Commutators and the Heisenberg Uncertainty Principle
Compatible Observables and Quantum Uncertainty
Commutators and the Heisenberg Uncertainty Principle: Overview and Plan
In the previous section, we saw that an ideal quantum measurement generally changes the state of the system.
Suppose a measurement of an observable \(\hat A\) produces the value \(a_n\). For a nondegenerate eigenvalue, the state afterward is
\[ \left\lvert a_n \right\rangle. \]
If we now measure a different observable \(\hat B\), the state may change again. A subsequent measurement of \(A\) need not reproduce the original result.
This raises an important question:
When can two observables possess definite values simultaneously?
The answer is encoded in the algebra of the corresponding operators.
If two observables are represented by operators that commute, they are compatible and can—under the usual conditions encountered here—be described using common eigenstates.
If they do not commute, the observables are generally incompatible.
The Heisenberg uncertainty principle provides a quantitative connection between this noncommutativity and the statistical uncertainties of measurements.
The specific goals for this chapter are:
- Understand why successive measurements of different observables can depend on the order in which they are performed.
- Introduce compatible and incompatible observables.
- Define the commutator and learn its most important algebraic properties.
- Understand the connection between commuting operators and common eigenstates.
- Introduce complete sets of commuting observables.
- Connect the uncertainty of an observable to the spread of measurement outcomes introduced in the previous section.
- Derive the general form of the Heisenberg uncertainty principle.
- Apply the uncertainty principle to position and momentum.
- Preview the commutation relations of angular momentum.
The commutator of two operators is defined as
\[ \boxed{ [\hat A,\hat B] = \hat A\hat B-\hat B\hat A. } \]
Two observables are compatible if their operators commute,
\[ \boxed{ [\hat A,\hat B]=0. } \]
For the Hermitian operators considered here, commuting observables can be described using a common set of eigenstates.
A set of mutually commuting observables whose eigenvalues uniquely specify a basis state is called a complete set of commuting observables (CSCO).
For two observables \(\hat A\) and \(\hat B\), the general Heisenberg uncertainty relation is
\[ \boxed{ \sigma_A\sigma_B \geq \frac12 \left| \left\langle [\hat A,\hat B] \right\rangle \right|. } \]
Position and momentum satisfy the canonical commutation relation
\[ \boxed{ [\hat r_i,\hat p_j] = i\hbar\delta_{ij}. } \]
In particular,
\[ \boxed{ [\hat x,\hat p_x]=i\hbar, } \]
which gives the position–momentum uncertainty relation
\[ \boxed{ \sigma_x\sigma_{p_x} \geq \frac{\hbar}{2}. } \]
The Cartesian components of angular momentum do not commute. For example,
\[ \boxed{ [\hat L_x,\hat L_y] = i\hbar\hat L_z, } \]
with the corresponding cyclic relations.
However,
\[ \boxed{ [\hat{\mathbf L}^2,\hat L_i]=0, } \]
so the magnitude of the angular momentum and one chosen component can be specified simultaneously.
You should understand that
- successive measurements of different observables can depend on the order in which they are performed because a quantum measurement generally changes the state;
- commuting observables are compatible: they can be assigned simultaneous definite values in a common eigenstate;
- noncommuting observables are generally incompatible, and a complete common eigenbasis does not exist;
- a CSCO is needed when one observable alone does not uniquely identify the quantum state because of degeneracy;
- the uncertainty \(\sigma_A\) describes the statistical spread of possible outcomes of repeated measurements of \(A\) on identically prepared systems;
- the Heisenberg uncertainty principle connects the possible simultaneous uncertainties of two observables to the commutator of their operators;
- a nonzero commutator does not necessarily imply a nonzero lower bound from the Robertson relation in every state, because the bound contains the state-dependent expectation value \(\langle[\hat A,\hat B]\rangle\);
- the position–momentum uncertainty relation describes a limitation on the quantum state itself, not merely experimental error or disturbance caused by imperfect measurements;
- for angular momentum, the noncommutativity of the Cartesian components explains why only the magnitude and one chosen component can be specified simultaneously.
The derivations of the commutator identities, the general uncertainty relation, and the angular-momentum commutators are included to show where these results come from, but the intermediate algebra is not intended as material to memorize.
The stronger Schrödinger–Robertson uncertainty relation is included as additional detail and is not part of the core knowledge expected for this course.
1. Successive Measurements and Compatibility
1.1 Measuring two different observables
Suppose the system is initially in the state
\[ \left\lvert \psi \right\rangle, \]
and we first measure the observable \(\hat A\).
Assume the result is the nondegenerate eigenvalue \(a_n\).
The state becomes
\[ \left\lvert \psi \right\rangle\longrightarrow \left\lvert a_n \right\rangle. \]
Now measure another observable \(\hat B\), whose eigenstates satisfy
\[ \hat B\left\lvert b_m \right\rangle= b_m\left\lvert b_m \right\rangle. \]
We can expand the state \(\left\lvert a_n \right\rangle\) in the eigenbasis of \(\hat B\):
\[ \left\lvert a_n \right\rangle= \sum_m \left\lvert b_m \right\rangle\left\langle b_m \middle| a_n \right\rangle. \]
The probability that the measurement of \(B\) gives the result \(b_m\) is therefore
\[ \boxed{ P(b_m|a_n) = \left| \left\langle b_m \middle| a_n \right\rangle\right|^2. } \]
After obtaining \(b_m\), the state becomes
\[ \left\lvert a_n \right\rangle\longrightarrow \left\lvert b_m \right\rangle. \]
1.2 Measuring \(A\) again
Now suppose we immediately measure \(A\) once more.
The probability of recovering the original value \(a_n\) is
\[ P(a_n|b_m) = \left| \left\langle a_n \middle| b_m \right\rangle\right|^2. \]
Unless \(\left\lvert b_m \right\rangle\) is also an eigenstate of \(\hat A\), this probability need not be one.
Thus the intermediate measurement of \(B\) may destroy the definite value of \(A\) established by the first measurement.
Schematically,
\[ \boxed{ A \longrightarrow B \longrightarrow A } \]
need not reproduce the result of the first measurement.
1.3 Compatible observables
There is, however, an important special case.
Suppose \(\hat A\) and \(\hat B\) possess a common set of eigenstates,
\[ \left\lvert a,b \right\rangle, \]
such that
\[ \hat A\left\lvert a,b \right\rangle= a\left\lvert a,b \right\rangle\]
and
\[ \hat B\left\lvert a,b \right\rangle= b\left\lvert a,b \right\rangle. \]
The same state then possesses definite values of both observables.
A measurement of one observable does not necessarily destroy the definite value of the other.
Such observables are called compatible observables.
We therefore need a mathematical criterion that tells us whether two observables are compatible.
That criterion is provided by the commutator.
2. Commutators and Compatible Observables
2.1 The commutator
The commutator of two operators \(\hat A\) and \(\hat B\) is defined by
\[ \boxed{ [\hat A,\hat B] = \hat A\hat B-\hat B\hat A. } \]
If
\[ [\hat A,\hat B]=0, \]
the operators commute.
If
\[ [\hat A,\hat B]\neq0, \]
they do not commute.
For the Hermitian operators encountered in ordinary quantum mechanics, commuting observables can be simultaneously diagonalized and therefore possess a common orthonormal eigenbasis.
Thus,
\[ \boxed{ [\hat A,\hat B]=0 \quad\Longrightarrow\quad \text{$A$ and $B$ are compatible}. } \]
Noncommuting observables are called incompatible observables.
2.2 Useful properties of commutators
Commutators obey a number of useful algebraic relations.
| Property | Relation |
|---|---|
| Antisymmetry | \([\hat A,\hat B]=-[\hat B,\hat A]\) |
| Self-commutator | \([\hat A,\hat A]=0\) |
| Linearity | \([\hat A,\hat B+\hat C]=[\hat A,\hat B]+[\hat A,\hat C]\) |
| Scalar multiplication | \([\hat A,c\hat B]=c[\hat A,\hat B]\) |
| Product rule | \([\hat A,\hat B\hat C]=[\hat A,\hat B]\hat C+\hat B[\hat A,\hat C]\) |
| Product rule | \([\hat A\hat B,\hat C]=\hat A[\hat B,\hat C]+[\hat A,\hat C]\hat B\) |
Starting with
\[ [\hat A,\hat B\hat C] = \hat A\hat B\hat C - \hat B\hat C\hat A, \]
add and subtract \(\hat B\hat A\hat C\):
\[ [\hat A,\hat B\hat C] = \hat A\hat B\hat C - \hat B\hat A\hat C + \hat B\hat A\hat C - \hat B\hat C\hat A. \]
Grouping the terms,
\[ [\hat A,\hat B\hat C] = (\hat A\hat B-\hat B\hat A)\hat C + \hat B(\hat A\hat C-\hat C\hat A). \]
Therefore,
\[ \boxed{ [\hat A,\hat B\hat C] = [\hat A,\hat B]\hat C + \hat B[\hat A,\hat C]. } \]
2.3 Why do commuting observables have common eigenstates?
Suppose
\[ \hat A\left\lvert a \right\rangle= a\left\lvert a \right\rangle\]
and
\[ [\hat A,\hat B]=0. \]
Then
\[ \hat A\hat B\left\lvert a \right\rangle= \hat B\hat A\left\lvert a \right\rangle= a\hat B\left\lvert a \right\rangle. \]
Thus \(\hat B\left\lvert a \right\rangle\) is itself contained in the eigenspace of \(\hat A\) with eigenvalue \(a\).
First consider a nondegenerate eigenvalue \(a\):
\[ \hat A\left\lvert a \right\rangle= a\left\lvert a \right\rangle. \]
Because
\[ [\hat A,\hat B]=0, \]
we have
\[ \hat A\hat B\left\lvert a \right\rangle= a\hat B\left\lvert a \right\rangle. \]
Therefore \(\hat B\left\lvert a \right\rangle\) is an eigenvector of \(\hat A\) with eigenvalue \(a\).
If the eigenvalue is nondegenerate, its eigenspace is one-dimensional. Consequently,
\[ \hat B\left\lvert a \right\rangle= b\left\lvert a \right\rangle. \]
Thus \(\left\lvert a \right\rangle\) is also an eigenstate of \(\hat B\).
If \(a\) is degenerate, \(\hat B\) may mix different vectors inside the eigenspace corresponding to \(a\).
However, because \(\hat B\) leaves that eigenspace invariant, we can diagonalize \(\hat B\) within it.
The resulting basis vectors are simultaneous eigenstates of both \(\hat A\) and \(\hat B\).
Thus commuting Hermitian operators can be represented using a common orthonormal eigenbasis.
For finite-dimensional Hermitian operators, simultaneous diagonalization of commuting operators is straightforward.
For general unbounded operators in infinite-dimensional Hilbert spaces, domains and continuous spectra require additional mathematical care.
For the operators considered in this course, we will use the standard quantum-mechanical statement that compatible observables can be represented by mutually commuting operators with a common set of eigenstates.
2.4 Complete sets of commuting observables
One observable does not always uniquely identify a quantum state.
If an eigenvalue is degenerate, several linearly independent states may have the same value of that observable.
We can then introduce additional commuting observables to distinguish them.
A collection of mutually commuting observables whose eigenvalues uniquely specify a basis state is called a complete set of commuting observables, or CSCO.
This idea will become particularly important for angular momentum and for the hydrogen atom.
3. The Heisenberg Uncertainty Principle
In the section on States and Operators, we introduced the variance and standard deviation of an observable.
For an observable \(\hat A\) in the state \(\left\lvert \psi \right\rangle\),
\[ \sigma_A^2 = \langle A^2\rangle - \langle A\rangle^2 \]
describes the spread of the possible outcomes of repeated measurements.
We will now ask how small the uncertainties of two different observables can be at the same time.
3.1 Centered operators
It is convenient to define the operator describing the deviation from the expectation value,
\[ \boxed{ \Delta\hat A = \hat A-\langle A\rangle, } \]
and similarly,
\[ \boxed{ \Delta\hat B = \hat B-\langle B\rangle. } \]
The variances can then be written compactly as
\[ \sigma_A^2 = \left\langle \psi \middle| (\Delta\hat A)^2 \middle| \psi \right\rangle\]
and
\[ \sigma_B^2 = \left\langle \psi \middle| (\Delta\hat B)^2 \middle| \psi \right\rangle. \]
3.2 The general Heisenberg uncertainty relation
For any two Hermitian observables \(\hat A\) and \(\hat B\),
\[ \boxed{ \sigma_A\sigma_B \geq \frac{1}{2} \left| \left\langle [\hat A,\hat B] \right\rangle \right|. } \]
This is the general Heisenberg uncertainty relation.
More precisely, this form of the relation is often called the Robertson uncertainty relation, after Howard Percy Robertson, who derived the general inequality for arbitrary pairs of observables in 1929.
It expresses a central consequence of quantum mechanics:
The uncertainties of two observables are constrained by the noncommutativity of their operators.
The familiar position–momentum uncertainty principle,
\[ \sigma_x\sigma_{p_x} \geq \frac{\hbar}{2}, \]
is a special case of this general relation.
Thus, the Heisenberg uncertainty principle is not an independent rule added to quantum mechanics. It follows naturally from the operator structure of the theory.
The following derivation gives the relation commonly called the Robertson uncertainty relation.
Let
\[ \Delta\hat A = \hat A-\langle A\rangle, \qquad \Delta\hat B = \hat B-\langle B\rangle. \]
For any real number \(\lambda\), define
\[ \left\lvert \chi \right\rangle= \left( \Delta\hat A + i\lambda\Delta\hat B \right) \left\lvert \psi \right\rangle. \]
The norm of any vector is non-negative:
\[ \left\langle \chi \middle| \chi \right\rangle\geq0. \]
Therefore,
\[ \left\langle \psi \middle| (\Delta\hat A-i\lambda\Delta\hat B) (\Delta\hat A+i\lambda\Delta\hat B) \middle| \psi \right\rangle\geq0. \]
Expanding gives
\[ \langle(\Delta\hat A)^2\rangle + \lambda^2 \langle(\Delta\hat B)^2\rangle + i\lambda \langle[ \Delta\hat A,\Delta\hat B ]\rangle \geq0. \]
Since constants commute with all operators,
\[ [ \Delta\hat A,\Delta\hat B ] = [\hat A,\hat B]. \]
Using
\[ \sigma_A^2 = \langle(\Delta\hat A)^2\rangle \]
and
\[ \sigma_B^2 = \langle(\Delta\hat B)^2\rangle, \]
we obtain
\[ f(\lambda) = \sigma_A^2 + \lambda^2\sigma_B^2 + i\lambda \langle[\hat A,\hat B]\rangle \geq0. \]
Because \(\hat A\) and \(\hat B\) are Hermitian,
\[ [\hat A,\hat B]^\dagger = -[\hat A,\hat B], \]
so the expectation value of the commutator is purely imaginary. Thus \(f(\lambda)\) is real.
The minimum occurs at
\[ \lambda_{\min} = -\frac{ i\langle[\hat A,\hat B]\rangle }{ 2\sigma_B^2 }. \]
Substituting this value gives
\[ \sigma_A^2 - \frac{ \left| \langle[\hat A,\hat B]\rangle \right|^2 }{ 4\sigma_B^2 } \geq0. \]
Multiplying by \(\sigma_B^2\),
\[ \sigma_A^2\sigma_B^2 \geq \frac14 \left| \langle[\hat A,\hat B]\rangle \right|^2. \]
Taking the square root gives
\[ \boxed{ \sigma_A\sigma_B \geq \frac12 \left| \langle[\hat A,\hat B]\rangle \right|. } \]
3.3 A subtle but important point
A nonzero commutator,
\[ [\hat A,\hat B]\neq0, \]
means that the observables are incompatible.
However, the lower bound in the Heisenberg uncertainty relation contains the expectation value
\[ \langle[\hat A,\hat B]\rangle, \]
which depends on the quantum state.
It is therefore possible that
\[ [\hat A,\hat B]\neq0 \]
but
\[ \langle[\hat A,\hat B]\rangle=0 \]
for a particular state.
In that state, the relation gives only
\[ \sigma_A\sigma_B\geq0. \]
This does not mean that the two operators commute. It only means that this particular form of the uncertainty relation provides no nonzero lower bound for that state.
The general Heisenberg relation above contains only the commutator of the two observables.
An even stronger inequality also includes correlations between them:
\[ \sigma_A^2\sigma_B^2 \geq \frac14 \left| \langle[\hat A,\hat B]\rangle \right|^2 + \frac14 \left| \left\langle \{ \Delta\hat A,\Delta\hat B \} \right\rangle \right|^2, \]
where
\[ \{\hat A,\hat B\} = \hat A\hat B+\hat B\hat A \]
is the anticommutator.
This is commonly called the Schrödinger–Robertson uncertainty relation.
The simpler relation
\[ \sigma_A\sigma_B \geq \frac12 \left| \langle[\hat A,\hat B]\rangle \right| \]
follows immediately by dropping the second, non-negative term.
4. Position and Momentum
4.1 The canonical commutation relation
The position and momentum operators satisfy
\[ \boxed{ [\hat x,\hat p_x] = i\hbar. } \]
More generally,
\[ \boxed{ [\hat r_i,\hat p_j] = i\hbar\delta_{ij}, } \]
where \(i\) and \(j\) label Cartesian components.
Thus,
\[ [\hat x,\hat p_y]=0, \]
whereas
\[ [\hat x,\hat p_x]=i\hbar. \]
The momentum operator in the position representation is
\[ \hat p_x = -i\hbar\frac{d}{dx}. \]
Because \(\hat p_x\) is a differential operator, we evaluate the commutator by letting it act on an arbitrary suitable function \(\psi(x)\):
\[ [\hat x,\hat p_x]\psi = \hat x\hat p_x\psi - \hat p_x\hat x\psi. \]
The first term is
\[ \hat x\hat p_x\psi = -i\hbar x\frac{d\psi}{dx}. \]
The second term is
\[ \hat p_x\hat x\psi = -i\hbar \frac{d}{dx}(x\psi). \]
Using the product rule,
\[ \hat p_x\hat x\psi = -i\hbar \left( \psi + x\frac{d\psi}{dx} \right). \]
Therefore,
\[ [\hat x,\hat p_x]\psi = -i\hbar x\frac{d\psi}{dx} + i\hbar \left( \psi+x\frac{d\psi}{dx} \right), \]
so
\[ [\hat x,\hat p_x]\psi = i\hbar\psi. \]
Since this holds for every suitable \(\psi\),
\[ \boxed{ [\hat x,\hat p_x] = i\hbar. } \]
4.2 The Heisenberg position–momentum uncertainty principle
Substituting
\[ [\hat x,\hat p_x] = i\hbar \]
into the general Heisenberg uncertainty relation gives
\[ \sigma_x\sigma_{p_x} \geq \frac12 |\langle i\hbar\rangle|. \]
Thus,
\[ \boxed{ \sigma_x\sigma_{p_x} \geq \frac{\hbar}{2}. } \]
This is the familiar Heisenberg position–momentum uncertainty principle.
Unlike the general uncertainty bound, the right-hand side is independent of the quantum state because the commutator is simply a constant times the identity operator.
4.3 What the uncertainty principle means
The relation
\[ \sigma_x\sigma_{p_x} \geq \frac{\hbar}{2} \]
does not say that position cannot be measured precisely.
Nor does it say that measuring position necessarily introduces an experimental error into momentum.
Instead, it says that no quantum state can have arbitrarily narrow probability distributions for both position and momentum simultaneously.
A state may have very small \(\sigma_x\).
A different state may have very small \(\sigma_{p_x}\).
But the product of the two uncertainties must satisfy
\[ \sigma_x\sigma_{p_x} \geq \frac{\hbar}{2}. \]
This limitation belongs to the structure of the quantum state itself.
5. Angular Momentum: A Preview
Angular momentum provides one of the most important examples of incompatible observables.
The Cartesian components satisfy
\[ \boxed{ [\hat L_x,\hat L_y] = i\hbar\hat L_z, } \]
together with the cyclic permutations
\[ [\hat L_y,\hat L_z] = i\hbar\hat L_x, \]
and
\[ [\hat L_z,\hat L_x] = i\hbar\hat L_y. \]
Thus the three Cartesian components cannot, in general, possess sharp values simultaneously.
For example, the general Heisenberg uncertainty relation gives
\[ \boxed{ \sigma_{L_x}\sigma_{L_y} \geq \frac{\hbar}{2} |\langle L_z\rangle|. } \]
For orbital angular momentum,
\[ \hat L_x = \hat y\hat p_z-\hat z\hat p_y, \]
\[ \hat L_y = \hat z\hat p_x-\hat x\hat p_z, \]
and
\[ \hat L_z = \hat x\hat p_y-\hat y\hat p_x. \]
Using the canonical commutation relations,
\[ [\hat r_i,\hat p_j] = i\hbar\delta_{ij}, \]
we calculate
\[ [\hat L_x,\hat L_y] = [ \hat y\hat p_z-\hat z\hat p_y, \hat z\hat p_x-\hat x\hat p_z ]. \]
Expanding,
\[ [\hat L_x,\hat L_y] = [\hat y\hat p_z,\hat z\hat p_x] - [\hat y\hat p_z,\hat x\hat p_z] - [\hat z\hat p_y,\hat z\hat p_x] + [\hat z\hat p_y,\hat x\hat p_z]. \]
The middle two terms vanish.
For the first term,
\[ [\hat y\hat p_z,\hat z\hat p_x] = -i\hbar\hat y\hat p_x. \]
For the last term,
\[ [\hat z\hat p_y,\hat x\hat p_z] = i\hbar\hat x\hat p_y. \]
Therefore,
\[ [\hat L_x,\hat L_y] = i\hbar ( \hat x\hat p_y-\hat y\hat p_x ), \]
and hence
\[ \boxed{ [\hat L_x,\hat L_y] = i\hbar\hat L_z. } \]
5.1 Angular-momentum magnitude and one component
Although the Cartesian components do not commute with one another,
\[ \boxed{ [\hat{\mathbf L}^2,\hat L_z]=0. } \]
In fact,
\[ [\hat{\mathbf L}^2,\hat L_i]=0 \]
for every Cartesian component \(i\).
It is therefore possible to choose states that are simultaneous eigenstates of
\[ \hat{\mathbf L}^2 \]
and one component, conventionally
\[ \hat L_z. \]
This is why angular-momentum states can be labeled by two quantum numbers.
Using
\[ \hat{\mathbf L}^2 = \hat L_x^2+\hat L_y^2+\hat L_z^2, \]
we obtain
\[ [\hat{\mathbf L}^2,\hat L_z] = [\hat L_x^2,\hat L_z] + [\hat L_y^2,\hat L_z] + [\hat L_z^2,\hat L_z]. \]
The last term vanishes.
Using the product rule,
\[ [\hat A\hat B,\hat C] = \hat A[\hat B,\hat C] + [\hat A,\hat C]\hat B, \]
we find
\[ [\hat L_x^2,\hat L_z] = \hat L_x[\hat L_x,\hat L_z] + [\hat L_x,\hat L_z]\hat L_x. \]
Since
\[ [\hat L_x,\hat L_z] = -i\hbar\hat L_y, \]
this gives
\[ [\hat L_x^2,\hat L_z] = -i\hbar ( \hat L_x\hat L_y + \hat L_y\hat L_x ). \]
Similarly,
\[ [\hat L_y^2,\hat L_z] = i\hbar ( \hat L_y\hat L_x + \hat L_x\hat L_y ). \]
The two contributions cancel, so
\[ \boxed{ [\hat{\mathbf L}^2,\hat L_z] = 0. } \]
5.2 Angular momentum and rotational symmetry
For a rotationally invariant Hamiltonian,
\[ \boxed{ [\hat H,\hat{\mathbf L}^2]=0. } \]
For a central potential,
\[ V(\vec r)=V(r), \]
the Hamiltonian is rotationally invariant.
Angular momentum therefore provides a natural set of quantum numbers for central-potential problems, including the hydrogen atom.
We will develop orbital angular momentum in detail in the next section.
6. The Central Idea
A measurement generally changes the quantum state. Because of this, successive measurements of different observables need not be independent of one another.
Whether two observables can possess simultaneous sharp values is determined by the algebra of their operators.
For compatible observables,
\[ [\hat A,\hat B]=0, \]
we can choose common eigenstates.
For incompatible observables,
\[ [\hat A,\hat B]\neq0, \]
a complete common eigenbasis generally does not exist.
The general Heisenberg uncertainty principle
\[ \boxed{ \sigma_A\sigma_B \geq \frac12 \left| \langle[\hat A,\hat B]\rangle \right| } \]
quantifies one consequence of this incompatibility.
For position and momentum,
\[ [\hat x,\hat p_x]=i\hbar, \]
giving
\[ \boxed{ \sigma_x\sigma_{p_x} \geq \frac{\hbar}{2}. } \]
The Heisenberg uncertainty principle is therefore not an isolated restriction on measurement precision. It follows naturally from the noncommutative operator structure of quantum mechanics.
The same structure will lead us directly to the quantum theory of angular momentum in the next section.