Re-Examining the Crisis in Quantum Theory

Part 1: Zeeman Spectroscopy

by David C. Cassidy, Hofstra University, Hempstead (NY)

Max Planck (1858–1947)

Thomas S. Kuhn, The Structure of Scientific Revolutions (1962), pp. 67-68

Because it leads to large-scale paradigm destruction....the emergence of

new theories is generally preceded by a period of pronounced professional

insecurity. As one might expect, that insecurity is generated by the

persistent failure of the puzzles of normal science to come out as they

should.

Paul Forman, “Weimar Culture, Causality, and Quantum Theory, 1918-

1927”, HSPS (1972), p. 62.

It now seems evident to me that these internal developments were not in

themselves sufficient conditions. The possibility of the crisis of the old

quantum theory was, I think, dependent upon the physicists’ own craving

for crises, arising from participation in, and adaptation to, the Weimar

intellectual milieu.

Gap

Suman Seth, “Crisis and the construction of modern theoretical physics,”

British Journal for History of Science, March 2007, p. 25

• Sommerfeld school: emphasis on solving problems,

“neither saw a crisis nor acknowledged its resolution.”

• Bohr and Copenhagen school: emphasis on finding and

using new principles, “openly advocated a crisis even before decisive

anomalies arose.”

Some historical studies of the quantum crisis

M. Jammer, The conceptual development of quantum mechanics. 1966.

P. Forman, “The doublet riddle and atomic physics circa 1924,” Isis, 1968.

P. Forman. “Alfred Landé and the anomalous Zeeman effect,” HSPS, 1970.

H. Small. The helium atom in the old quantum theory. PhD diss., 1971.

R. Stuewer. The Compton effect: Turning point in physics, 1971.

D. Cassidy. Werner Heisenberg and the crisis in quantum theory, 1920-25.

PhD diss., 1976.

D. Serwer. “Unmechanischer Zwang: Pauli, Heisenberg, and the rejection of

the mechanical atom, 1923-1925.” HSPS, 1977.

J. Hendry. “Bohr-Kramers-Slater: A virtual theory of virtual oscillators.”

Centaurus, 1981.

Gap

O. Darrigol. From c-numbers to q-numbers. 1992. Chapter 8, “A Crisis”

H. Kragh. Quantum generations. 1999. Section: “Quantum anomalies”

Historical Implications of Bohr Atom

• Extraordinarily successful visualizable mechanical model of orbiting

electrons in states obeying classical mechanics but not electrodynamics.

Historical Implications of Bohr Atom

• Extraordinarily successful visualizable mechanical model of orbiting

electrons in states obeying classical mechanics but not electrodynamics.

• Despite the success, many statements appeared over the next decade

on the fundamentally unsatisfactory nature of the theory, starting with Bohr

himself in 1913.

Historical Implications of Bohr Atom

• Extraordinarily successful visualizable mechanical model of orbiting

electrons in states obeying classical mechanics but not electrodynamics.

• Despite the success, many statements appeared over the next decade

on the fundamentally unsatisfactory nature of the theory, starting with Bohr

himself in 1913.

• Yet the Bohr atom (with extensions by Sommerfeld) set the standard for a

successful quantum theory of atomic phenomena for a decade:

“Eine modellmäßige Deutung”--- “a model-based interpretation”--- of atomic

phenomena.

The Onset of the Crisis

• Beginning about 1918, new and more precise data and

mechanical calculations resulted by the early 1920s in an

increasing failure to achieve the ideal set by the Bohr atom

as the standard for normal science.

The Onset of the Crisis

• Beginning about 1918, new and more precise data and

mechanical calculations resulted by the early 1920s in an

increasing failure to achieve the ideal set by the Bohr atom

as the standard for normal science.

• This failure magnified the sense of insecurity about the old

quantum theory within the quantum physics community.

The Onset of the Crisis

• Beginning about 1918, new and more precise data and

mechanical calculations resulted by the early 1920s in an

increasing failure to achieve the ideal set by the Bohr atom

as the standard for normal science.

• This failure magnified the sense of insecurity about the old

quantum theory within the quantum physics community.

• The insecurity reached such proportions by 1923 that it

came close to what Kuhn described as a crisis situation.

At the same time, as Seth suggested, noticeable

differences did appear among different groups.

However….

• The old quantum theory did work quite well for many other

phenomena, such as molecular band spectra.

However….

• The old quantum theory did work quite well for many other

phenomena, such as molecular band spectra.

• Forman thesis and related issues regarding the quantum

crisis require a separate paper.

3 Problems, 3 Groups, 3 Places- Roughly Defined

1. Problem: Multiplet (complex) line spectra of atoms and their

anomalous Zeeman effects.

Research group: Sommerfeld school in Munich, including Tübingen

spectroscopists Paschen and Back with theorist Alfred Landé

3 Problems, 3 Groups, 3 Places- Roughly Defined

1. Problem: Multiplet (complex) line spectra of atoms and their

anomalous Zeeman effects.

Research group: Sommerfeld school in Munich, including Tübingen

spectroscopists Paschen and Back with theorist Alfred Landé

2. Problem: Simple 3-body atomic/molecular systems beyond

hydrogen atom: H2+, normal He, excited He

Research group: Born school in Göttingen, including Pauli and

Heisenberg; also Kramers, Kemble, Van Vleck

Both of these problem areas concerned atomic structure.

3. Problem: Nature of light: Interaction of radiation and matter, including

dispersion theory and the problem of light quanta

Research group: Bohr school in Copenhagen: including Pauli,

Heisenberg, Kramers.

Sommerfeld an Bohr, 4. Sept. 1913Werden Sie Ihr Atommodell auch auf den Zeeman-Effekt anwenden?Ich wollte mich damit beschäftigen.

Sommerfeld & Bohr, ca. 1919

Some important results of Sommerfeld, 1916

• The Sommerfeld quantum conditions, involving only integer

numbers of quanta

• a relativistic treatment of the Kepler orbits of the electrons in the

Bohr model led to angular momentum as a degree of freedom,

and...

• two new quantum numbers for the orbital motion of an electron,

giving 3 altogether.

Sommerfeld’s quantum numbers:

n for orbital state

k for azimuthal angular momentum

m for space quantization, if z axis defined.

0,1,2,..., yielding 21 statesmkk= ± ± ± +

• Sommerfeld intended to define the z-axis by a weak magnetic

field and thus obtain the Zeeman effect, but he did not do so in that

paper.

Normal Zeeman Effect for singlet lines by Sommerfeld and Debye,1916

0022

L

e

mheHEEEmh

mc!

"= + # = + 1,0m! = ±

Gives splitting of a singlet line into 3 lines in weak magnetic field, but ….

• Origin of selection rule unknown

• Could not account for anomalous Zeemaneffect

0E

E Energy of Zeeman term

Energy of unperturbed optical term

L! Larmor frequency

Anomalous Zeeman Effect, about 1920

• Associated with optical multiplet lines: doublets and triplets

• Splitting into more than 3 lines or not at Larmor frequency.

From Sommerfeld, Atombau u. Spektrallinien, 1921

Paschen-Back Effect, 1913

As magnetic field H increases, the anomalous Zeeman effect transforms

continuously into the normal Zeeman effect. Shown below for doublet

sodium D-lines.

From Sommerfeld, Atombau, 1924; my diss.

doublets: j = k, k – 1triplets: j = k, k – 1, k - 2

Today: fine structure owingto spin-orbit coupling

Sommerfeld, “Ein Zahlenmysterium in der Theorie des

Zeemaneffektes”. Naturwissenschaften, 1920.

Der musikalischen Schönheit unserer Zahlentafel wird dadurch kein

Abbruch getan, daß sie einstweilen ein Zahlenmysterium darstellt. In

der Tat sehe ich bisher keinen Weg zur modellmäßigen Erklärung

weder der Dublett-Triplett-Tatsachen noch ihrer magnetischen

Beeinflussung.

Landé, 1921

0 LEEmh != + normal Zeeman effect

iLEEgmh != + anomalous Z.E.

1g =

2

21

jg

k=

!

11/

11/1(1)

11/(1)

k

gkk

k

+! "# #

= $ + $% &# #$ $' (

singlets

doublets

triplets

0,1,2,...,mj= ± ± ±

135 1,,,...,()2222

mj= ± ± ± ± !

triplets, 2j+1 states, odd

doublets, 2j states, even

j = k

j = k - 1

j = k - 2

Sommerfeld to Einstein, Okt. 1921

Es kommt wirklich Licht oder besser gesagt Dämmerung

in die Spektroskopie.

Sommerfeld to Landé, Feb. 1921

Bravo! Sie können hexen! Ihre Construktion der Dublett-

Zeeman-Typen ist sehr schön.

Heisenberg, 1924. Photo by F. Hund.

Sommerfeld, Zs. f. Physik, 8 (1922)

__21

2(*1(2*/*))LEEhmmk! " "= + ± + +

The Sommerfeld-Voigt equation for doublets, 1922

E is the energy of the Zeeman term

__

E is the average of the doublet energies

1

LH

!"

!

#= $ k* = k – ½, j* = j – ½

*1/2,3/2,...,*mk= ± ± ±

Landé’s g-factors obtained from dominant terms in an

expansion of square root as ! "#

**mj !

Vector addition in Heisenberg’s core model, 1921

1122

1122

()

()1

jkk

jkk

= ! + =

= ! ! = !

Sommerfeld to Einstein, 11 January 1922

Ein Schüler von mir (Heisenberg, 3. Semester!) hat diese Gesetze

u. die der anomalen Zeemaneffekte sogar modellmässig gedeutet

(Z. f. Ph., im Druck). Alles klappt, bleibt aber doch im tiefsten Grunde unklar.

Ich kann nur die Technik der Quanten fördern, Sie müssen

Ihre Philosophie machen. .....Setzen Sie ihr nur ordentlich zu!

1. Born and Heisenberg in Göttingen–a rigorous and systematic

application of Hamilton-Jacobi mechanics to quantum models of

the excited helium atom in order to determine if Bohr-Sommerfeld

theory worked or not.

Two research programs spawned by the Bohr Festival

2. Bohr and Pauli in Copenhagen–“a desperate attempt to

remain true to the integer quantum numbers, in which we

hoped to see even in the paradoxes a hint for the way in

which we might seek the solution of the anomalous Zeeman

effect.”

Bohr to Landé, 3 March 1923

Es war…ein Verzweiflungsversuch, den ganzen Quantenzahlen treu

zu bleiben, indem wir hofften, eben in den Paradoxien einen

Fingerzeig zu sehen für die Wege, auf denen man die Lösung des

anomalen Zeemaneffektes suchen dürfte.”

Pauli an Bohr, 21 Feb. 1924

Ich selbst kann an dieser Sorte von theoretischer Physik

keinerlei Geschmack gewinnen und ziehe mich von ihr zu

meiner Wärmeleitung im festen Körper zurück.

Born, Naturwissenschaften, Bohr-Heft, July 1923

Es wird immer wahrscheinlicher, daß nicht nur neue Annahmen

im gewöhnlichen Sinne physikalischer Hypothesen erforderlich

sein werden, sondern daß das ganze System der Begriffe der Physik

von Grund aus umgebaut werden muß.

Thank you!

Einstein and Bohr, late 1920s