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«75 Years Schrödinger Equation Erwin Schrödinger and the Discovery of Wave Mechanics Siegmund Brandt Seminar at Universidad Autonoma de Madrid July ...»

75 Years Schrödinger Equation

Erwin Schrödinger and the Discovery of

Wave Mechanics

Siegmund Brandt

Seminar at Universidad Autonoma de Madrid

July 5th, 2001

Abstract

The formulation of the laws of nature on the microscopic scale belongs to the greatest achievements of the

twentieth century.

Only a few months after the creation of quantum mechanics by Werner Heisenberg and its mathematical

formulation as “matrix mechanics” by Max Born, Pascual Jordan, and Heisenberg a completely different approach to the physical description of small-scale systems like atoms was successfully developed by Erwin Schrödinger. Using a concept proposed by Louis de Broglie he assumed that particles (in an atom in particular the electrons) have wave properties and early in 1926 he published a wave equation for particle waves. A little later he showed that his “wave mechanics” is mathematically equivalent to matrix mechanics. Because of its simpler immediate interpretation and computational methods in practice now nearly always wave mechanics is used.

The talk sketches Schrödinger’s eventful life and reports on his discoveries within the scientific developments of the time. The interpretation of Schrödinger’s wave function in terms of probabilities was first given by Max Born. It is now universally accepted. Schrödinger himself, however, always rejected it.

1887– First Vienna Time* 1887–1920 1887 Born in Vienna, August 12th.

1898 Pupil in the “Akademische Gymnasium” in Vienna (always first of his grade).

1906 Begins to study physics and mathematics at the University of Vienna, in particular theoretical physics with Fritz Hasenöhrl.

1910 Ph.D. with an experimental dissertation (advised by Egon von Schweidler).

Assistant with Franz Exner at “Physikalisches Institut”| at Vienna University.

1914 “Habilitation” at Vienna University.

1914–1918 Soldier.

1918 Appointment as professor at Tschernowitz (now Ukraine) is not realized because of the collapse of the Austro-Hungarian empire.

1920 Marriage to Annemarie (Anny) Bertel.

* The names used for the different periods of his life are taken from the short section “My Life” in Schrödinger’s book “My View of the World“, Vienna 1961.

In his Mothers Arm 1888 Fritz Hasenöhrl Schrödinger followed a course in theoretical physics given by Hasenöhrl over eight semesters, five hours per week. He admired Hasenöhrl and often expressed his gratitude for this detailed and thorough education.

Hasenöhrl was killed in action in October 1915.

Hasenöhrl in a paper „Theorie der Strahlung bewegter Körper“ discussed the equivalence of mass and energy even before Einstein.

–  –  –

1905 Einstein transfers Planck’s discovery of energy quantization from the material “resonators“ to the electromagnetic radiation and he proposes an experimental method to check his hypothesis of a light

quantum:

„If now monochromatic radiation behaves like a discontinuous ν medium, which is composed of energy quanta of magnitude hν, it suggests itself to study whether the laws of creation and transformation of light are such as if light were composed of such energy quanta.“ Light of frequency ν is composed of individual light quanta of energy

–  –  –

Schrödinger learns about de Broglie’s matter waves from a paper by Einstein.

Following a suggestion by Pieter Debye he gives a talk on de Broglie waves in the Zurich Physics Colloquium.

While taking a cure in Arosa over Christmas 1925 and New Year 1926 Schrödinger has a break-through in the development of wave mechanics: He discovers a wave equation for matter waves, later called stationary Schrödinger equation.

Back in Zurich he consults with Hermann Weyl on mathematical questions.

„Villa Herwig” in Arosa, where Schrödinger discovered his stationary wave equation

In 1926 Schrödinger publishes 6 papers:

•„Quantisierung als Eigenwertproblem I“ (Stationary Schrödinger equation, hydrogen spectrum) •„Quantisierung als Eigenwertproblem II“ („Derivation“ from the Hamilton formalism (wave mechanics)) •„Über das Verhältnis der Heisenberg-Born-Jordanschen Quantenmechanik und der meinen“ (Relation to matrix mechanics) •„Quantisierung als Eigenwertproblem III“ (Perturbation theory) •„Quantisierung als Eigenwertproblem IV“ (Time-dependent Schrödinger equation) •„Der stetige Übergang von der Mikro- zur Makromechanik“ (Wave packet, particle localization) Pieter Debye Hermann Weyl The „Abhandlungen“, a Collection of Reprints in Book Form Schrödinger’s Original Path to his Wave Equation for Matter Waves Stationary Schrödinger Equation Schrödinger expected that his first (relativistic) equation would describe the hydrogen spectrum including fine structure. Only when that turned out not to be the case he turned to a simplified, non relativistic equation.

In his first paper he gave a rather nebulous motivation for his equation which he “did not follow up” in his second paper.

–  –  –

William Rowan Hamilton (1805–1865) Derivation from the Hamilton Formalism Relation to the Hamilton Formalism Relation to the Hamilton Formalism, Contd.





Relation to Matrix Mechanics Contd.

Translation of the marked section on the previous page The decisive point [der springende Punkt] in constructing the matrices lies in the simple remark that

Heisenberg’s strange computational rules for functions of the two times n quantities:

q1, q2,Kqn; p1, p2,Kpn (position and canonic conjugate momentum coordinates) completely coincide with the computational rules which, according to ordinary analysis apply to linear differential operators in the field of one times n variables q1, q2 Kqn

–  –  –

A. E. Haas asked, whether the new waves of mechanics would be permanent waves. They were.

The mood of the time is shown in an anecdote then coming from Göttingen. P. Ehrenfest from Leiden was visiting there and heard a seminar talk by the young E. Wigner. He liked especially, that Wigner in his talk used Schrödinger’s formalism. Max Born, one of the three authors of matrix mechanics, remarked that this, after all, was only a matter of habit. One could express the same with matrices. Ehrenfest answered: “ I will believe that. But there are good and bad habits.“ Wave Group (or Wave Packet) Wave packet (in front), composed of harmonic matter waves of different velocities (in the back), shown for two different moments in time (top and bottom).

Displayed is the real part of the complex wave function.

The wave function of a wave packet practically vanishes everywhere except in certain regions of space (ideally only in one region around the position of the “classical” particle).

Figure from Schrödinger’s last paper of 1926 Wave Packet in the Potential of an Harmonic Oscillator The absolute square ρ = ψψ * = ψ of the wave function in some cases has a simple form.

For the motion of a particle in the force field of an harmonic oscillation it may have the bell shape shown. Its width may change periodically with time (top) but may even stay unchanged (bottom). It is always concentrated around the position of the „classical particle“ (shown as small red circle). Schrödinger therefore interpreted it as a density distribution of the particle.

Splitting of a Wave Packet through Tunnel Effect Upon impact on a potential barrier the classical particle is reflected. The function ρ = ψψ * = ψ splits into two parts which completely separate at large times. Therefore, the function obviously cannot be the density distribution of a single particle.

Born’s Probability Interpretation

–  –  –

W = ρ∆V Schrödinger’s Rejection of the Probability Interpretation The probability interpretation turned out more and more to be the only one compatible with experiment. It is the basis of the „Copenhagen Interpretation“ of quantum mechanics, so called after the many discussions in the circle around Niels Bohr in Copenhagen. Some famous physicists,

however, rejected it. Born reports on a letter by Schrödinger written in the autumn of 1960:

... from one of these letters: „Du Maxel, Du weißt, ich hab Dich lieb und daran kann nichts etwas ändern. Aber ich habe das Bedürfnis, Dir mal gründlich den Kopf zu waschen. Also halt her.“ [“Maxel [Austrian diminutive of Max], you know, I like you very much and nothing can change that. But I have an urge to give you a good talking-to. So listen.”] And then came strong words about the „impudence“, with which I always assured the „Copenhagen Interpretation“ of quantum mechanics were generally accepted, although I well knew that Einstein, Planck, de Broglie, von Laue, and he, Erwin, were not satisfied by it. To my enumeration of some good researchers which did not share these reservations he replied: „Seit wann wird übrigens eine wissenschaftliche These durch Mehrheit entschieden? (Du könntest freilich erwidern: mindestens schon seit Newton).“ [“Since when, by the way, a scientific thesis is decided by majority? (You might, of course answer:

at least already since Newton).”] And so on for pages.... But his next letter begins with the words:

„Dank Dir für die reizende lange Antwort auf den Kopfwasch.“ [“Thank you for the charming long answer on my talking-to.”] Like this it was always in the long years of our correspondence: a mixture of rude and tender; sharpest exchange of opinions, never a feeling of being hurt.

Time of Learning and Teaching in Berlin 1927— 1927—1933 1927 Call to the University of Berlin as successor of Max Planck.

1929 Member of the Prussian Academy of Sciences.

1933 Emigrates outraged because of the anti-Semitism of the Nazi government.

Papers on:

• Wave mechanics

• Relativistic quantum mechanics

Essay:

„What is a Law of Nature?“ (Write-up of his Inaugural Lecture of 1922 in Zurich) Berlin Nobel Prize Winners with Guest

–  –  –

Being introduced by Max Planck in 1929 as member of the Prussian Academy Schrödinger’s Signatures in the “Studienbuch” (Lecture Record) of...

... the Berlin Student Maria Brandt, Mother of the Lecturer of the Present Talk

–  –  –

„To create a true picture of my life I am lacking the gift storyof a story-teller but actually I am also lacking the possibility, possibility, since omitting the relations to women would, in my case, on the one hand, lead to a large gap, on the other hand for, seems to be called for, firstly because of gossip, secondly because they are not of sufficient interest, and thirdly because in these matters no human being is completely honest and truthful or is allowed to be.“ be.“ From Max Born’s Obituary of Erwin Schrödinger As a physicist he belongs doubtlessly to very very great. For, even though his wave mechanics is based on the work of predecessors, above all on that of William R. Hamilton and Louis de Broglie, it is nevertheless original in the highest degree and completely independent of the Göttingen and Cambridge quantum mechanics.

Schrödinger then found the correct link between the two methods.

But from the beginning his procedure was popular and has remained so. His name probably is the most widely quoted in physics publications. Who of us has not written down the words Schrödinger Equation or Schrödinger Function an uncounted number of times? Probably the next generations will do the same and keep his name alive.

Physikalische Blätter 17 (1961) 87 Literature Erwin Schrödinger, Abhandlungen zur Wellenmechanik, Leipzig 1928 Erwin Schrödinger, Meine Weltansicht, Wien 1961 Erwin Schrödinger, Die Wellenmechanik (Dokumente der Naturwissenschaften, Bd. 3) Stuttgart 1963 Erwin Schrödinger, Gesammelte Abhandlungen, 4 Bde., Wien 1984 Walter Moore, Schrödinger, Life and Thought, Cambridge University Press, Cambridge 1989 G. Kerber, A. Dick, W. Kerber, Dokumente, Materialien und Bilder zur 100. Wiederkehr des Geburtstages von Erwin Schrödinger, Wien 1987 Abraham Pais, Inward Bound, Oxford University Press, Oxford 1986 Friedrich Hund, Geschichte der Quantentheorie, BI, Mannheim 1967 J. Mehra, H. Rechenberg, The Historical Development of Quantum Theory, 5 Bände, Springer, New York 1982



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