Fortschritte der Physik / Progress of Physics: Volume 34, Number 7 [Reprint 2022 ed.] 9783112613887, 9783112613870

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FORTSCHRITTE DER PHYSIK PROGRESS OF PHYSICS

Volume 34-1986 Number 7

Board of Editors

F. Kaschluhn A. Lösche R. Rompe

Editor-in-Chief F. Kaschluhn

Advisory Board A. M. Baldin, Dubna J. Fischer, Prague G. Höhler, Karlsruhe K. Lanius, Berlin J. Lopuszanski, Wroclaw A. Salam, Trieste D. Y. Shirkov, Dubna A. N. Tavkhelidze, Moscow I. Todorov, Sofia J. Zinn-Justin, Saclay

CONTENTS:

A. Fkidman Introduction to T Quarkonium Physics

397-439

Nguyen van Hiett Spontaneous Compactification of Extra Dimensions in Eleven-Dimensional Quantum Gravity

441-455

P. Gbaneau The Ampere-Neumann Electrodynamics of Metallic Conductors

457-501

AKADEMIE-VERLAG • BERLIN ISSN 0015 - 8208

Fortschr. Phys., Berlin 84 (1986) 7, 397-601

Instructions to Authors 1. Only papers not published and not submitted for publication elsewhere will be accepted. 2. Manuscripts should be submitted in English, with an abstract in English. Two copies are desired. 3. Manuscripts should be no less than 30 and preferably no more than about 100 pages in length. 4. All manuscripts should be typewritten on one side only, double-spaced and with a margin 4 cm wide. Manuscript sheets should be numerated consecutively from "1" onwards. Footnotes should be avoided. 5. The titel of the paper should be followed by the author's name (with first name abbreviated), by the institution and its address from which the manuscript originates. 6. Figures and tables should be restricted to the minimum needed to clarify the text. They should be numbered consecutively and must be referred too in the text and on the margin. Figures and tables should be added to the manuscript on separate, consecutively numerated sheets. The tables should have a headline. Legends of figures should be submitted on a separate sheet. All figures should bear the author's name and number of figure overleaf. Photographs for half-tone reproduction should be in the form of highly glazed prints. Line drawings should be in a form suitable for reproduction. The lettering should be sufficiently large and bold to permit reduction. If requested, original drawings and photographs will be returned to the author upon publication of the paper. 7. Formulae should not be written so small and not with pencil. Separate lines for formulae are desirable. Si-units should be used. Letters in formulae are normally printed in italics, numbers in ordinary upright typeface. Underlining to denote special typefaces should be done in accordance with the following code: Italics: wavy underlined with pencil (only necessary for type written symbols in the text) Boldface italics (vectors): straight forward and wavy underlined Upright letters (all abbreviations like all units (cm, g, ...), all elements and particles (H, He, ..., n, p, ...), elementary mathematical functions like Be, Im, sin, cos, exp,...): black underlined Greek letters: red underlined Boldface Greek letters: red underlined twice Upright Greek letters (symbols of elementary particles): red and black underlined Large letters: underlined with pencil twice Small letters: overlined with pencil twice (This will be necessary for handwritten letters that do not differ in shape, as c C, k K, o 0, p P, s 8, u U, v V, w W, x X, y Y, z Z). I t will help the printer if position of subscripts and superscripts is marked with pencil in the following way: at, M$j, Wn\ Please differentiate between following symbols: a, «; a, a , oo; a, d; c, C, a; e, I; I ?) =

one gets: E =

(y,\H\y>).

By differentiating this equation with respect to p, one obtains 8E

¡8H\

„

d

,

,

,

i and dE _ 8p

18H \ 8p

which is the Feynman-Hellmann theorem. Using expression (2.1), one can write: hW2

8H_8^ dp

2p

8p

=

- - [ H -

V{r)}

from which the expectation value yields, using the theorem just demonstrated, p! - H - < P > ] .

tip

The right hand side is always negative as the average kinetic energy E — (F) is a positive number. Since p = m 0 /2, one has 8E T~ 8viq

< 0

which establishes the property 1. In order to demonstrate the virial theorem (point (2)), we use the relation ih j f ) = (V\ [A, H] \y>) + ih(y>\ M

\v)

(2.2)

where A is any operator in the Hilbert space. This expression is derived using the timedependent Schrodinger equation

g

ifl

8t

=

H

together with its hermitian conjugate

An elementary calculation gives then formula (2.2). For a bound state the derivative jf

{w\ A |y> = 0

402

A . FRIDMAN, Q u a r k o n i u m P h y s i o s

vanishes for any operator which does not depend explicitely on the time Indeed in the case of a bound state, the wave function has the form: ip(r, t)

=

= 0).

Et —i •

exp

p\ A \y>) = f

rV(V) J

ipdh =

0

or finally ^ = < r . P F > which is the virial theorem. For central potential [F(r) = In this case the theorem is simply IP

dV

V{r)}.

One has

rVV

= r

8Vj8r.

(2.4)

This formula allows one to calculate the average of the squared velocity of the bound quark ¡3Q = \p\lmQ, yielding:

403

Fortschr. Phys. 34 (1986) 7

Finally let us derive the expression relating \y{0)\2 with the potential (point 3). To this end one assumes that the potential is central. Then the wave function can be expressed as a product of a radial wave function R{r) and the spherical harmonics Yim{8,