Matrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925. Fig. ... Werner Karl Heisenberg (December 5, 1901 – February 1, 1976) was a celebrated German physicist and Nobel laureate, one of the founders of quantum mechanics and acknowledged to be one of the most important physicists of the twentieth century. ... Max Born (December 11, 1882 in Breslau – January 5, 1970 in Göttingen) was a mathematician and physicist. ... Pascual Jordan (October 18, 1902 in Hanover - July 31, 1980 in Hamburg) was a German physicist. ... Year 1925 (MCMXXV) was a common year starting on Thursday (link will display the full calendar) of the Gregorian calendar. ...

Matrix mechanics was the first complete definition of quantum mechanics, its laws, and properties that described fully the behavior of subatomic particles by associating their properties with matrices. It has been shown to be exactly equivalent to the Schrödinger wave formulation of quantum mechanics and is the basis of the bra-ket notation used to summarize quantum mechanical wave functions. In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. ... For a non-technical introduction to the topic, please see Introduction to quantum mechanics. ... Bra-ket notation is the standard notation for describing quantum states in the theory of quantum mechanics. ... A wave function is a mathematical tool that quantum mechanics uses to describe any physical system. ...

Development of matrix mechanics

In 1925, Werner Heisenberg, Max Born, and Pascual Jordan formulated the matrix mechanics representation of quantum mechanics. Werner Karl Heisenberg (December 5, 1901 – February 1, 1976) was a celebrated German physicist and Nobel laureate, one of the founders of quantum mechanics and acknowledged to be one of the most important physicists of the twentieth century. ... Max Born (December 11, 1882 in Breslau – January 5, 1970 in Göttingen) was a mathematician and physicist. ... Pascual Jordan (October 18, 1902 in Hanover - July 31, 1980 in Hamburg) was a German physicist. ...

Epiphany at Heligoland

In 1925 Werner Heisenberg was working, in Göttingen, on the problem of calculating the spectral lines of hydrogen. By May 1925 Heisenberg began to describe atomic systems by observables only. On June 7, to escape the effects of a bad attack of hay fever, Heisenberg left for the pollen free North Sea island of Heligoland. Whilst there Heisenberg, in between mountain climbing and learning by heart poems from Goethe's West Osticher Divan, continued to ponder the spectral issue and eventually realised that adopting non-commuting observables might solve the problem, and he later wrote^[1] Göttingen marketplace with old city hall, Gänseliesel fountain and pedestrian zone Göttingen ( ) is a city in Lower Saxony, Germany. ... A spectral line is a dark or bright line in an otherwise uniform and continuous spectrum, resulting from an excess or deficiency of photons in a narrow frequency range, compared with the nearby frequencies. ... General Name, Symbol, Number hydrogen, H, 1 Chemical series nonmetals Group, Period, Block 1, 1, s Appearance colorless Atomic mass 1. ... In physics, particularly in quantum physics, a system observable is a property of the system state that can be determined by some sequence of physical operations. ... For the play, see Hay Fever. ... The North Sea is a sea of the Atlantic Ocean, located between the coasts of Norway and Denmark in the east, the coast of the British Isles in the west, and the German, Dutch, Belgian and French coasts in the south. ... For the landscape in Norway, see Helgeland. ...  , IPA: , (28 August 1749 – 22 March 1832) was a German polymath. ... A map or binary operation from a set to a set is said to be commutative if, (A common example in school-math is the + function: , thus the + function is commutative) Otherwise, the operation is noncommutative. ...

"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 Papers

After Heisenberg returned to Gottingen, he showed Wolfgang Pauli his calculations, commenting at one point:^[2] Map of Germany showing Göttingen 1 External links Coat of Arms University of Göttingen Top: The old Auditorium Maximum (1862-65) Bottom: New library building Göttingen is a city in Lower Saxony, Germany. ... This article is about Austrian-Swiss physicist Wolfgang Pauli. ...

"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 on it.

prior to publication. Heisenberg then departed for awhile, leaving Born to analyse the paper.^[3] In the paper, Heisenberg formulated quantum theory avoiding the concrete but unobservable representations of electron orbits by using parameters such as transition probabilities for quantum jumps, which necessitated using two indexes corresponding to the initial and final states.^[4] When Born read the paper, he recognized the formulation as one which could be transcribed and extended to the systematic language of matrices,^[5] which he had learned from his study under Jakob Rosanes^[6] 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.^[7] A follow-on paper was submitted for publication before the end of the year by all three authors.^[8] (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-commutivity of the probability amplitudes can be found in an article by Jeremy Bernstein.^[9] 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.^[10]) The main building of WrocÅ‚aw University, seen from the University bridge (Most Uniwersytecki) spanning the Oder River. ... Pascual Jordan (October 18, 1902 in Hanover - July 31, 1980 in Hamburg) was a German physicist. ...

Up until this time, matrices were seldom used by physicists, they were considered to belong to the realm of pure mathematics. 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.^[11] 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 einter allgemeinen Theroire der Linearen Integralgleichungen published in 1912.^{[12] [13]} 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.^[14] 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.^{[15] [16]} Gustav Mie (September 29, 1869 Rostock – February 13, 1957 Freiburg im Breisgau) was a German physicist. ... Richard Courant (born January 8, 1888 at Lublinitz, today Poland, died January 27, 1972 at New York/USA) was a German and American mathematician. ... David Hilbert (January 23, 1862, Königsberg, East Prussia – February 14, 1943, Göttingen, Germany) was a German mathematician, recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. ... Year 1924 (MCMXXIV) was a leap year starting on Tuesday (link will display the full calendar) of the Gregorian calendar. ... For other persons named John Neumann, see John Neumann (disambiguation). ... David Hilbert (January 23, 1862, Königsberg, East Prussia – February 14, 1943, Göttingen, Germany) was a German mathematician, recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. ... The mathematical concept of a Hilbert space (named after the German mathematician David Hilbert) generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the plane and three-dimensional space to spaces of functions. ...

Heisenberg's Reasoning

Before matrix mechanics, quantum theory described the motion of a particle by a classical orbit 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 Planck's constant A commemoration plaque for Max Planck on his discovery of Plancks constant, in front of Humboldt University, Berlin. ...

While this restriction correctly selects orbits with the right energy values E_n, the old quantum mechanical formalism did not say how you should 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 which repeats itself every orbital period. The frequencies which 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 2πn / T only.

The coefficients X_n 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,

.

A quantum mechanical particle, on the other hand, can't emit radiation continuously, it can only emit photons. Assuming that the quantum particle started in orbit number n, emitted a photon, then ended up in orbit number m, the energy of the photon is E_n − E_m, which means that its frequency is (E_n − E_m) / h for large n and m, but with n-m relatively small, we recover the classical frequencies by Bohr's correspondence principle Niels Bohr Niels Henrik David Bohr (October 7, 1885 – November 18, Danish physicist who made essential contributions to understanding atomic structure and quantum mechanics. ... In physics, the correspondence principle is a principle, first invoked by Niels Bohr in 1923, which states that the behavior of quantum mechanical systems reduce to classical physics in the limit of large quantum numbers. ...

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 n and m small, or if n − m is large, the frequencies are not integer multiples of any single frequency.

Since the frequencies which the particle emits are the same as the frequencies in the fourier description of its motion, this suggests that something in the time-dependent description of the particle is oscillating with frequency (E_n − E_m) / h . Heisenberg called this quantity X_nm, and demanded that it should reduce to the classical fourier coefficient in the classical limit. For large values of n, m but with n-m relatively small, X_nm is the (n-m)th fourier coefficient of the classical motion at orbit n. Since X_nm has opposite frequency to X_mn, the condition that X is real becomes:

.

By definition, X_nm only has the frequency (E_n − E_m) / h, so its time evolution is simple:

.

This is the original form of Heisenberg's equation of motion.

Given two arrays X_nm and P_nm describing two physical quantities, Heisenberg could form a new array of the same type by combining the terms X_nkP_km, which also oscillate with the right frequency. Since the Fourier coefficients of the product of two quantities is the convolution of the Fourier coefficients of each one separately, the correspondence with Fourier series allowed Heisenberg to deduce the rule by which the arrays should be multiplied: In mathematics and, in particular, functional analysis, convolution is a mathematical operator which takes two functions f and g and produces a third function that in a sense represents the amount of overlap between f and a reversed and translated version of g. ...

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. Because of the multiplication rule, the product depends on the order: XP is different from PX.

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.

Further Discussion

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 great controversy. Schrödinger's later introduction of wave mechanics was favored. Werner Karl Heisenberg (December 5, 1901 – February 1, 1976) was a celebrated German physicist and Nobel laureate, one of the founders of quantum mechanics and acknowledged to be one of the most important physicists of the twentieth century. ... Max Born (December 11, 1882 in Breslau – January 5, 1970 in Göttingen) was a mathematician and physicist. ... Pascual Jordan (October 18, 1902 in Hanover - July 31, 1980 in Hamburg) was a German physicist. ... The wave equation is an important partial differential equation which generally describes all kinds of waves, such as sound waves, light waves and water waves. ...

Part of the reason was that Heisenberg's formulation was in an alien mathematical language, 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 under the direction of Einstein and the other under the direction of Bohr. Einstein emphasized wave-particle duality, while Bohr emphasized the discrete energy states and quantum jumps. DeBroglie had shown how to reproduce the discrete energy states in 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 which pictured electrons as waves, or as anything at all. They preferred to focus on the quantities which 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 which 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 which 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. Extremely high resolution spectrogram of the Sun showing thousands of elemental absorption lines (fraunhofer lines) Spectroscopy is the study of the interaction between radiation (electromagnetic radiation, or light, as well as particle radiation) and matter. ... In physics quanta is the plural of quantum. ...

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. In physics, particularly in quantum physics a system observable is a property of the system state that can be determined by some sequence of physical operations. ... In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ... A number of mathematical entities are named Hermitian, after the mathematician Charles Hermite: Hermitian matrix Hermitian operator Hermitian adjoint Hermitian form Hermitian metric See also: self-adjoint This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ...

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 you measure two observerables simultaneously, the state of the system should collapse 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. In linear algebra, the eigenvectors (from the German eigen meaning own) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ... In quantum physics, the Heisenberg uncertainty principle is a mathematical property of a pair of canonical conjugate quantities - usually stated in a form of reciprocity of spans of their spectra. ...

If two matrices share their eigenvalues, 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 then is a consequence of the fact that two matrices A and B do not always commute, A B - B A does not necessarily equal 0. The famous commutation relation of matrix mechanics:

shows that there are no states which simultaneously have a definite position and momentum. But the principle of uncertainty (also called complementarity by Bohr) holds for most other pairs of observables too. The momentum and position of an electron in an atom do not commute, so it is impossible to precisely determine the momentum and the position of an electron in its orbit. Complementarity is a concept in a number of fields: Economics In economics is a concept similar to that of externality. ...

In 1925, Werner Heisenberg was not yet 24 years old.

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 X(t) and P(t) are sinusoidal. In physics, the correspondence principle is a principle, first invoked by Niels Bohr in 1923, which states that the behavior of quantum mechanical systems reduce to classical physics in the limit of large quantum numbers. ... In classical mechanics, a Harmonic oscillator is a system which, when displaced from its equilibrium position, experiences a restoring force proportional to the displacement according to Hookes law: where is a positive constant. ...

In units where the mass and frequency of the oscillator are equal to one, the energy of the oscillator is

The level sets of H are the orbits, and they are nested circles. The classical orbit with energy E is: In mathematics, a level set of a real-valued function f of n variables is a set of the form { (x1,...,xn) | f(x1,...,xn) = c } where c is a constant. ...

The old quantum condition says 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 Planck's constant. The area of the circle of radius is 2πE So A commemoration plaque for Max Planck on his discovery of Plancks constant, in front of Humboldt University, Berlin. ...

or, in units of length where is one, the energy is an integer.

The fourier components of X(t) and P(t) are very simple, even more so if they are combined into the quantities:

both and have only a single frequency, and X and P can be recovered from their sum and difference.

Since A(t) has a classical fourier series with only the lowest frequency, and the matrix element A_mn is the (m-n)th fourier coefficient of the classical orbit, the matrix for is nonzero only on the line just above the diagonal, where it is equal to . The matrix for is likewise only nonzero on the line below the diagonal, with the same elements. Reconstructing X and P from and :

which, up to the choice of units, are the Heisenberg matrices for the harmonic oscillator. Notice that both matrices are hermitian, since they are constructed from the fourier coefficients of real quantities. To find X(t) and P(t) is simple, since they are quantum fourier coefficients so they evolve simply with time. A Hermitian matrix (or self-adjoint matrix) is a square matrix with complex entries which is equal to its own conjugate transpose — that is, the element in the ith row and jth column is equal to the complex conjugate of the element in the jth row and ith column, for...

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 (XP + PX), while the imaginary part is proportional to the commutator [X,P] = (XP − PX). It is easy to verify explicitly that (XP − PX) in the case of the harmonic oscillator, is i, multiplied by the identity. It is also easy to verify that the matrix In linear algebra, the identity matrix of size n is the n-by-n square matrix with ones on the main diagonal and zeros elsewhere. ...

is a diagonal matrix, with eigenvalues E_i. In linear algebra, a diagonal matrix is a square matrix in which the entries outside the main diagonal are all zero. ... In linear algebra, a scalar λ is called an eigenvalue (in some older texts, a characteristic value) of a linear mapping A if there exists a nonzero vector x such that Ax=λx. ...

The harmonic oscillator is too special. It is too easy to find the matrices exactly, and it is too hard to discover general conditions from these special forms. For this reason, Heisenberg investigated the anharmonic oscillator, with Hamiltonian In physics, Hamiltonian has distinct but closely related meanings. ...

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:

He noticed that if this could be done then H considered as a matrix function of X and P, will have zero time derivative.

Where A * B is the symmetric product. In mathematics, a Jordan algebra is defined in abstract algebra as an algebra over a field with multiplication satisfying the following axioms: (commutative law) (Jordan identity) Jordan algebras were first introduced by Pascual Jordan in quantum mechanics. ...

.

A constant matrix is the same as a diagonal matrix, because all the off diagonal elements have a nonzero frequency, so this also establishes 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.

But demanding that the classical equations of motion are preserved is not a strong enough condition to determine the matrix elements. Planck's constant does not appear in the classical equations, so that the matrices could be constructed for many different values of h and still satisfy the equations of motion, but with different energy levels.

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. This is clearly not satisfactory. The old quantum conditions refer to the area enclosed by the sharp classical orbits, which do not exist in the new formalism.

Heisenberg discovered how to translate the old quantum condition into a simple statement in matrix mechanics. To do this, he investigated the action integral as a matrix quantity:

There are several problems with this integral, all stemming from the incompatibility of the matrix formalism with the old picture of orbits. Which period T should you use? Semiclassically, it should be either m or n, but the difference is order h and we want an answer to order h. The quantum condition tells us that J_mn is 2πn on the diagonal, then the fact that J is classically constant tells us that the off diagonal elements are zero.

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 using the correspondence principle. In classical mechanics, action-angle coordinates are a set of canonical coordinates useful in solving many integrable systems. ...

In the classical setting, the derivative is the derivative with respect to J of the integral which defines J, so it is tautologically equal to 1.

Where the derivatives dp/dJ dx/dJ should be interpreted as differences with respect to J at corresponding times on nearby orbits, exactly what you would get if you differentiated the Fourier coefficients of the orbital motion. These derivatives are symplectically orthogonal in phase space to the time derivatives dP/dt dX/dt. The final expression is clarified by introducing the variable canonically conjugate to J, which is called the angle variable θ. The derivative with respect to time is a derivative with respect to θ, up to a factor of 2π / T. In classical mechanics, action-angle coordinates are a set of canonical coordinates useful in solving many integrable systems. ...

So the quantum condition integral is the average value over one cycle of the Poisson bracket of X and P. An analogous differentiation of the Fourier series of P dX demonstrates that the off diagonal elements of the Poisson bracket are all zero. The Poisson bracket of two canonically conjugate variables, such as X and P, is the constant value 1, so this integral really is the average value of 1, so it is 1, as we knew all along, because it is dJ/dJ after all. But Heisenberg, Born and Jordan weren't familiar with the theory of Poisson brackets, so for them, the differentiation effectively evaluated {X,P} in J θ coordinates. In mathematics and classical mechanics, the Poisson bracket is an important operator in Hamiltonian mechanics, playing a central role in the definition of the time-evolution of a dynamical system in the Hamiltonian formulation. ...

The Poisson Bracket, unlike the action integral, has a simple translation to matrix mechanics--- it is the imaginary part of the product of two variables, the commutator. To see this, examine the 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.

In the correspondence limit, when indices m n are large and nearby, while k,r are small, the rate of change of the matrix elements in the diagonal direction is the matrix element of the J derivative of the corresponding classical quantity. So we can shift any matrix element diagonally using the formula:

Where the right hand side is really only the (m-n)'th Fourier component of (dA/dJ) at orbit near m to this semiclassical order, not a full well defined matrix.

The semiclassical time derivative of a matrix element is obtained up to a factor of i by multiplying by the distance from the diagonal,

Since the coefficient A_{m(m + k)} is semiclassically the k'th Fourier coefficient of the m-th classical orbit.

The imaginary part of the product of A and B can be evaluated by shifting the matrix elements around so as to reproduce the classical answer, which is zero. The leading nonzero residual is then given entirely by the shifting. 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 r'th Fourier components of classical quantities.

Flipping the summation variable in the first sum from r to r'=k-r, the matrix element becomes:

and it is clear that the main part cancels. The leading quantum part, neglecting the higher order product of derivatives, is

which can be identified as i times the k-th classical Fourier component of the Poisson bracket. Heisenberg's original differentiation trick of was eventually extended to a full semiclassical derivation of the quantum condition in collaboration with Born and Jordan.

Once they were able to establish that:

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. The new quantization rule was assumed to be universally true, even though the derivation from the old quantum theory required semiclassical reasoning.

To make the transition to modern quantum mechanics, the most important further addition was the quantum state vector, now written , which is the vector that the matrices act on. Without the state vector, it is not clear which particular motion the Heisenberg matrices are describing, since they include all the motions somewhere.

The interpretation of the state vector components ψ_m was given by Born, and it is that the value of a matrix A(t) corresponding to the classical quantity A is random, with an average value equal to

Alternatively and equivalently, the state vector gives the probability amplitude ψ_i for the quantum system to be in the energy state i. Once the state vector was introduced, matrix mechanics could be rotated to any basis, where the H matrix was no longer diagonal. The Heisenberg equation of motion in its original form states that A_mn evolves in time like a Fourier component In quantum mechanics, a probability amplitude is a complex-valued function that describes an uncertain or unknown quantity. ...

which can be recast in differential form

and it can be restated so that it is true in an arbitrary basis by noting that the H matrix is diagonal with diagonal values E_m:

This is now a matrix equation, so it holds in any basis. This is the modern form of the Heisenberg equation of motion. The formal solution is:

All the forms of the equation of motion above say the same thing, that A(t) is equal to A(0) up to a basis rotation by the unitary matrix e^iHt. By rotating the basis for the state vector at each time by e^iHt, you can undo the time dependence in the matrices. The matrices are now time independent, but the state vector rotates: In government, see Unitary state In mathematics, see Unitary matrix Unitary operator Unitary group Unitary representation This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ...

This is the Schroedinger equation for the state vector, and the time dependent change of basis is the transformation to the Schroedinger picture. In physics, the Schrödinger equation, proposed by the Austrian physicist Erwin Schrödinger in 1925, describes the time-dependence of quantum mechanical systems. ... In quantum mechanics, a state function is a linear combination (a superposition) of eigenstates. ...

In quantum mechanics in the Heisenberg picture the state vector, does not change with time, and an observable A satisfies Fig. ... The Heisenberg Picture of quantum mechanics is also known as Matrix mechanics. ... Quite literally, quantum state describes the state of a quantum system. ...

The extra term is for operators like A = (X + t²P) which have an explicit time dependence in addition to the time dependence from unitary evolution. The Heisenberg picture does not distinguish time from space, so it is nicer for relativistic theories. Werner Karl Heisenberg (December 5, 1901 – February 1, 1976) was a celebrated German physicist and Nobel laureate, one of the founders of quantum mechanics and acknowledged to be one of the most important physicists of the twentieth century. ... Albert Einsteins theory of relativity is a set of two theories in physics: special relativity and general relativity. ...

Moreover, the similarity to classical physics is more obvious: the Hamiltonian equations of motion for classical mechanics are recovered by replacing the commutator above by the Poisson bracket. Classical physics is physics based on principles developed before the rise of quantum theory, usually including the special theory of relativity and general theory of relativity. ... In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. ... In mathematics and classical mechanics, the Poisson bracket is an important operator in Hamiltonian mechanics, playing a central role in the definition of the time-evolution of a dynamical system in the Hamiltonian formulation. ...

By the Stone-von Neumann theorem, the Heisenberg picture and the Schrödinger picture are unitarily equivalent. In mathematics and in theoretical physics, the Stone-von Neumann theorem is any one of a number of different formulations of the uniqueness of the canonical commutation relations between position and momentum operators. ... Heisenbergs form for the equations of motion We have seen that in Schrödingers scheme the dynamical variables of the system remain fixed during a period of undisturbed motion. ...

See also Schrödinger picture. Heisenbergs form for the equations of motion We have seen that in Schrödingers scheme the dynamical variables of the system remain fixed during a period of undisturbed motion. ...

Nobel Prize

In 1928, Albert Einstein nominated Heisenberg, Born, and Jordan for the Nobel Prize in Physics,^[17] but it was not to be. The announcement of the Nobel Prize in Physics for 1932 was delayed until November 1933.^[18] 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”^[19] and Erwin Schrödinger and Paul Adrien Maurice Dirac shared the 1933 Prize "for the discovery of new productive forms of atomic theory".^[20] One can rightly ask why Born was not awarded the Prize in 1932 along with Heisenberg? Bernstein gives some speculations on this matter. One of them is related to Jordan joining the Nazi Party on May 1, 1933 and becoming a Storm Trooper.^[21] Hence, Jordan’s Party affiliations and Jordan’s links to Born may have affected Born’s chance at the Prize at that time. Bernstein also notes that when Born won the Prize in 1954, Jordan was still alive, and the Prize was awarded for the statistical interpretation of quantum mechanics, attributable alone to Born.^[22] “Einstein” redirects here. ... Hannes Alfvén (1908–1995) accepting the Nobel Prize for his work on magnetohydrodynamics [1]. List of Nobel Prize laureates in Physics from 1901 to the present day. ... Year 1932 (MCMXXXII) was a leap year starting on Friday (the link will display full 1932 calendar) of the Gregorian calendar. ... Year 1933 (MCMXXXIII) was a common year starting on Sunday (link will display full calendar) of the Gregorian calendar. ... Bust of Schrödinger, in the courtyard arcade of the main building, University of Vienna, Austria. ... Paul Adrien Maurice Dirac, OM, FRS (IPA: [dɪræk]) (August 8, 1902 – October 20, 1984) was a British theoretical physicist and a founder of the field of quantum physics. ... The Nazi Party (German: , or NSDAP, English: National Socialist German Workers Party), was a far-right, racist political party in Germany between 1920 and 1945. ... is the 121st day of the year (122nd in leap years) in the Gregorian calendar. ... Year 1933 (MCMXXXIII) was a common year starting on Sunday (link will display full calendar) of the Gregorian calendar. ... The seal of SA SA propaganda poster. ...

Heisenberg’s reaction to Born for Heisenberg receiving the Prize for 1932 and to Born for Born receiving the Prize in 1954 are also instructive in evaluating whether Born should have shared the Prize with Heisenberg. 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.”^[23] 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.”^[24] Max Karl Ernst Ludwig Planck (April 23, 1858 in Kiel, Germany – October 4, 1947 in Göttingen, Germany) was a German physicist. ...

The Three Formulating Papers

• W. Heisenberg, Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen, Zeitschrift für Physik, 33, 879-893, 1925 (received July 29, 1925). [English translation in: B. L. van der Waerden, editor, Sources of Quantum Mechanics (Dover Publications, 1968) ISBN 0-486-61881-1 (English title: Quantum-Theoretical Re-interpretation of Kinematic and Mechanical Relations).]

• M. Born and P. Jordan, Zur Quantenmechanik, Zeitschrift für Physik, 34, 858-888, 1925 (received September 27, 1925). [English translation in: B. L. van der Waerden, editor, Sources of Quantum Mechanics (Dover Publications, 1968) ISBN 0-486-61881-1 (English title: On Quantum Mechanics II).]

• M. Born, W. Heisenberg, and P. Jordan, Zur Quantenmechanik II, Zeitschrift für Physik, 35, 557-615, 1925 (received November 16, 1925). [English translation in: B. L. van der Waerden, editor, Sources of Quantum Mechanics (Dover Publications, 1968) ISBN 0-486-61881-1]

Ãœber quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen (English Quantum theoretical re-interpretation of kinematic and mechanical relations) was a breakthrough paper in quantum mechanics written by Werner Heisenberg. ...

Bibliography

• Jeremy Bernstein Max Born and the Quantum Theory, Am. J. Phys. 73 (11) 999-1008 (2005). Department of Physics, Stevens Institute of Technology, Hoboken, New Jersey 07030. Received 14 April 2005; accepted 29 July 2005.

• Max Born The statistical interpretation of quantum mechanics. Nobel Lecture – December 11, 1954.

• Nancy Thorndike Greenspan, "The End of the Certain World: The Life and Science of Max Born" (Basic Books, 2005) ISBN 0-7382-0693-8. Also published in Germany: Max Born - Baumeister der Quantenweld. Eine Biographie (Spektrum Akademischer Verlag, 2005), ISBN 3-8274-1640-X.

• Max Jammer The Conceptual Development of Quantum Mechanics (McGraw-Hill, 1966)

• Jagdesh Mehra and Helmut Rechenberg The Historical Development of Quantum Theory. Volume 3. The Formulation of Matrix Mechanics and Its Modifications 1925–1926. (Springer, 2001) ISBN 0-387-95177-6

• B. L. van der Waerden, editor, Sources of Quantum Mechanics (Dover Publications, 1968) ISBN 0-486-61881-1

Footnotes

1. ^ W. Heisenberg, "Der Teil und das Ganze", Piper, Munich, (1969)The Birth of Quantum Mechanics.
2. ^ The Birth of Quantum Mechanics
3. ^ W. Heisenberg, Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen, Zeitschrift für Physik, 33, 879-893, 1925 (received July 29, 1925). [English translation in: B. L. van der Waerden, editor, Sources of Quantum Mechanics (Dover Publications, 1968) ISBN 0-486-61881-1 (English title: “Quantum-Theoretical Re-interpretation of Kinematic and Mechanical Relations”).]
4. ^ Emilio Segrè, From X-Rays to Quarks: Modern Physicists and their Discoveries (W. H. Freeman and Company, 1980) ISBN 0-7167-1147-8, pp 153 - 157.
5. ^ Abraham Pais, Niels Bohr’s Times in Physics, Philosophy, and Polity (Clarendon Press, 1991) ISBN 0-19-852049-2, pp 275 - 279.
6. ^ Max Born – Nobel Lecture (1954)
7. ^ M. Born and P. Jordan, Zur Quantenmechanik, Zeitschrift für Physik, 34, 858-888, 1925 (received September 27, 1925). [English translation in: B. L. van der Waerden, editor, Sources of Quantum Mechanics (Dover Publications, 1968) ISBN 0-486-61881-1]
8. ^ M. Born, W. Heisenberg, and P. Jordan, Zur Quantenmechanik II, Zeitschrift für Physik, 35, 557-615, 1925 (received November 16, 1925). [English translation in: B. L. van der Waerden, editor, Sources of Quantum Mechanics (Dover Publications, 1968) ISBN 0-486-61881-1]
9. ^ Jeremy Bernstein Max Born and the Quantum Theory, Am. J. Phys. 73 (11) 999-1008 (2005)
10. ^ Mehra, Volume 3 (Springer, 2001)
11. ^ Jammer, 1966, pp. 206-207.
12. ^ van der Waerden, 1968, p. 51.
13. ^ The citation by Born was in Born and Jordan's paper, the second paper in the trilogy which launched the matrix mechanics formulation. See van der Waerden, 1968, p. 351.
14. ^ Constance Ried Courant” (Springer, 1996) p. 93.
15. ^ John von Neumann Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren, Mathematische Annalen 102 49–131 (1929)
16. ^ When von Neumann left Göttingen in 1932, his book on the mathematical foundations of quantum mechanics, based on Hilbert’s mathematics, was published under the title Mathematische Grundlagen der Quantenmechanik. See: Norman Macrae, John von Neumann: The Scientific Genius Who Pioneered the Modern Computer, Game Theory, Nuclear Deterrence, and Much More (Reprinted by the American Mathematical Society, 1999) and Constance Reid, Hilbert (Springer-Verlag, 1996) ISBN 0-387-94674-8.
17. ^ Bernstein, 2004, p. 1004.
18. ^ Greenspan, 2005, p. 190.
19. ^ Nobel Prize in Physics and 1933 – Nobel Prize Presentation Speech.
20. ^ Nobel Prize in Physics and 1933 – Nobel Prize Presentation Speech.
21. ^ Bernstein, 2005, p. 1004.
22. ^ Bernstein, 2005, p. 1006.
23. ^ Greenspan, 2005, p. 191.
24. ^ Greenspan, 2005, pp. 285-286.

Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen (English Quantum theoretical re-interpretation of kinematic and mechanical relations) was a breakthrough paper in quantum mechanics written by Werner Heisenberg. ... Hannes Alfvén (1908–1995) accepting the Nobel Prize for his work on magnetohydrodynamics [1]. List of Nobel Prize laureates in Physics from 1901 to the present day. ... Hannes Alfvén (1908–1995) accepting the Nobel Prize for his work on magnetohydrodynamics [1]. List of Nobel Prize laureates in Physics from 1901 to the present day. ...

See also

• Interaction picture
• Quantum mechanics
• Schrödinger equation
• Bra-ket notation
• Basic quantum mechanics

In quantum mechanics, the Interaction picture (or Dirac picture) is an intermediate between the Schrödinger picture and the Heisenberg picture. ... Fig. ... For a non-technical introduction to the topic, please see Introduction to quantum mechanics. ... Bra-ket notation is the standard notation for describing quantum states in the theory of quantum mechanics. ... Quantum mechanics is a physical science dealing with the behaviour of matter and electromagnetic waves on the scale of atoms and subatomic particles. ...

External links

• An Overview of Matrix Mechanics
• Matrix Methods in Quantum Mechanics
• Heisenberg Quantum Mechanics (The theory's origins and its historical developing 1925-27)
• Werner Heisenberg 1901 - 1976
• Werner Karl Heisenberg Co-founder of Quantum Mechanics