Matrix mechanics

From Wikipedia, the free encyclopedia

Matrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925.

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.

1 Development of matrix mechanics
- 1.1 The Three Papers
- 1.2 Further Discussion
2 Mathematical details
3 Deriving Heisenberg's equation
4 Nobel Prize
5 The Three Formulating Papers
6 Bibliography
7 Footnotes
8 See also
9 External links

[edit] Development of matrix mechanics

[edit] The Three Papers

In 1925, Werner Heisenberg, Max Born, and Pascual Jordan formulated the matrix mechanics representation of quantum mechanics. On July 9, Heisenberg gave Born a paper to review and submit for publication.^[1] 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.^[2] When Born read the paper, he recognized the formulation as one which could be transcribed and extended to the systematic language of matrices,^[3] which he had learned from his study under Jakob Rosanes^[4] 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.^[5] A follow-on paper was submitted for publication before the end of the year by all three authors.^[6] (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.^[7] 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.^[8])

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.^[9] 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.^[10] ^[11] 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.^[12] 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.^[13] ^[14]

[edit] 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 because there were no visual aids to fall back on in matrix mechanics and the mathematics were unfamiliar to most physicists.

Matrix mechanics consists of an array of quantities which when appropriately manipulated gave the observed frequencies and intensities of spectral lines. Heisenberg said himself that once and for all he had gotten rid of all electron orbits that did not exist. However, in Heisenberg's theory, the result of multiplication changes depending on its order. This means that the physical quantities in Heisenberg's theory are not ordinary variables but mathematical matrices. Heisenberg developed matrix mechanics by interpreting the electron as a particle with quantum behavior. It is based on sophisticated matrix computations which introduce discontinuities and quantum jumps.

In atomic physics, through spectroscopy, it was known that observational data related to atomic transitions arise from interactions of the atoms with light quanta. Heisenberg was the first to say that the atomic spectrum which showed spectral lines only in places where photons were being absorbed or emitted as electrons changed orbitals were the only relevant objects to be defined. Heisenberg recognized that the matrix formulation was built on the premise that all physical observables must be represented by matrices. The set of eigenvalues of the matrix representing an observable is the set of all possible values that could arise as outcomes of experiments conducted on a system to measure the observable. Since the outcome of an experiment to measure a real observable must be a real number, Hermitian matrices would represent such observables as their eigenvalues are real. If the result of a measurement is a certain eigenvalue, the corresponding eigenvector represents the state of the system immediately after the measurement.

Instead of using three dimensional orbitals, Heisenberg's matrix mechanics described the space in which the state of a quantum system inhabits as being one-dimensional as in the case of an anharmonic oscillator. To illustrate, consider the simple example of a point particle like an electron that is free to move on a line. An observable in this case could be the position of the particle, represented by the matrix X. Since the particle could be anywhere on the line, the possible outcome of a measurement of its position could be any one of an infinite set of eigenvalues of X, denoted by x. Thus X must be an infinite-dimensional matrix, and hence so is the corresponding linear vector space. Thus even one-dimensional motion could have an infinite-dimensional linear vector space associated with it. This made operators, functions, and spaces necessary to describe quantum mechanics.

The act of measurement in matrix mechanics is taken to 'collapse' the state of the system to that eigenvector (or eigenstate). Anyone familiar with Schrödinger's wave equation which came later in 1925 will be familiar with this concept in the form of wavefunction collapse. If one were to make simultaneous measurements of two or more observables, the system will collapse to a common eigenstate of these observables right after the measurement.

Further, from matrix theory we know that eigenvectors corresponding to distinct eigenvalues of a Hermitian matrix are orthogonal to each other which is analogous to the x, y, z axes of the Cartesian coordinate system except with an infinite number of distinct eigenvalues, and hence as many mutually orthogonal eigenvectors directed along different independent directions in the linear vector space.

Prior to measurement, the system could have been in a linear superposition of different eigenstates, with coefficients that might not be known. The Copenhagen interpretation is concerned only with outcomes of experiments.

Since a single measurement of any observable A yields one of the eigenvalues of A as the outcome, and collapses the state of the system to the corresponding eigenstate, subsequent measurements made immediately thereafter would continue to yield the same eigenvalue. So the correct thing to do would be to prepare a collection of a very large number of identical copies of the system and conduct a single trial on each copy. The arithmetic mean of all the results thus obtained is the average value we see, denoted by (A).

The Uncertainty Principle in matrix mechanics stems from the fact that, in general, two matrices A and B do not obey the arithmetical law of commutation. The commutator A B - B A = [A, B] does not equal 0. The famous commutation relation that is the basis for Quantum Mechanics and the later Uncertainty Principle is:

$\begin{matrix} \sum_{k}\end{matrix} p(n,k) q(k,n) - q(n,k) p(k,n) = h/2\pi i$

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

[edit] Mathematical details

In quantum mechanics in the Heisenberg picture the state vector, $| \psi \rangle$ does not change with time, and an observable A satisfies

$\frac{dA}{dt} = {i \over \hbar } [ H , A(t) ] + \left(\frac{\partial A}{\partial t}\right)_{classical}$

In some sense, the Heisenberg picture is more natural and fundamental than the Schrödinger picture, especially for relativistic theories. Lorentz invariance is manifest in the Heisenberg picture.

Moreover, the similarity to classical physics is easily seen: by replacing the commutator above by the Poisson bracket, the Heisenberg equation becomes an equation in Hamiltonian mechanics.

By the Stone-von Neumann theorem, the Heisenberg picture and the Schrödinger picture are unitarily equivalent.

[edit] Deriving Heisenberg's equation

Suppose we have an observable A (which is a Hermitian linear operator). The expectation value of A for a given state |ψ(t)> is given by:

$\lang A \rang _{t} = \lang \psi (t) | A | \psi(t) \rang$

or if we write following the Schrödinger equation

$| \psi (t) \rang = e^{-iHt / \hbar} | \psi (0) \rang$

(where H is the Hamiltonian and ħ is Planck's constant divided by 2*π) we get

$\lang A \rang _{t} = \lang \psi (0) | e^{iHt / \hbar} A e^{-iHt / \hbar} | \psi(0) \rang$

and so we define

$A(t) := e^{iHt / \hbar} A e^{-iHt / \hbar}$

Now,

${d \over dt} A(t) = {i \over \hbar} H e^{iHt / \hbar} A e^{-iHt / \hbar} + \left(\frac{\partial A}{\partial t}\right)_{classical} + {i \over \hbar}e^{iHt / \hbar} A \cdot (-H) e^{-iHt / \hbar}$

(differentiating according to the product rule),

$= {i \over \hbar } e^{iHt / \hbar} \left( H A - A H \right) e^{-iHt / \hbar} + \left(\frac{\partial A}{\partial t}\right)_{classical} = {i \over \hbar } \left( H A(t) - A(t) H \right) + \left(\frac{\partial A}{\partial t}\right)_{classical}$

(the last passage is valid since exp(-iHt/hbar) commutes with H)

$= {i \over \hbar } [ H , A(t) ] + \left(\frac{\partial A}{\partial t}\right)_{classical}$

(where [X,Y] is the commutator of two operators and defined as $[X, Y]: = X Y - Y X$ )

So we get

${d \over dt} A(t) = {i \over \hbar } [ H , A(t) ] + \left(\frac{\partial A}{\partial t}\right)_{classical}$

Though matrix mechanics does not include concepts such as the wave function of Erwin Schrödinger's wave equation, the two approaches were proven to be mathematically equivalent by the mathematician John von Neumann.

[edit] Nobel Prize

In 1928, Albert Einstein nominated Heisenberg, Born, and Jordan for the Nobel Prize in Physics,^[15] but it was not to be. The announcement of the Nobel Prize in Physics for 1932 was delayed until November 1933.^[16] 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”^[17] and Erwin Schrödinger and Paul Adrien Maurice Dirac shared the 1933 Prize "for the discovery of new productive forms of atomic theory".^[18] 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.^[19] 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.^[20]

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.”^[21] 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.”^[22]

[edit] The Three Formulating Papers

W. Heisenberg, Über quantentheoretishe Umdeutung kinematisher 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]

[edit] 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

[edit] Footnotes

^ W. Heisenberg, Über quantentheoretishe Umdeutung kinematisher 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”).]
^ 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.
^ Abraham Pais, Niels Bohr’s Times in Physics, Philosophy, and Polity (Clarendon Press, 1991) ISBN 0-19-852049-2, pp 275 - 279.
^ Max Born – Nobel Lecture (1954)
^ 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]
^ 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]
^ Jeremy Bernstein Max Born and the Quantum Theory, Am. J. Phys. 73 (11) 999-1008 (2005)
^ Mehra, Volume 3 (Springer, 2001)
^ Jammer, 1966, pp. 206-207.
^ van der Waerden, 1968, p. 51.
^ 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.
^ Constance Ried Courant” (Springer, 1996) p. 93.
^ John von Neumann Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren, Mathematische Annalen 102 49–131 (1929)
^ 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.
^ Bernstein, 2004, p. 1004.
^ Greenspan, 2005, p. 190.
^ Nobel Prize in Physics and 1933 – Nobel Prize Presentation Speech.
^ Nobel Prize in Physics and 1933 – Nobel Prize Presentation Speech.
^ Bernstein, 2005, p. 1004.
^ Bernstein, 2005, p. 1006.
^ Greenspan, 2005, p. 191.
^ Greenspan, 2005, pp. 285-286.

[edit] See also

[edit] 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)

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Categories: Quantum mechanics | Fundamental physics concepts

Matrix mechanics

From Wikipedia, the free encyclopedia

Contents

[edit] Development of matrix mechanics

[edit] The Three Papers

[edit] Further Discussion

[edit] Mathematical details

[edit] Deriving Heisenberg's equation

[edit] Nobel Prize

[edit] The Three Formulating Papers

[edit] Bibliography

[edit] Footnotes

[edit] See also

[edit] External links

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