Matrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925. It was the first conceptually autonomous and logically consistent formulation of quantum mechanics. Its account of quantum jumps supplanted the Bohr model's electron orbits. It did so by interpreting the physical properties of particles as matrices that evolve in time. It is equivalent to the Schrödinger wave formulation of quantum mechanics, as manifest in Dirac's bra–ket notation. In some contrast to the wave formulation, it produces spectra of (mostly energy) operators by purely algebraic, ladder operator methods. Relying on these methods, Wolfgang Pauli derived the hydrogen atom spectrum in 1926, before the development of wave mechanics.
Development of matrix mechanics In 1925, Werner Heisenberg, Max Born, and Pascual Jordan formulated the matrix mechanics representation of quantum mechanics.
Epiphany at Heligoland
In 1925 Werner Heisenberg was working in Göttingen on the problem of calculating the spectral lines of hydrogen. By May 1925 he began trying to describe atomic systems by observables only. On June 7, after weeks of failing to alleviate his hay fever with aspirin and cocaine, Heisenberg left for the pollen-free North Sea island of Heligoland. While there, in between climbing and memorizing poems from Goethe's West-östlicher Diwan, he continued to ponder the spectral issue and eventually realised that adopting non-commuting observables might solve the problem. He later wrote:
It was about three o' clock at night when the final result of the calculation lay before me. At first I was deeply shaken. I was so excited that I could not think of sleep. So I left the house and awaited the sunrise on the top of a rock.
The three fundamental papers After Heisenberg returned to Göttingen, he showed Wolfgang Pauli his calculations, commenting at one point:
Everything is still vague and unclear to me, but it seems as if the electrons will no more move on orbits. On July 9 Heisenberg gave the same paper of his calculations to Max Born, saying that "he had written a crazy paper and did not dare to send it in for publication, and that Born should read it and advise him" prior to publication. Heisenberg then departed for a while, leaving Born to analyse the paper. In the paper, Heisenberg formulated quantum theory without sharply-defined electron orbits, directly advocating for a re-interpretation of quantum theory that only focused on experimental observables like frequencies and transition probabilities. Before Heisenberg's paper, Hendrik Kramers had calculated the relative intensities of spectral lines in the Sommerfeld model by interpreting the Fourier coefficients of the orbits as intensities. But his answer, like all other calculations in the old quantum theory, was only correct for large orbits. Heisenberg, after a collaboration with Kramers, began to believe that the transition probabilities describing quantum transitions would need a new interpretation different from classical mechanics because Heisenberg believed that the frequencies that should appear in a series describing the position of the electron should only be the ones that are experimentally observed in quantum transitions (like through spectral lines), not the complete set of spatial frequencies that come from making a traditional Fourier series of classical orbits. The quantities in Heisenberg's original formulation involved a series that described position as a series of "virtual oscillators" with two indices, with the two indices representing the initial and final states of a quantum transition. Rather than following the multiplication rule as expected from multiplying Fourier series, Heisenberg formed a non-commutative multiplication rule to ensure that multiplying position states would preserve the frequencies that are only found in the quantum transitions. When Born read the paper, he recognized the formulation, particularly the non-commutative multiplication rule, as one which could be transcribed and extended to the systematic language of matrices, which he had learned from his study under Jakob Rosanes at Breslau University. Born, with the help of his assistant and former student Pascual Jordan, began immediately to make the transcription and extension, and they submitted their results for publication; the paper was received for publication just 60 days after Heisenberg's paper. A follow-on paper was submitted for publication before the end of the year by all three authors. (A brief review of Born's role in the development of the matrix mechanics formulation of quantum mechanics along with a discussion of the key formula involving the non-commutativity of the probability amplitudes can be found in an article by Jeremy Bernstein. A detailed historical and technical account can be found in Mehra and Rechenberg's book The Historical Development of Quantum Theory. Volume 3. The Formulation of Matrix Mechanics and Its Modifications 1925–1926.)
Up until this time, matrices were seldom used by physicists; they were considered to belong to the realm of pure mathematics, thus requiring Born and Jordan's paper to introduce matrix algebra to physicists unaware of their use. Gustav Mie had used them in a paper on electrodynamics in 1912 and Born had used them in his work on the lattices theory of crystals in 1921. While matrices were used in these cases, the algebra of matrices with their multiplication did not enter the picture as they did in the matrix formulation of quantum mechanics. Born, however, had learned matrix algebra from Rosanes, as already noted, but Born had also learned Hilbert's theory of integral equations and quadratic forms for an infinite number of variables as was apparent from a citation by Born of Hilbert's work Grundzüge einer allgemeinen Theorie der Linearen Integralgleichungen published in 1912. Jordan, too, was well equipped for the task. For a number of years, he had been an assistant to Richard Courant at Göttingen in the preparation of Courant and David Hilbert's book Methoden der mathematischen Physik I, which was published in 1924. This book, fortuitously, contained a great many of the mathematical tools necessary for the continued development of quantum mechanics. In 1926, John von Neumann became assistant to David Hilbert, and he would coin the term Hilbert space to describe the algebra and analysis which were used in the development of quantum mechanics. A linchpin contribution to this formulation was achieved in Dirac's reinterpretation/synthesis paper of 1925, which invented the language and framework usually employed today, in full display of the noncommutative structure of the entire construction.
Heisenberg's reasoning Before matrix mechanics, the old quantum theory described the motion of a particle by a classical orbit, with well defined position and momentum X(t) , P(t) , with the restriction that the time integral over one period T of the momentum times the velocity must be a positive integer multiple of the Planck constant ( h ) as described by the Sommerfeld-Wilson quantization condition
∫ 0 T P d X d t d t = ∫ 0 T P d X = n h . {\displaystyle \ \int _{0}^{T}P\;{\frac {\ \mathrm {d} X\ }{\mathrm {d} t}}\;\mathrm {d} t\ =\ \int _{0}^{T}P\;\mathrm {d} X=n\ h~.}
While this restriction correctly selects orbits with the right energy values En , the old quantum formalism did not describe time dependent processes, such as the emission or absorption of radiation. When a classical particle is weakly coupled to a radiation field, so that the radiative damping can be neglected, it will emit radiation in a pattern that repeats itself every orbital period. The frequencies that make up the outgoing wave are then integer multiples of the orbital frequency, and this is a reflection of the fact that X(t) is periodic, so that its Fourier representation has frequencies 2 π n / T only.
X ( t ) = ∑ n = − ∞ ∞ e i 2 π n t / T X n . {\displaystyle X(t)\ =\sum _{n=-\infty }^{\infty }e^{i\ 2\pi nt/T}X_{n}~.}
The coefficients Xn are complex numbers. The ones with negative frequencies must be the complex conjugates of the ones with positive frequencies, so that X(t) will always be real,
X n = X − n ∗ . {\displaystyle X_{n}=X_{-n}^{*}~.}
A quantum mechanical particle, on the other hand, cannot emit radiation continuously; it can only emit photons. Under the Bohr model, for a quantum particle starting in quantum number n that then emits a photon by transitioning to orbit number m , the energy of the photon is En − Em , that gives a photon of frequency is En − Em / h . For large n and m , but with n − m relatively small, Bohr's correspondence principle expects the same classical frequencies
E n − E m ≈ h ( n − m ) T . {\displaystyle E_{n}-E_{m}\approx {\frac {\ h\ (n-m)\ }{T}}~.}
In the formula above, T is the classical period of either orbit n or orbit m , since the difference between them is higher order in h . But for small n and m , or if n − m is large, the frequencies are not integer multiples of any single frequency. Since in classical mechanics, the frequencies that the particle emits are the same as the frequencies in the Fourier description of its motion, Heisenberg inferred that in the time-dependent description of the particle, there should be something oscillating with frequency En − Em / h . Heisenberg called this quantity Xnm , and demanded that it should reduce to the classical Fourier coefficients in the classical limit. For large values of n and m but with n − m relatively small, Xnm is the (n − m)th Fourier coefficient of the classical motion at orbit n . Since Xnm has opposite frequency to Xmn , the condition that X is real becomes
X n m = X m n ∗ . {\displaystyle X_{nm}=X_{mn}^{*}~.}
By definition, Xnm only has the frequency En − Em /h , so its time evolution may be described as:
X n m ( t ) = e i 2 π ( E n − E m ) t / h X n m ( 0 ) = e i ( E n − E m ) t / ℏ X n m ( 0 ) . {\displaystyle \ X_{nm}(t)\ =\ e^{i\ 2\pi \left(E_{n}-E_{m}\right)t/h}\ X_{nm}(0)\ =\ e^{i\ \left(E_{n}-E_{m}\right)t/\hbar }\ X_{nm}(0)~.}
This is the original form of Heisenberg's equation of motion. Given two arrays Xnm and Pnm describing two physical quantities, when modeling each as classical Fourier series, it is expected that their multiplication Xnk Pkm should also result in a new frequency as part of a new Fourier series. Whilst the Fourier coefficients of the product of two quantities is the convolution of the Fourier coefficients of each one separately, Heisenberg changed the multiplication rule to ensure that when multiplying each component, the new frequencies would only correspond to frequencies that already existed in the quantum orbit:
( X P ) m n = ∑ k = 0 ∞ X m k P k n . {\displaystyle \ \left(X\ P\right)_{mn}\ =\ \sum _{k=0}^{\infty }\ X_{mk}\ P_{kn}~.}
Born noticed that this is the law of matrix multiplication, so that the position, the momentum, the energy, all the observable quantities in the theory, are interpreted as matrices. Under this multiplication rule, the product depends on the order: X P is different from P X . The X matrix is a complete description of the motion of a quantum mechanical particle. Because the frequencies in the quantum motion are not multiples of a common frequency, the matrix elements cannot be interpreted as the Fourier coefficients of a sharp classical trajectory. Nevertheless, as matrices, X(t) and P(t) satisfy the classical equations of motion; also see Ehrenfest's theorem, below.
Matrix basics When it was introduced by Werner Heisenberg, Max Born and Pascual Jordan in 1925, matrix mechanics was not immediately accepted and was a source of controversy, at first. Schrödinger's later introduction of wave mechanics was greatly favored. Part of the reason was that Heisenberg's formulation was in an odd mathematical language, for the time, while Schrödinger's formulation was based on familiar wave equations. But there was also a deeper sociological reason. Quantum mechanics had been developing by two paths, one led by Einstein, who emphasized the wave–particle duality he proposed for photons, and the other led by Bohr, that emphasized the discrete energy states and quantum jumps that Bohr discovered. De Broglie had reproduced the discrete energy states within Einstein's framework – the quantum condition is the standing wave condition, and this gave hope to those in the Einstein school that all the discrete aspects of quantum mechanics would be subsumed into a continuous wave mechanics. Matrix mechanics, on the other hand, came from the Bohr school, which was concerned with discrete energy states and quantum jumps. Bohr's followers did not appreciate physical models that pictured electrons as waves, or as anything at all. They preferred to focus on the quantities that were directly connected to experiments. In atomic physics, spectroscopy gave observational data on atomic transitions arising from the interactions of atoms with light quanta. The Bohr school required that only those quantities that were in principle measurable by spectroscopy should appear in the theory. These quantities include the energy levels and their intensities but they do not include the exact location of a particle in its Bohr orbit. It is very hard to imagine an experiment that could determine whether an electron in the ground state of a hydrogen atom is to the right or to the left of the nucleus. It was a deep conviction that such questions did not have an answer. The matrix formulation was built on the premise that all physical observables are represented by matrices, whose elements are indexed by two different energy levels. The set of eigenvalues of the matrix were eventually understood to be the set of all possible values that the observable can have. Since Heisenberg's matrices are Hermitian, the eigenvalues are real. If an observable is measured and the result is a certain eigenvalue, the corresponding eigenvector is the state of the system immediately after the measurement. The act of measurement in matrix mechanics collapses the state of the system. If one measures two observables simultaneously, the state of the system collapses to a common eigenvector of the two observables. Since most matrices don't have any eigenvectors in common, most observables can never be measured precisely at the same time. This is the uncertainty principle. If two matrices share their eigenvectors, they can be simultaneously diagonalized. In the basis where they are both diagonal, it is clear that their product does not depend on their order because multiplication of diagonal matrices is just multiplication of numbers. The uncertainty principle, by contrast, is an expression of the fact that often two matrices A and B do not always commute, i.e., that AB − BA does not necessarily equal 0. The fundamental commutation relation of matrix mechanics,
∑ k ( X n k P k m − P n k X k m ) = i ℏ δ n m {\displaystyle \sum _{k}\left(X_{nk}P_{km}-P_{nk}X_{km}\right)=i\hbar \,\delta _{nm}}
implies then that there are no states that simultaneously have a definite position and momentum. This principle of uncertainty holds for many other pairs of observables as well. For example, the energy does not commute with the position either, so it is impossible to precisely determine the position and energy of an electron in an atom.
Nobel Prize In 1928, Albert Einstein nominated Heisenberg, Born, and Jordan for the Nobel Prize in Physics. The announcement of the Nobel Prize in Physics for 1932 was delayed until November 1933. It was at that time that it was announced Heisenberg had won the Prize for 1932 "for the creation of quantum mechanics, the application of which has, inter alia, led to the discovery of the allotropic forms of hydrogen" and Erwin Schrödinger and Paul Adrien Maurice Dirac shared the 1933 Prize "for the discovery of new productive forms of atomic theory". On November 25, 1933, Born received a letter from Heisenberg in which he said he had been delayed in writing due to a "bad conscience" that he alone had received the Prize "for work done in Göttingen in collaboration – you, Jordan and I". Heisenberg went on to say that Born and Jordan's contribution to quantum mechanics cannot be changed by "a wrong decision from the outside". In 1954, Heisenberg wrote an article honoring Max Planck for his insight in 1900. In the article, Heisenberg credited Born and Jordan for the final mathematical formulation of matrix mechanics and Heisenberg went on to stress how great their contributions were to quantum mechanics, which were not "adequately acknowledged in the public eye".
Mathematical development Once Heisenberg introduced the matrices for X and P, he could find their matrix elements in special cases by guesswork, guided by the correspondence principle. Since the matrix elements are the quantum mechanical analogs of Fourier coefficients of the classical orbits, the simplest case is the harmonic oscillator, where the classical position and momentum, X(t) and P(t), are sinusoidal.
Harmonic oscillator In units where the mass and frequency of the oscillator are equal to one (see nondimensionalization), the energy of the oscillator is
H = 1 2 ( P 2 + X 2 ) . {\displaystyle H={\tfrac {1}{2}}\left(\ P^{2}+X^{2}\right)~.}
The level sets of H are the clockwise orbits, and they are nested circles in phase space. The classical orbit with energy E is
X ( t ) = + 2 E cos ( t ) {\textstyle \ X(t)=+{\sqrt {2E\ }}\ \cos(t)\qquad } and P ( t ) = − 2 E sin ( t ) . {\textstyle \qquad P(t)=-{\sqrt {2E\ }}\ \sin(t)~.}
The old quantum condition dictates that the integral of P dX over an orbit, which is the area of the circle in phase space, must be an integer multiple of the Planck constant. The area of the circle of radius √2 E is 2π E . So
E = n h 2 π = n ℏ , {\displaystyle E={\frac {\ n\ h\ }{2\pi }}=n\ \hbar \ ,}
or, in natural units where ħ ≡ 1 , the energy becomes some whole number. The Fourier components of X( t ) and P( t ) are simple, and even more so when they are re-expressed as a sum and difference of position X and momentum P :
A ( t ) = X ( t ) + i P ( t ) = 2 E e − i t {\textstyle \ A(t)=X(t)+i\ P(t)={\sqrt {2E\ }}\ e^{-i\ t}\qquad } and A † ( t ) = X ( t ) − i P ( t ) = 2 E e + i t . {\textstyle \qquad A^{\dagger }(t)=X(t)-i\ P(t)={\sqrt {2E\ }}\ e^{+i\ t}~.}
Both A and A† have only a single frequency, and X and P can be recovered from the similar sum and difference of A and A† . Since A( t ) has a classical Fourier series with only the lowest frequency, and the matrix element Amn is the (m − n)th Fourier coefficient of the classical orbit, the matrix for A is nonzero only on the line just above the diagonal, where it is equal to √2En . The matrix for A† is likewise only nonzero on the line below the diagonal, with the same elements. Thus, from A and A† , reconstruction yields
2 X ( 0 ) = ℏ [ 0 1 0 0 0 ⋯ 1 0 2 0 0 ⋯ 0 2 0 3 0 ⋯ 0 0 3 0 4 ⋯ ⋮ ⋮ ⋮ ⋮ ⋮ ⋱ ] , {\displaystyle {\sqrt {2\ }}\ X(0)={\sqrt {\hbar \ }}\;{\begin{bmatrix}0&{\sqrt {1\ }}&0&0&0&\cdots \\{\sqrt {1\ }}&0&{\sqrt {2\ }}&0&0&\cdots \\0&{\sqrt {2\ }}&0&{\sqrt {3\ }}&0&\cdots \\0&0&{\sqrt {3\ }}&0&{\sqrt {4\ }}&\cdots \\\vdots &\vdots &\vdots &\vdots &\vdots &\ddots \\\end{bmatrix}}\ ,}
and
2 P ( 0 ) = ℏ [ 0 − i 1 0 0 0 ⋯ i 1 0 − i 2 0 0 ⋯ 0 i 2 0 − i 3 0 ⋯ 0 0 i 3 0 − i 4 ⋯ ⋮ ⋮ ⋮ ⋮ ⋮ ⋱ ] , {\displaystyle {\sqrt {2\ }}\ P(0)={\sqrt {\hbar \ }}\;{\begin{bmatrix}0&-i{\sqrt {1\ }}&0&0&0&\cdots \\i{\sqrt {1}}&0&-i{\sqrt {2\ }}&0&0&\cdots \\0&i{\sqrt {2\ }}&0&-i{\sqrt {3\ }}&0&\cdots \\0&0&i{\sqrt {3\ }}&0&-i{\sqrt {4\ }}&\cdots \\\vdots &\vdots &\vdots &\vdots &\vdots &\ddots \\\end{bmatrix}}\ ,}
which, up to the choice of units, are the Heisenberg matrices for the harmonic oscillator. Both matrices are Hermitian, since they are constructed from the Fourier coefficients of real quantities. Finding X( t ) and P( t ) is direct, since they are quantum Fourier coefficients so they evolve simply with time, as
X m n ( t ) = X m n ( 0 ) e i ( E m − E n ) t {\textstyle \ X_{mn}(t)\ =\ X_{mn}(0)\ e^{i\ (E_{m}-E_{n})t}\qquad } and P m n ( t ) = P m n ( 0 ) e i ( E m − E n ) t . {\textstyle \qquad P_{mn}(t)\ =\ P_{mn}(0)\ e^{i\ (E_{m}-E_{n})t}~.}
The matrix product of X and P is not hermitian, but has a real and imaginary part. The real part is one half the symmetric expression X P + P X , while the imaginary part is proportional to the commutator, which is written as
[ X , P ] ≡ X P − P X . {\displaystyle \ {\bigl [}\ X\ ,\ P\ {\bigr ]}\ \equiv \ X\ P-P\ X~.}
In the special case of the harmonic oscillator, it is simple to verify explicitly that X P − P X is i ħ I , where I is the identity matrix. It is likewise simple to verify that the matrix
H = 1 2 ( X 2 + P 2 ) {\displaystyle H={\tfrac {1}{2}}\left(\ X^{2}+P^{2}\right)}
is a diagonal matrix, with eigenvalues Ei .
Conservation of energy
The harmonic oscillator is an important case. Finding the matrices is easier than determining the general conditions from these special forms. For this reason, Heisenberg investigated the anharmonic oscillator, with Hamiltonian
H = 1 2 P 2 + 1 2 X 2 + ε X 3 . {\displaystyle H={\tfrac {1}{2}}P^{2}+{\tfrac {1}{2}}X^{2}+\varepsilon X^{3}~.}
In this case, the X and P matrices are no longer simple off-diagonal matrices, since the corresponding classical orbits are slightly squashed and displaced, so that they have Fourier coefficients at every classical frequency. To determine the matrix elements, Heisenberg required that the classical equations of motion be obeyed as matrix equations,
d X d t = P , d P d t = − X − 3 ε X 2 . {\displaystyle {\frac {dX}{dt}}=P~,\qquad {\frac {dP}{dt}}=-X-3\varepsilon X^{2}~.}
He noticed that if this could be done, then H, considered as a matrix function of X and P, will have zero time derivative.
d H d t = P ∗ d P d t + ( X + 3 ε X 2 ) ∗ d X d t = 0 , {\displaystyle {\frac {dH}{dt}}=P*{\frac {dP}{dt}}+\left(X+3\varepsilon X^{2}\right)*{\frac {dX}{dt}}=0~,}
where A∗B is the anticommutator,
A ∗ B = 1 2 ( A B + B A ) . {\displaystyle A*B={\tfrac {1}{2}}(AB+BA)~.}
Given that all the off diagonal elements have a nonzero frequency; H being constant implies that H is diagonal. It was clear to Heisenberg that in this system, the energy could be exactly conserved in an arbitrary quantum system, a very encouraging sign. The process of emission and absorption of photons seemed to demand that the conservation of energy will hold at best on average. If a wave containing exactly one photon passes over some atoms, and one of them absorbs it, that atom needs to tell the others that they can't absorb the photon anymore. But if the atoms are far apart, any signal cannot reach the other atoms in time, and they might end up absorbing the same photon anyway and dissipating the energy to the environment. When the signal reached them, the other atoms would have to somehow recall that energy. This paradox led Bohr, Kramers and Slater to abandon exact conservation of energy. Heisenberg's formalism, when extended to include the electromagnetic field, was obviously going to sidestep this problem, a hint that the interpretation of the theory will involve wavefunction collapse.
Differentiation trick — canonical commutation relations Demanding that the classical equations of motion are preserved is not a strong enough condition to determine the matrix elements. The Planck constant does not appear in the classical equations, so that the matrices