $\sigma_1 + \sigma_2$ has eigenvalues $\pm\sqrt2$, not the $-2, 0, +2$ you would get by adding $\pm1$ to $\pm1$. Its test is a linear-polarisation test with axes at 22.5°.

This finally disposes of the puzzle of lesson 1.5. For three magnet directions 120° apart, the three matrices that represent the three measurements really do add up to the zero matrix: $\mu_1 + \mu_2 + \mu_3 = 0$ is true as a statement about observables, and so about their means in any state. It is false as a statement about individual results, each of which is $\pm\mu$, because the three do not commute and their results were never meant to be added.

This lesson follows §3-6 and §3-7. The book goes on to functions of an observable (same test, new labels), to how the matrix changes when the axes are changed, and to several results about eigenvalues that are set as exercises.