Born and Jordan’s Matrix Formulation

Heisenberg’s 1925 paper introduced arrays of transition quantities and a noncommutative multiplication rule. Born and Jordan supplied the crucial mathematical recognition: those arrays behave as matrices. That recognition made the new mechanics less mysterious algebraically and led quickly to the canonical commutation relation.

For the modern definitions used below, see Canonical Commutation Relations and Operators.

Matrix Interpretation

Heisenberg’s transition quantities were indexed by two state labels. Born and Jordan recognized that this is exactly the structure of a matrix. If $A$ and $B$ are two such arrays, their product is

$$(AB)_{mn} = \sum_k A_{mk}B_{kn}.$$

This rule was not a decorative import from linear algebra. It matched the way transition frequencies compose through intermediate states. Matrix multiplication supplied a precise algebra for the new quantities.

In modern notation, a chosen basis turns an operator $A$ into matrix elements

$$A_{mn} = \langle m\vert A\vert n\rangle.$$

That notation is later than the original work, but it captures why the matrix interpretation was powerful. A physical quantity is not represented by a single numerical value independent of state. It is represented by a structured set of transition and diagonal elements.

For a quantity corresponding to a real classical observable, the matrix should have the finite-dimensional Hermitian pattern

$$A_{mn} = A_{nm}^*,$$

or its appropriate infinite-dimensional analogue. This later became part of the route toward self-adjoint operators as observables. The historical matrix step did not yet settle every domain and spectral question, but it made the algebraic shape visible.

Canonical Commutation Relation

The most famous algebraic outcome is the canonical commutation relation. In modern convention,

$$[Q,P] = QP-PQ = i\hbar I.$$

Original papers may write equivalent formulas with the opposite product order or with $h/(2\pi i)$; these differ by sign conventions and by whether the commutator is written as $QP-PQ$ or $PQ-QP$. The physical content is that the coordinate and conjugate momentum quantities do not commute, and the noncommutativity is set by $\hbar$.

This relation is not a small correction to classical algebra. It says that canonical position and momentum cannot be represented by ordinary commuting numerical variables. Their product order matters.

There is also an important mathematical caveat. Exact canonical commutation cannot hold for finite-dimensional matrices. For finite matrices,

$$\operatorname{Tr}(AB-BA) = 0,$$

because the trace is cyclic. But if $[Q,P]=i\hbar I_N$ in an $N$-dimensional space, then

$$\operatorname{Tr}(i\hbar I_N) = i\hbar N,$$

which is nonzero when $\hbar\ne0$. Therefore the exact position-momentum commutation relation requires infinite-dimensional operators and domain care. Finite matrices can model spin systems, truncated oscillators, and approximations, but not the exact canonical pair without qualification.

Relation to Hamiltonian Mechanics

Born and Jordan’s formulation also sharpened the relation between the new algebra and Hamiltonian mechanics. Classical Hamiltonian dynamics uses canonical variables and Poisson brackets. Quantum matrix mechanics replaces the classical bracket structure by a commutator structure.

A compact modern comparison is

$$\{A,H\}_{\mathrm{cl}} \quad\longrightarrow\quad \frac{1}{i\hbar}[A,H].$$

This correspondence is a guide, not a universal algorithm. Operator ordering, domains, spectra, and representation choices all matter. Still, it explains why the matrix formulation could reproduce a mechanics rather than only fit spectral lines.

For an observable $A$ with no explicit time dependence, the Heisenberg-picture equation is

$$\frac{dA}{dt} = \frac{i}{\hbar}[H,A].$$

If $H$ is diagonal in the energy basis, $H_{mn}=E_m\delta_{mn}$, then

$$([H,A])_{mn} = (E_m-E_n)A_{mn}.$$

Therefore

$$\frac{dA_{mn}}{dt} = i\omega_{mn}A_{mn}, \qquad \omega_{mn} = \frac{E_m-E_n}{\hbar}.$$

This recovers the same pair-indexed transition frequencies that motivated matrix mechanics in the first place. The algebra and the spectroscopy fit together.

A familiar equation from a new product rule

Take $H=P^2/(2m)+V(Q)$ on a common domain where the following products are defined. The product rule $[A,BC]=[A,B]C+B[A,C]$ gives

$$[Q,P^2]=2i\hbar P, \qquad [V(Q),P]=i\hbar V'(Q).$$

Consequently,

$$\dot Q=\frac{P}{m}, \qquad \dot P=-V'(Q).$$

These resemble Hamilton’s equations, but $Q$ and $P$ are operators. Their expectation values need not follow a closed classical trajectory: in general $\langle V'(Q)\rangle\ne V'(\langle Q\rangle)$.

Role in Modern Operator Formalism

Born and Jordan’s matrix interpretation helped turn Heisenberg’s transition scheme into a systematic mechanics. Its modern descendants include:

• observables represented by self-adjoint operators;
• states represented by vectors, rays, or density operators;
• basis choices represented by matrix elements;
• time evolution expressed through Hamiltonians and commutators;
• compatibility and uncertainty expressed through noncommuting operator products.

The modern language is broader than the original matrix mechanics. It includes continuous spectra, wavefunctions, distributional eigenstates, unbounded operators, spin degrees of freedom, tensor products, and density matrices. But the conceptual core remains recognizable: the algebra of quantities is not the commutative algebra of classical functions on phase space.

Matrix mechanics also changed the meaning of “observable.” It did not mean “whatever can be pictured.” It meant a quantity whose measurable consequences are encoded in spectra, transition elements, and algebraic relations.

What Born and Jordan Clarified

Their contribution can be summarized in four linked steps:

1. Heisenberg’s arrays are matrices.
2. Heisenberg’s multiplication law is matrix multiplication.
3. Matrix multiplication is generally noncommutative.
4. A further quantum condition fixes the canonical position–momentum commutator to $i\hbar I$. Noncommutativity alone does not determine this coefficient or this particular algebra.

This did not finish quantum mechanics. Dirac’s transformation theory, Schrödinger’s wave mechanics, Born’s probability interpretation, and later Hilbert-space formalism were all needed. But Born and Jordan made the algebraic skeleton explicit.

Common Mistakes

• Treating Born and Jordan as merely naming Heisenberg’s arrays. They clarified the multiplication law, commutator structure, and Hamiltonian-mechanics connection.
• Writing the canonical commutator without specifying the order convention.
• Assuming finite matrices can satisfy $[Q,P]=i\hbar I$ exactly.
• Treating the Poisson-bracket-to-commutator rule as a foolproof quantization algorithm.
• Reading modern self-adjoint-operator domain theory back into 1925 without historical caution.
• Confusing matrix mechanics with the later Heisenberg picture of time evolution.

Exercises

1. Show that finite-dimensional matrices cannot satisfy $[Q,P]=i\hbar I_N$ exactly.

Solution

For finite matrices, cyclicity of trace gives

$$\operatorname{Tr}([Q,P]) = \operatorname{Tr}(QP-PQ) = \operatorname{Tr}(QP)-\operatorname{Tr}(PQ) = 0.$$

But

$$\operatorname{Tr}(i\hbar I_N) = i\hbar N,$$

which is nonzero for $N>0$ and $\hbar\ne0$. Therefore the exact canonical commutator cannot hold in finite dimension.

2. Let $H$ be diagonal with $H_{mn}=E_m\delta_{mn}$. Show that $([H,A])_{mn}=(E_m-E_n)A_{mn}$.

Solution

Compute the two products:

$$(HA)_{mn} = \sum_k H_{mk}A_{kn} = E_m A_{mn},$$

and

$$(AH)_{mn} = \sum_k A_{mk}H_{kn} = A_{mn}E_n.$$

Thus

$$([H,A])_{mn} = (HA-AH)_{mn} = (E_m-E_n)A_{mn}.$$

3. Why is the rule $\{A,H\}_{\mathrm{cl}}\to [A,H]/(i\hbar)$ only a correspondence guide rather than a complete quantization algorithm?

Solution

Classical functions commute, while quantum operators may not. A classical expression such as $q^2p$ has ordering choices after quantization, such as $Q^2P$, $QPQ$, or $PQ^2$. In infinite-dimensional systems, domains and self-adjointness also matter. The commutator rule captures an important structural analogy but does not uniquely define every quantum operator.

References

• M. Born and P. Jordan, “Zur Quantenmechanik,” Zeitschrift für Physik 34, 858-888, 1925, DOI: 10.1007/BF01328531.
• M. Born, W. Heisenberg, and P. Jordan, “Zur Quantenmechanik II,” Zeitschrift für Physik 35, 557-615, 1926, DOI: 10.1007/BF01379806.
• P. A. M. Dirac, “The fundamental equations of quantum mechanics,” Proceedings of the Royal Society A 109, 642-653, 1925, DOI: 10.1098/rspa.1925.0150.
• W. Heisenberg, “Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen,” Zeitschrift für Physik 33, 879-893, 1925, DOI: 10.1007/BF01328377.
• M. Jammer, The Conceptual Development of Quantum Mechanics, 2nd ed., American Institute of Physics, 1989.
• J. Mehra and H. Rechenberg, The Historical Development of Quantum Theory, Springer, 1982-2001.
• B. L. van der Waerden, ed., Sources of Quantum Mechanics, Dover, 1968.