X Consequences of the measurement postulate

We define the expectation value of an observable $A$ via $⟨A⟩=\int aP(a)𝑑a$ (for a continuous observable) or $⟨A⟩={\sum }_{a}aP(a)$ (for a discrete observable). This represents the averaged outcome of many experiments on identically prepared quantum systems.

Inserting the expression (125) for the probability $P(a)$ and making use of the form $\widehat{A}={\sum }_n{a}_n|n⟩⟨n|$ of the operator in the eigenrepresentation, we find the important result

$$⟨A⟩=⟨\psi |\widehat{A}|\psi ⟩.$$

(131)

This allows us to calculate an expectation value without determining all eigenfunctions and eigenvalues of $\widehat{A}$. In particular, for a particle with wave function $\psi (x)$, the expectation value is given by

$$⟨A⟩=\int {\psi }^{*}(x)[\widehat{A}\psi (x)]𝑑x.$$

(132)

The expectation value

$$⟨A^m⟩=⟨\psi |{\widehat{A}}^m|\psi ⟩$$

(133)

is known as the $m$th moment of $\widehat{A}$. The uncertainty of an observable is defined as the root-mean square standard deviation

$$\mathrm{\Delta }A=\sqrt{⟨A^2⟩-{⟨A⟩}^2}.$$

(134)

When the system state is identical to an eigenstate $|n⟩$ of $\widehat{A}$, it follows from the orthonormality of these states that $P(a)=0$ vanishes for all $a\ne a_n$, while $P(a_n)=1$. In this special case, the value of the observable is well defined, and $\mathrm{\Delta }A=0$. The final part $IVc$ of the measurement postulate asserts that this situation is realised right after a measurement. Therefore, in immediately repeated measurements of the same observable one will obtain a sequence of identical results. However, while the post-measurement state is an eigenstate of the measured observable, it is in general not an eigenstate of other relevant observables. Thus, if one consecutively measures the values of different observables one will in general obtain a random sequence of outcomes with finite probabilities. The following considerations provide a simple rule that quantifies these observations in terms of the uncertainties of the different observables.

For each pair of operators ${\widehat{A}}^{(1)}$, ${\widehat{A}}^{(2)}$ we can introduce a new operator, called the commutator, which is denoted by $[{\widehat{A}}^{(1)},{\widehat{A}}^{(2)}]$. This operator is defined as

$$[{\widehat{A}}^{(1)},{\widehat{A}}^{(2)}]={\widehat{A}}^{(1)}{\widehat{A}}^{(2)}-{\widehat{A}}^{(2)}{\widehat{A}}^{(1)}.$$

(135)

In other words, the commutator is an operator which acts as

$$[{\widehat{A}}^{(1)},{\widehat{A}}^{(2)}]\psi ={\widehat{A}}^{(1)}{\widehat{A}}^{(2)}\psi -{\widehat{A}}^{(2)}{\widehat{A}}^{(1)}\psi .$$

(136)

If ${\widehat{A}}^{(1)}$ and ${\widehat{A}}^{(2)}$ were numbers the commutator would clearly vanish. For operators, however, the order generally matters. For instance, applying the product rule one finds $\frac{d}{dx}(x\psi )=\psi +x\frac{d}{dx}(\psi )$. Since $\widehat{p}=-i\mathrm{\hslash }\frac{d}{dx}$ this gives

$$[\widehat{x},\widehat{p}]=i\mathrm{\hslash },$$

which is the most important of all commutators.

For some operators, $[{\widehat{A}}^{(1)},{\widehat{A}}^{(2)}]=0$. One then say that the operators ${\widehat{A}}^{(1)}$ and ${\widehat{A}}^{(1)}$ commute. An example is found in three-dimensional systems: Since the momentum operators are defined by partial derivatives we have, e.g., $[\widehat{x},{\widehat{p}}_y]=0$.

The importance of the commutator results from the following mathematical theorem: if two operators commute (i.e., their commutator $[{\widehat{A}}^{(1)},{\widehat{A}}^{(2)}]=0$ vanishes) then the eigenstates of ${\widehat{A}}^{(1)}$ are identical to the eigenstates of ${\widehat{A}}^{(2)}$. This holds even though the eigenvalues $a_n^{(1)}$, $a_n^{(2)}$ of both operators are generally different. In the joint eigenbasis, the operators then read ${\widehat{A}}^{(1)}={\sum }_n{a}_n^{(1)}|n⟩⟨n|$, ${\widehat{A}}^{(2)}={\sum }_n{a}_n^{(2)}|n⟩⟨n|$.

Heisenberg’s uncertainty principle can be generalised to quantify the extent of incompatibility of two arbitrary observables $A^{(1)}$, $A^{(2)}$ in terms of their uncertainties $\mathrm{\Delta }A=\sqrt{⟨A^2⟩-{⟨A⟩}^2}$:

$$\mathrm{\Delta }{A}^{(1)}\mathrm{\Delta }{A}^{(2)}\ge \frac{1}{2}\left|⟨[{\widehat{A}}^{(1)},{\widehat{A}}^{(2)}]⟩\right|.$$

(137)

This gives a lower bound for the product of uncertainties of two observables, and expresses this lower bound by the expectation value of their commutator. For momentum and position, $[\widehat{x},\widehat{p}]=i\mathrm{\hslash }$, which is just a constant. Therefore $⟨i\mathrm{\hslash }⟩=i\mathrm{\hslash }$, and $|i\mathrm{\hslash }|=\mathrm{\hslash }$. This recovers the conventional uncertainty relation $\mathrm{\Delta }x\mathrm{\Delta }p\ge \mathrm{\hslash }/2$.

According to Heisenberg’s uncertainty principle, for commuting observables we find

$$\mathrm{\Delta }{A}^{(1)}\mathrm{\Delta }{A}^{(2)}\ge 0.$$

(138)

But since the uncertainty of any observable $\widehat{A}$ is never negative, $\mathrm{\Delta }A\ge 0$, this uncertainty relation poses no restriction whatsoever.

Indeed, if the wave function of a system is identical to a joint eigenfunction of two commuting operators ${\widehat{A}}^{(1)}$ and ${\widehat{A}}^{(2)}$, then the values of these operators are both defined precisely. Because of this, observables with vanishing commutator are also called simultaneous variables. An example is the position $\widehat{x}$ and the momentum ${\widehat{p}}_y$, which can be both determined at the same time (the wave function $\psi (x,y)$ is then well localised in $x$ direction, while it looks like a plane wave in $y$ direction).

On the other hand, pairs of observables such as $\widehat{x}$ and ${\widehat{p}}_x$ which do not commute are called incompatible, as they cannot be both fixed at the same time.

We can also understand this difference by reconsidering the effects of consecutive measurements of two observables ${\widehat{A}}^{(1)}$ and ${\widehat{A}}^{(2)}$.

• •

  If the observables commute, then the order of measurements does not affect the probabilities, which are both calculated with the same (joint) eigenstates $|n⟩$. After the two experiments, the state of system is a joint eigenstate, in which both observables are well defined.

• •

  If the two observables are incompatible, the order of measurements matters. This is because the generalised wave function of both observables has to be calculated with different eigenstates. Moreover, the state of the system after the two experiments is an eigenfunction of the observable that has been measured last. In this state, only the second observable is well defined, but the first observable is not (so if one carries out a third experiment which measures again the first observable, one generally finds a different value than in the first measurement).

In this sense, a measurement of an observable destroys the information obtained in earlier measurements of incompatible observables, but does not destroy the information obtained in earlier measurements of simultaneous observables.

In many situations the values of one observable alone do not suffice to uniquely determine the state of a system. This occurs whenever eigenvalues are degenerate, and thus are associated with several eigenstates. In such situations the complete determination of a state requires to measure a larger set ${\widehat{A}}^{(l)}$ of simultaneous observables that all commute with each other, $[{\widehat{A}}^{(l)},{\widehat{A}}^{(m)}]=0$. This property guarantees that one can find a joint eigenbasis for all of these observables, given by states $|𝐚⟩=|a^{(1)},a^{(2)},a^{(3)},\mathrm{\dots }⟩$ fulfilling ${\widehat{A}}^{(l)}|𝐚⟩=a^{(l)}|𝐚⟩$. These states are only fully specified by knowledge of the eigenvalues $a^{(l)}$ of the full set of simultaneous observables.