Heisenberg’s Matrix Mechanics

Heisenberg’s matrix mechanics was a sharp break with the old attempt to picture electrons moving on classical orbits. It began from a disciplined demand: build the theory from quantities connected to observations, especially spectral transition frequencies and intensities, rather than from unobservable orbital coordinates.

The formulas below use modern notation to explain the 1925 reasoning. For the mathematical definitions, see Operators and Commutators.

Spectral Transitions

Atomic spectra were not pictures of electron paths. They were lists of sharply defined frequencies and intensities. The Ritz combination principle and Bohr frequency condition suggested that spectral lines connect pairs of stationary states:

$$h\nu_{mn} = E_m-E_n, \qquad \omega_{mn} = \frac{E_m-E_n}{\hbar}.$$

The signed frequency $\omega_{mn}$ describes the phase of a matrix element. For emission from a higher level $n$ to a lower level $m$, the photon frequency is positive: $(E_n-E_m)/h=\lvert\nu_{mn}\rvert$.

Old quantum theory tried to preserve classical orbits while imposing quantization rules on them. That strategy worked surprisingly well for some hydrogen-like patterns, but it did not provide a general mechanics for intensities, transitions, multi-electron systems, or non-periodic motion.

Heisenberg’s 1925 move was to stop treating an electron orbit $x(t)$ as the basic object. Instead, the theory should use quantities tied to transitions between states. A classical periodic coordinate could be expanded in Fourier components,

$$x(t) = \sum_{\alpha} x_\alpha e^{i\alpha\omega t}.$$

The old quantum correspondence suggested that the Fourier harmonics of a classical orbit should be replaced by transition quantities between quantum states:

$$x_\alpha \quad\leadsto\quad X_{mn}, \qquad \alpha\omega \quad\leadsto\quad \omega_{mn}.$$

This was not merely a change of symbols. It changed what counts as a legitimate quantity in the theory.

Observable Quantities Over Electron Orbits

In a classical model, an electron has a position $x(t)$ at every time. Even if one cannot directly see the orbit, the model treats it as a well-defined path. Heisenberg’s reinterpretation was more austere: use quantities associated with observable radiation, such as transition frequencies and strengths.

For an allowed transition $n\to m$, one may associate a transition frequency $\omega_{mn}$ and a transition quantity $X_{mn}$. In modern language, $X_{mn}$ is a matrix element of an observable-like quantity $X$:

$$X_{mn} = \langle m\vert X\vert n\rangle.$$

Historically, the bra-ket notation and abstract Hilbert-space interpretation came later. The important early step was the array itself:

$$X \sim \begin{pmatrix} X_{11} & X_{12} & X_{13} & \cdots \\ X_{21} & X_{22} & X_{23} & \cdots \\ X_{31} & X_{32} & X_{33} & \cdots \\ \vdots & \vdots & \vdots & \ddots \end{pmatrix}.$$

Diagonal entries describe quantities associated with a stationary state. Off-diagonal entries encode transitions between different states. This immediately fits spectroscopy better than an orbit picture does, because spectral lines are already indexed by pairs of states.

Matrix mechanics replaces a visualizable orbit with transition quantities indexed by pairs of stationary states. Off-diagonal entries encode transitions; multiplication of such arrays naturally sums over intermediate states.

Transition Amplitudes

It is tempting to call every off-diagonal entry a probability amplitude, but that would read later language too quickly into the early theory. In Heisenberg’s setting, the entries were transition quantities tied to radiation and correspondence arguments. Their magnitudes were related to line strengths, while their phases and multiplication rules carried dynamical information.

A useful modern reconstruction is:

• frequencies come from energy differences, $\omega_{mn}=(E_m-E_n)/\hbar$;
• transition strengths are related to squared moduli such as $\lvert X_{mn}\rvert^2$;
• time dependence can be attached to entries through factors $e^{i\omega_{mn}t}$;
• products of quantities must respect the way transition frequencies compose.

The last point is decisive. For any intermediate index $k$, the signed frequencies obey:

$$\omega_{mk} + \omega_{kn} = \omega_{mn}.$$

This identity makes products of the time-dependent entries carry the correct final frequency. Summing over $k$ is an algebraic composition rule; it need not describe two successive real transitions.

Noncommutative Multiplication

Suppose $X$ and $Y$ are two arrays of transition quantities. Their product is not obtained by multiplying entries independently. The natural composition rule sums over intermediate states:

$$(XY)_{mn} = \sum_k X_{mk}Y_{kn}.$$

Reversing the order gives

$$(YX)_{mn} = \sum_k Y_{mk}X_{kn}.$$

There is no general reason for these two sums to agree:

$$XY\ne YX.$$

This was the structural shock. In classical mechanics, ordinary functions of position and momentum commute under pointwise multiplication. In matrix mechanics, the quantities that replace them need not commute.

Born and Jordan recognized that Heisenberg’s multiplication law was matrix multiplication. In modern notation, noncommutativity is measured by the commutator

$$[X,Y] = XY-YX.$$

For canonical position and momentum, the formal descendant is

$$[X,P] = i\hbar I.$$

The exact relation requires infinite-dimensional operators, with products defined on a suitable common domain. Finite matrices cannot satisfy it: the trace of a commutator is zero, whereas the trace of $i\hbar I_N$ is $i\hbar N$. Born and Jordan’s formulation explains this distinction.

From Transition Arrays to Operators

In modern quantum mechanics, a pure state is a ray and a sharp observable is a self-adjoint operator. Choosing a basis turns that operator into a matrix. Changing basis changes the entries without changing the physical quantity. This explains why matrix mechanics survives as part of the general formalism.

The 1925 theory did not yet contain the later state, probability, and measurement concepts in their textbook form. Dirac’s transformation theory explains the next step toward a representation-independent description, while matrix–wave equivalence shows why the wave equation preserved the noncommutative structure.

Common Mistakes

• Treating matrix mechanics as an awkward notation for wave mechanics.
• Thinking Heisenberg simply guessed matrices from linear algebra. The multiplication rule arose from transition quantities and frequency-combination logic; Born and Jordan then recognized the matrix structure.
• Reading modern bra-ket notation back into the 1925 paper without historical caution.
• Assuming noncommutativity is only a technical complication. It is a physical and algebraic feature of quantum observables.
• Saying matrix mechanics ignored experiment. It was built from spectral data and the demand to use observable quantities.
• Confusing the Heisenberg historical formulation with the modern Heisenberg picture of time evolution.

Exercises

1. Explain why spectral lines naturally suggest quantities indexed by two states rather than by one classical orbit.

Solution

A spectral line corresponds to a transition between two stationary states, with frequency determined by an energy difference:

$$h\nu_{mn} = E_m-E_n.$$

The observed quantity is therefore associated with a pair of labels $m,n$, not with a single point on a classical trajectory. Matrix mechanics builds this pair-label structure into the basic quantities.

2. Show that the frequency-combination rule supports matrix multiplication. If $\omega_{ab}=(E_a-E_b)/\hbar$, compute $\omega_{mk}+\omega_{kn}$.

Solution

Using the definition,

$$\omega_{mk} + \omega_{kn} = \frac{E_m-E_k}{\hbar} + \frac{E_k-E_n}{\hbar} = \frac{E_m-E_n}{\hbar} = \omega_{mn}.$$

The intermediate label $k$ disappears from the final frequency, matching the idea that products of transition arrays should sum over intermediate states.

3. For two $2\times 2$ arrays

$$X = \begin{pmatrix} 0 & 1\\ 0 & 0 \end{pmatrix}, \qquad Y = \begin{pmatrix} 0 & 0\\ 1 & 0 \end{pmatrix},$$

compute $XY$ and $YX$. Do they commute?

Solution

Multiplying in the first order gives

$$XY = \begin{pmatrix} 1 & 0\\ 0 & 0 \end{pmatrix}.$$

Multiplying in the reverse order gives

$$YX = \begin{pmatrix} 0 & 0\\ 0 & 1 \end{pmatrix}.$$

Since $XY\ne YX$, the arrays do not commute.

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.
• 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.
• H. Kragh, Quantum Generations: A History of Physics in the Twentieth Century, Princeton University Press, 1999.
• 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.