On June 7, after weeks of failing to alleviate his hay fever with aspirin and cocaine,[3] Heisenberg left for the pollen-free North Sea island of Helgoland.
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.
[4]: 275 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.
[8] When Born read the paper, he recognized the formulation as one which could be transcribed and extended to the systematic language of matrices,[9] which he had learned from his study under Jakob Rosanes[10] at Breslau University.
[19][20] A linchpin contribution to this formulation was achieved in Dirac's reinterpretation/synthesis paper of 1925,[21] which invented the language and framework usually employed today, in full display of the noncommutative structure of the entire construction.
While this restriction correctly selects orbits with more or less the right energy values En, the old quantum mechanical formalism did not describe time dependent processes, such as the emission or absorption of radiation.
Born pointed out 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.
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.
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.
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.
[24] 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"[25] and Erwin Schrödinger and Paul Adrien Maurice Dirac shared the 1933 Prize "for the discovery of new productive forms of atomic theory".
[29] 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.
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.
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.
So, in order to implement his program, Heisenberg needed to use the old quantum condition to fix the energy levels, then fill in the matrices with Fourier coefficients of the classical equations, then alter the matrix coefficients and the energy levels slightly to make sure the classical equations are satisfied.
His crucial insight was to differentiate the quantum condition with respect to n. This idea only makes complete sense in the classical limit, where n is not an integer but the continuous action variable J, but Heisenberg performed analogous manipulations with matrices, where the intermediate expressions are sometimes discrete differences and sometimes derivatives.
In the following discussion, for the sake of clarity, the differentiation will be performed on the classical variables, and the transition to matrix mechanics will be done afterwards, guided by the correspondence principle.
But Heisenberg, Born and Jordan, unlike Dirac, were not familiar with the theory of Poisson brackets, so, for them, the differentiation effectively evaluated {X, P} in J,θ coordinates.
The Poisson Bracket, unlike the action integral, does have a simple translation to matrix mechanics – it normally corresponds to the imaginary part of the product of two variables, the commutator.
To see this, examine the (antisymmetrized) product of two matrices A and B in the correspondence limit, where the matrix elements are slowly varying functions of the index, keeping in mind that the answer is zero classically.
Since all the matrix elements are at indices which have a small distance from the large index position (m,m), it helps to introduce two temporary notations: A[r,k] = A(m+r)(m+k) for the matrices, and dA/dJ[r] for the rth Fourier components of classical quantities,
Heisenberg's original differentiation trick was eventually extended to a full semiclassical derivation of the quantum condition, in collaboration with Born and Jordan.
this condition replaced and extended the old quantization rule, allowing the matrix elements of P and X for an arbitrary system to be determined simply from the form of the Hamiltonian.
This interpretation is statistical: the result of a measurement of the physical quantity corresponding to the matrix A is random, with an average value equal to
All these forms of the equation of motion above say the same thing, that A(t) is equivalent to A(0), through a basis rotation by the unitary matrix eiHt, a systematic picture elucidated by Dirac in his bra–ket notation.
Assuming limits are defined sensibly, this extends to arbitrary functions−but the extension need not be made explicit until a certain degree of mathematical rigor is required,
Conversely, this suggests that it might be possible to find quantum systems of size N which physically compute the answers to problems which classically require 2N bits to solve.
The one-to-one association of infinitesimal symmetry generators and conservation laws was discovered by Emmy Noether for classical mechanics, where the commutators are Poisson brackets, but the quantum-mechanical reasoning is identical.
It was physically clear to Heisenberg that the absolute squares of the matrix elements of X, which are the Fourier coefficients of the oscillation, would yield the rate of emission of electromagnetic radiation.
This then allowed the magnitude of the matrix elements to be interpreted statistically: they give the intensity of the spectral lines, the probability for quantum jumps from the emission of dipole radiation.