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The Virginia Journal of Science

Volume 4, (New Series), 1953 EDITORIAL BOARD

Boyd Harshbarger, Blacksburg . . . . . . Editor-in-Chief

Horton H. Hobbs, Jr., Charlottesville . .„ . . . . . Technical Editor

Mary E. Humphreys, Staunton . . . . . . . Assistant Technical Editor

Clinton W. Baber . . . . ...' . . . . Advertising Manager

SECTION EDITORS

W. P. Judkins, Blacksburg . . . . . . . . . . . . Agricultural Science

Irving G. Foster, Lexington . . . Astronomy, Mathematics, and Physics

Robert T. Brumfield, Farmville . . . . . . . . Biology

J. Douglas Reid, Richmond . . . . . . . ..... . . . . Bacteriology

Carl J. Likes, Richmond _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ Chemistry

Francis G. Lankford, Jr., Charlottesville . . . . . . . . . Education

Robert M. Hubbard, Charlottesville .............................................. . . . Engineering

W. D. Lowry, Blacksburg . . . . . Geology

Ebbe C. Hoff, Richmond _ _ . _ Medical Sciences

Richard H. Henneman, Charlottesville ................................. _ ...................... Psychology

L. W. Jarman, Richmond _ _ _ _ Science Teachers

Walter A. Hendricks, Bethesda, Maryland _ _ _ Statistics

Published by the Virginia Academy of Science

CONTENTS

The Science of Colloids . . . Ernst A. Hauser 1

The Spectrum of the Harmonic Oscillator . . . E. J. McShane 7

Lipotropic Action of Vitamin Bi2 in the White Rat . . .

. - . . . . . . J. C. F. Forbes and O. Petterson 11

Compressible Viscous Fluids . V. G. Szebehely 14

Experiences of the Multiphasic Screening Program from a Laboratory

Standpoint in Richmond, Virginia . . ., .

. . . W. A. Dorsey and W. J. Williams 19

News and Notes . . . . . J, . . . . . . . . . . . . . . 23

No. 2, April, 1953 (Mailed April 27, 1953)

Fundamental Characteristics of Jet Propulsion Engines...M. J. Zucrow 41

Characters of Systematic Importance in the Family Branchiobdellidae

(Oligochaeta) . . . . . . . Perry C. Holt 57

The Chromosomes of the Polygyrid Snail, Allogona profunda .

. . . Ladley Husted and Paul Randolph Burch 62

Note on Egg-laying of the Four-toed Salamander, Hemidactylium

scutatum (Schlegel), in Eastern Virginia . . . . .

. J . Ollie King Goodwin and John Thornton Wood 65

News and Notes . . . . 67

Program of the Thirty-First Annual Meeting of the Virginia Academy

of Science . . . . . . . . . . . . . . 76

No. 3, July, 1953 (Mailed August 10, 1953)

Recent Experiments on the Ferromagnetic Deflection of Mu-Mesons

and Electrons . . . ... Stephan Berko and Frank L. Hereford 101

Observations on the Life History of Hesperocorixa interrupta (Say)

. . . . . . . . . . . . . Marvin L. Bobb 111

Large Elastic Bending of Cantilevers with Hydrodynamic Loading

. . . . . :. . . Phillip Eisenberg and L. Foger Whicker 116

ii

Si>S. 73 » V 2 1

Virginia Bryophytes Collected by Bernard Mikula _ _ _ _ _

. . . . . . . . . . . . . Paul M. Patterson 125

Electron Microscopic Analysis of Iron Oxide Pigments . . . . .

. . . . . Ralph R. Wright and William G. Dechant 129

dditional Records of the Occurrence of the Freshwater Jellyfish,

Craspedacusta sowerbii, in Virginia . . . . . . . . . . . .

. . . . . ... Horton H. Hobbs, Jr. and Charles H. Page 137

Justus Henry Cline, The Conservationist-Geologist of Virginia _ _

. . . . . . . . . . . Marcellus H. Stow 139

News and Notes . . . . . . . . . . . . 140

No. 4, September, 1953 Proceedings for the Year 1952-1953

Minutes of the Thirty-First Annual Meeting, May 6, 7, 8, 9, 1953 Detailed Table of Contents . . . . - . 162

in

SUBJECT INDEX

Allogona . 82, 63, 64

profunda . 62

ptycophora . 62

townsendiana . 62

Ambystoma Qpacum . 65

Annual Meeting, 31st

Notes . 23

Program . . . 76, 166

Atrichum crispum . . 126

Awards

Jefferson Medal . 164

J. Shelton Horsley ...... 24, 164

Bequest Form . 325

Bdellodrilus illuminatus . 60

Branchiobdella . 59i

Branchiobdellidae . . 57

Cambarincola . 57, 58, 59, 60

chirocephala . 58

macrodonta . 59

Cantilevers, bending . 116

Chamberlainia cyrtophylla . 127

Cirrodrilus . . . 57, 58

Cline, Justus Henry . 139

Colloids . 1-6

Committee Reports

Collegiate Members . 189

Flora . B . 192

Finance and Endowment _ _ 183

James River Project . 190

Journal . 23, 199

Local Arrangements 77, 157, 194

Junior Academy . 184

Long Range Planning . 181

Membership . 190

Place of Meeting . 194

Research . . . .1 . 23, 182

Resolutions . 193

Resource-Use Education . 191

Science Talent Search 187, 195

Science Teaching . 195

Speakers Bureau . 193

Craspedacusta sowerbii . 137

Dichelyma capillaceum . 127

Ditrichirm heteromalla . 126

heteromalla ortho carpa . 126

Drepanocladus aduncus _ 127

Elastic Equation _ 116

Electrons . 101

Elodea . Ill

Ferromagnetic Deflection . 101

Financial Statement

Academy . . 17^

Journal . 181, 200

Fluids, viscous . 14-18

Fontinalis novae-angliae . 127

Frullania Kunzei ...... . 125

F unaria flavicans . . . 127

Grant-in-Aid . . . 24

Harmonics . . 7-10

Helodium paludosum . 127

H emidactylium scutatum . 65-66

Hesperocorixa

interrupt a . >111-115

nitida . . Ill, 112

Histomonas meleagridis . . . 71

Institute for Scientific

Research . 198

Iron Oxide . 129

Jellyfish, freshwater . 137

Jet Engines . 43

Ramjet . 44

Pulse jet . 46

Turbojet . 47

Jet Propulsion . 41

Level flight . 41

Definition . 4-y

Laboratory Tests, program ... 19-22

Leskea arenicola . 127

Leucolejeunea conchifolia . 125

Lipotropic action . 11-13

Manuscripts, Suggestions for .

. . . . . inside back cover

Membership Application . 325

Membership List

Senior . .'. . 294

Student . 322

Mesodon . 62, 63

appressns . 63

rugeli . 63

Minutes

Academy Conference . 169

Annual Meeting . 202

Council . 141, 143, 168, 204

Junior Academy . 207

IV

Junior Academy Meeting . 207

Sections ............................................ 209

Multiphasic Screening . . 19, 22

Mu-mesons . V....7L...:- 101

News and Notes . . 23, 67, 140

Officers and Committeemen . 157

Oscillator, harmonic . 7-10

Oxide pigments . . . .C 129

Palmocorixa buenoi . Ill

gillettei . Ill

nana . Ill

Philonotis capillaris . . . 127

Polytrickum commune perigoniale

. . . I . . . 126

Presidents, List . 156

President’s Message .....BL . 140

Proceedings, 1952-1953 . 155

Table of Contents . 162

Pterodrilus . . 57, 58, 59, 60

alcicornis . 59

Rat . 11-13

Rhytidium rugosum . 127

Rocket Engine . 51

Salamander, egg-laying . 65-66

Secretary Treasurer Report . 171

Sections

Agricultural . 25, 81, 209 *

Astronomy, Mathematics

and Physics . 26, 82, 215

Bacteriology . 67, 71, 84, 225

Biology . 27, 67, 71, 85, 228

Chemistry . 28, 87, 145, 233

Education . 35, 90, 244

Engineering . 91, 147, 250

Geology . 92, 25$

Medical Sciences . 95, 269

Psychology . 37, 96, 151, 276

* Italicized numbers indicate abstracts.

Science Teachers . 1. . 97, 283

Statistics 37, 69, 98, 151, 286

Sheep, epidermis . . . :..... . . 71

Sigara signata .L.:...........:. . : . 112

•- rnodesta .1..,.,,.,,.....„i....„..........^......... 112

alternata . . 112

Sphagnum capillaceum . . 125

compactum . 125

cuspidatum serrulatum . 125

cuspidatum torreyi .. . 125

cyclophyllum . 125

henry ense . 126

imbricatum . 126

imbricatum affine . 126

magellanicum . 126

palustre . 126

recurvum . 126

strictum . 126

subsecundum . 126

tenerum . 126

Stenotrema . . . 62, 63, 64

Siephanodrilus . 58

Thuidium Allenii . 127

Tortula fragilis . 126

Trichocorixa calva . Ill

macroceps . Ill

Triodopsis . 62, 63, 64

appressus . 63

fraudulenta . 63, 64

tridentata edentilabris . 63

Vitamin Bi2 . 11-13

Viscous fluids . . 14-18

Xironodrilus . 57, 58, 59, 60

pulcherrimus . 60

Xironogiton . ; . 57, 58, 59, 60

occidentals . 60

AUTHOR INDEX

Berko, Stephan . . . . 101

Bobb, Marvin . . . . . . Ill

Burch, Paul Randolph . . . 62

Dechant, William G. .......... _ _ _ 101

Dorsey, W. A. . . . . . 19

Eisenberg, Phillip ............. _ _ _ _ 116

Forbes, J. C. F. . . . . 11

Goodwin, Ollie King . 65

Hauser, Ernst A - - - - 1

Hereford, Frank L. |.....| . . 101

Hobbs, Horton H., Jr. . 137

Holt, Perry C . . . . . 57

Husted, Ladley . . . . . . . . 62

McShane, E. J . 7

v

Page, Charles H .

Patterson, Paul M .

Petterson, O . . .

Stow, Marcellus H. .... _ _

Szebehely, V. G _ _

. 137 Whicker, L. Folger . 116

. 125 Williams, W. J _ _ _ 19

. . 11 Wood, John Thornton . 65

... _ 139 Wright, Ralph R. _ _ _ _ 129

_ 14 Zucrow, M. J _ _ _ _ _ 41

vi

THE VIRGINIA OITRNAL OF SCIENCE

A JOURNAL ISSUED QUARTERLY BY THE VIRGINIA ACADEMY OF SCIENCE

Vol. 4

No. 1

January, 1953

THE VIRGINIA JOURNAL OF SCIENCE

Published Four Times a Year: In January, April, July, and September, by The Virginia Academy of Science Printed by The Giles County Virginian , Pearisburg, Va.

CONTENTS

Pages

The Science of Colloids Ernst A. Hauser . . . . . . . . 1

The Spectrum of the Harmonic Oscillator E. J. McShane....... . . 7

Lipotropic Action of Vitamin in the White Rat—

J. C. F Forbes and O. Petterson. . . . . . . . II

Compressible Viscous Fluids— V. G. Szebehely. _ _ _ _ _ _ _ 14

Experiences of the Multiphasic Screening Program from a Laboratory Standpoint in Richmond, Virginia W. A. Dorsey and

W. J. Williams.. _ .... ... . .. . 19

News and Notes . . . . . . . . : . . . . . . 23

EDITORIAL BOARD Boyd Harshbarger, Editor-in-Chief Horton H. Hobbs, Jr., Technical Editor Mary E. Humphreys, Assistant Technical Editor Clinton W. Baber, Advertising Manager

Section Editors

W. P. Judkins Irving G. Foster

Robert T. Brumfield Carl J. Likes N. F. Murphy B. N. Cooper

Richard H. Henneman L. W. Jarman

J. Douglas Reid Francis G. Lankford, Jr. William Bickers Walter Hendricks

Entered as second-class matter January 15, 1950, at the post office at Blacksburg , Virginia, under the Act of March 3, 1879. Subscription $3 per volume . Published at Blacksburg , Va.

Mailed March 24, 1953

THE VIRGINIA JOURNAL OF SCIENCE

Vol. 4, New Series January, 1953 No. 1

The Science of Colloids

Ernst A. Hauser

Massachusetts Institute of Technology Sir William Dampier Whetham made the following statement in 1932:

“To the scholar, the man of science sometimes seems to be busy about little facts and trivial problems in an entirely superfi¬ cial way. On the other hand, to the philosopher or the man of science, if they ignore the underlying verities and look only to little interpretations, it seems that, as Hume said, 'Popular theo¬ logy has a positive appetite for absurdity/ . . . Here again the historical method enables us to get beneath surface trivialities, see the deepest secrets of nature that may lie hid in the move¬ ment of the needle of a galvanometer or the marking on a butter¬ fly’s wing and trace the groping of man’s soul . . . And so physi¬ cal science continually widens our knowledge of the phenomena of the natural world and of the relations between the concepts, final or proximate, that we use to interpret the phenomena . . .

As Newton said, ‘The business of natural philosophy is to argue from phenomena . . . and to deduce causes from effects till we come to the veiy first cause, which certainly is not mechanical’.”

Exactly twenty years before Sir Whetham made his statement, the Ger¬ man scientist, Wilhelm Ostwald, made the following remarks:

“Natural science and natural philosophy are not two fields which oppose each other, but they belong together just as two paths which lead to the same goal. This goal is: The mastery of nature by the human being. This is accomplished by the various natural sciences, all of which collate the actual circum¬ stances between natural phenomena and attempt to discover if and to what extent they depend on each other. This is done in order to be able to predict one phenomenon from what is known of another with more or less accuracy. Natural philosophy joins in such attempts, but only in a more general way . . . Every rep¬ resentation of science has its natural philosophical component.

In specific articles in which scientists report their progress, the natural philosophical components are generally present in the form of assumptions, of sentences which are not discussed, fre¬ quently not even referred to specifically, but on whose accept¬ ance the most important deductions are based. In all such cases these important sentences can never be found in the right places.

If they are contained in the introduction of the textbook they are of little value because the facts they actually should summarize are developed only later in the book. If they are contained in

Editor’s Note : We are pleased to present this invited article by Mr. Ernst A. Hauser, Professor of Colloid Chemistry at the Massachusetts Institute of Technology, and Visiting- Professor of Colloid Chemistry at the Worcester Polytechnic Institute in Worcester. Mass,

2 The Virginia Journal of Science [January

the summary at the end they are too late because they have been previously used already in many instances without, however, referring to their general importance. The best thing to do is to permit generalization only in such measure as the given indivi¬ dual facts call for and justify.

“Therefore any education in natural sciences is interpenetrated by natural philosophy. Just as one can obtain a survey over a complicated structure, as for example the confusion of streets in a large city, only if one has all information about the individual streets and also knows the relation of one to another after having studied a general city plan, it is advisable when studying a spe¬ cial branch of science to look at the general plan so that one does not become confused whenever one’s path leads through an unfamiliar section.”

Michael Faraday’s life history is one of the most striking examples of Ostwald’s and Whetham’s reasoning. When Faraday reached the age of thirteen he had to start earning his own living. In 1805 he became an apprentice bookbinder, and although he did not have an opportunity to attend an advanced school, he loved to read the scientific books which he had to bind. Occasionally his superior gave him permission to attend some evening lectures on science, and in 1812 a customer of the shop who had heard of Faraday’s interest in science gave him a ticket for a course of four lectures on chemistry offered by Humphry Davy. On March 1, 1813, Faraday entered the employment of the Royal Institution as Davy’s assistant. Shortly thereafter Faraday accompanied Davy on an extended trip to Europe where he became acquainted with several leaders in the field of electrical problems. During the years 1831 to 1840 Faraday devoted most of his time to research in electro-magnetic problems. At the age of forty he laid the foundation for what we today term a transformer and for what we consider as a modern dynamo.

On February 5, 1857, Michael Faraday presented the Bakerian Lec¬ ture. He had chosen as his title, “Experimental Relations of Gold (and Other Metals) to Light.” In this lecture he made the following state¬ ments :

“Agents competent to reduce gold from its solution are very numerous, and may be applied in many different ways, leaving it either in films, or in an excessively subdivided condition. Phos¬ phorus is a very favourable agent when the latter object is in view. If a piece of this substance be placed under the surface of a moderately strong solution of chloride of gold, the reduced metal adheres to the phosphorus, as a granular crystalline crust. If the solution be weak and the phosphorus clean, part of the gold is reduced in exceedingly fine particles, which becoming diffused, produce a beautiful ruby fluid. . . .

“Fluids thus prepared may differ much in appearance. Those from the basins, or from the stronger solutions of gold, are often evidently turbid, looking brown or violet in different lights. Those prepared with weaker solutions and in bottles, are fre¬ quently more amethystine or ruby in colour and apparently clear.

The latter, when in their finest state, often remain unchanged

1953]

Science of Colloids

3

for many months, and have all the appearance of solutions. But they never are such, containing in fact no dissolved, but only diffused gold. The particles are easily rendered evident, by gathering the rays of the sun (or a lamp) into a cone by a lens, and sending the part of the cone near the focus into the fluid; the cone becomes visible, and though the illuminated particles cannot be distinguished because of their minuteness, yet the light they reflect is golden in character, and seen to be abundant in proportion to the quantity of solid gold present. Portions of fluid so dilute as to show no trace of gold, by colour or appear¬ ance, can have the presence of the diffused solid particles render¬ ed evident by the sun in this way.”

Michael Faraday offered further evidence for his conclusion, but unfor¬ tunately his point of view was not accepted.

It cannot be denied that Faraday, on the basis of pure logical rea¬ soning, made most important contributions to the science of colloids. It was Thomas Graham, however, who laid the scientific foundation of colloid chemistry. Graham published two particular papers, the first of which must be regarded as the actual cornerstone of systematic colloidal research. He paid no attention to the argument which started with Faraday’s disclosure because he was primarily interested in studying the diffusive power of liquids and in establishing differences comparable to various degrees of volatility. The respective passage of Graham’s paper reads as follows:

“The comparatively fixed class, as regards diffusion, is repre¬ sented by a different order of chemical substances, marked out by the absence of the power to crystallize, which are slow in the extreme. Among the latter are hydrated silicic acid, hydrated alumina and other metallic peroxides of the aluminous class, when they exist in the soluble form; with starch, dextrin, and the gums, caramel, tannin, albumen, gelatin, vegetable and ani¬ mal extractive matters. Low diffusibility is not the only property which the bodies last enumerated possess in common. They are distinguished by the gelatinous character of their hydrates. Al¬ though often largely soluble in water, they are held in solution by a most feeble force. They appear singularly inert in the capacity of the acids and bases, and in all the ordinary chemical relations. But, on the other hand, their peculiar physical aggre¬ gation with the chemical indifference referred to appears to be required in substances that can intervene in the organic processes of life. The plastic elements of the animal body are found in this class. As gelatine appears to be its type, it is proposed to designate substances of the class as 'colloids,’ and to speak of their peculiar form of aggregation as the 'colloidal condition of matter.’ Opposed to the colloidal is the crystalline condition. Substances affecting the latter form will be classed as ‘crystal¬ loids’.”

Graham also pointed out that a dominating quality of colloids is the tendency of their particles to adhere, aggregate and contract. If it is

4

The Virginia Journal of Science

[January

borne in mind that the colloidal state of matter is the result of a peculiar attraction and aggregation of molecules, it is not surprising that colloidal characteristics spread on both sides into the liquid and solid condition. Unfortunately, however, Graham did not continue his research and paid no attention to the possibility that gold sols might exist.

About thirty years after Graham had published his basic contribution a few scientists claimed that colloids are true solutions and that only a few insoluble particles are present. In contrast thereto, a few other scientists, particularly Wilhelm Ostwald, claimed that this statement was not correct but that one is actually dealing with matter in an extremely fine degree of dispersion. Richard Zsigmondy was one of those who origi¬ nally had voiced serious objections to Faraday’s statements. He claimed that the so-called “heterogeneous” theory of colloidal solutions was incor¬ rect. Zsigmondy, however, decided to try to prove or disprove Faraday’s hypothesis. He figured that if he could place a solution of colloidal gold sol on the microscope table and then focus the sunlight at a right angle onto the container in which he had placed the sol, he might be able to discover whether or not such particles actually existed. In the publica¬ tion in which he discussed his experiment and in which he refuted the solution theory, he stated:

“How entirely erroneous was my idea! A swarm of dancing gnats in a sunbeam will give one an idea of the motion of the gold particles in the solution. This motion gives an indication of the. continuous mixing up of the fluid, and it lasts hours, weeks, months, and if the fluid is stable, even years.”

Immediately thereafter Zsigmondy contacted H. Siedentopf, who was the scientific adviser of the Garl Zeiss optical works in Jena, and they developed what still today is known as the “slit ultramicroscope.”

Not only did this invention increase the range of visibility from about 500 millimicrons down to dimensions of only a few millimicrons, but it also established the following facts of general importance to natural sci¬ ence:

1. That colloidal solutions must be classified as heterogeneous systems insofar as their noncoherent particles, microscopically invisible but detectable in the ultramicroscope, are suspended in a medium. The size of these particles is below TO"4 cm. and above 10"7 cm.

2. That since the Brownian motion of the ultramicroscopically visible particles can be considered as a thermal movement fol¬ lowing the same laws as deductible from the kinetic-gas theory, ultramicroscopy must be accepted as the first actual experimental proof that the kinetic-gas theory with all its conclusions stands to fact.

3. That experimental proof could be offered for the energy- distribution law of Boltzmann-Maxwell.

Fairly soon after the slit ultramicroscope had been developed special condensers were built which made it possible to use the standard type of microscope to determine the presence or absence of colloidal particles.

1953]

Science of Colloids

o

If the preparation under investigation is composed of nonspherical parti¬ cles, however, it will be difficult to determine their actual shape. The slit ultramicroscope made this possible because the light enters the prep¬ aration only from the side. Nonspherical particles will therefore twinkle, in contrast to spherical particles which would show no change from the point of view of illumination.

Richard Zsigmondy deserves special credit not only for his invention of the slit ultramicroscope but also for having admitted to himself that his original idea was not based on facts.

The science of colloids depends to a great extent on information and experience gained from other fields of science than chemistry. For exam¬ ple, we owe most of our knowledge of the structure of matter to the X-ray diffraction technique. It therefore seems only fitting to explain what actually led Wilhelm Roentgen to make his great discovery.

In 1895 while working in Wurzburg, Roentgen noticed that a barium platinocyanide screen which happened to be lying close to a highly ex¬ hausted vacuum tube which was connected with the electric current ex¬ hibited pronounced fluorescence. Further investigation showed him that the radiation had the power of passing through various substances which are opaque to ordinary light. He also found that this radiation, which he later termed “X rays”, would also cause the darkening of an unexposed photographic plate. For this discovery he received the Rumford Medal of the Royal Society in London in 1896. In 1901 Roentgen also received the Nobel Prize for Physics. His discovery very soon found its applica¬ tion in physics, particularly in evaluating the molecular structure of mat¬ ter. We owe the most important contributions in this respect to von Lane, and shortly thereafter, to Debye and Scherrer.

One of the greatest contributions in the field of X-ray studies was in studying the structure of rubber. It could readily be shown that if un¬ stretched rubber was subjected to X-ray diffraction, only a simple amor¬ phous band was visible. Upon stretching, however, it became evident that the rubber showed a pronounced alignment in the direction of stretch. Soon thereafter it was ascertained that rubber which has been subjected to freezing would show unoriented zones of “crystalline” structure. It was also possible to prove that if rubber was subjected to very fast expan¬ sion, it no longer gave any indication of what was known as an amorphous band. On the basis of this work the so-called multi-phase theory of rub¬ ber has been worked out. This theory has demonstrated the function of the plastic phase in rubber. The “plasticizer” is the low molecular weight fraction. In its absence the rubber (natural or synthetic) loses most of its elastic properties.

The science of colloids has also made it possible to develop entirely new techniques for the production of rubber articles. We owe this develop¬ ment entirely to what we have learned about the colloidal condition of rubber latex. It should also not be overlooked that we owe our knowl¬ edge of how to produce synthetic rubber emulsions to what the science of colloids has taught us about the natural rubber milk sap.

Ore flotation is another field of applied colloid science. We deal here with ground ore containing nonmetallic ingredients. Depending on the affinity of the ore to certain chemicals which are added to the slurry, it is readily possible to separate the ore from the worthless matter.

6 The Virginia Journal of Science [January

In another field, systematic research has revealed that the enzyme hyal- nronidase is quite active in the prevention of kidney stones. In the absence of a sufficient quantity of hyaluronidase, stone formation will be accelerated by the growth of crystals or by the agglomeration of the dispersed phase of inorganic colloids. This fact also offers excellent proof that many colloids owe their stability mainly to the electric charge they carry on their surface. Those colloids which are called “hydrophilic” owe their stability primarily to their affinity to water. They will therefore react quite differently from the so-called “hydrophobic,” water-hating, col¬ loids. All work carried out along these lines has demonstrated the need of most accurate studies of the electro-kinetic potential carried by colloidal matter.

One could continue almost indefinitely to offer further evidence of the importance of colloid science and why its history is so important. There is almost no modern industry which can afford to disregard the science of colloids. Just to mention a few, the following fields might be referred to: the soap industry; the production of natural and synthetic fibers; the natural and synthetic rubber industry; the entire field of antibiotics; the ceramic industry, where the condition of the clays depends primarily on their surface properties; the ink industry; the brewing of beer; and the colloid-chemical properties of all photographic processes.

Example after example could be offered to give a more detailed dis¬ cussion of what colloid science actually stands for. It must not be over¬ looked, however, that it includes far more than just chemistry alone. A basic knowledge of physics, too, for example, is essential for really valu¬ able contributions in these days. The modern colloid scientist needs a well-founded knowledge of both chemistry and physics.

Men working in other fields— in medicine, geology, biology, food tech¬ nology and many other branches of science also need at least a smat¬ tering of colloid science. Without this knowledge they cannot ascertain if, and to what extent, colloidal phenomena are involved in their problems and how they could be handled. The science of colloids is more far- reaching than most people realize, and from now on more cognizance should be taken of this undeniable fact.

1953]

Spectrum of Harmonic Oscillator

The Spectrum of the Harmonic Oscillator

E. J. McShane University of Virginia

In any text on quantum mechanics it is proved that with proper choice of units, the simple harmonic oscillator has eigenvalues This is usually

done by means of solving a boundary value problem. But this implies using a special way of representing states and observables. Closer to the heart of the system are the simple algebraic relations between operators, in particular the commutation rules; even if the scheme of representing states must be compli¬ cated, for example to obtain relativistic invariance, the algebra will survive. P. A. M. Dirac showed, in his famous book [Principles of Quantum Mechanics, Oxford Press, 1947] that if we assume that the Hamiltonian operator for the oscillator has a pure point spectrum, it can be proved from the algebraic relations alone that the spectrum consists of J, f , . . . . In this note we prove from the algebraic relations that the spectrum must necessarily be a pure point spectrum, consisting of exactly the points just mentioned.

From the theory of quantum mechanics we choose a few central concepts. We start with a Hilbert space S; usually this is taken to be the set of functions in the coordinate-space which have integrable squares, but to us this is at present irrelevant. The inner product in S is denoted by parentheses; thus (/,<?) is the inner product of / and g, which in the usual interpretation would be the integral of the product of the function / with the conjugate of g. To each physical observable corresponds a self-adjoint operator on S; to each state of a system corresponds a unit vector in *S; and when the system is in the state corresponding to the unit vector \J/, and the observable corresponding to operator B is observed, the expected value of the observable is the value of the inner product (B\p,\l/).

In a one-dimensional system consisting of an object elastically attracted to the origin, let p denote the momentum and q the coordinate of the object. If the units are properly chosen, the total energy is (p2 + q)/ 2. Let P and Q be the self-adjoint operators corresponding to the respective observables p and q; the Hamiltonian operator, corresponding to the total energy, is then

(1) H = (P2 + Q2)/ 2.

We do not need to know the special form of operators P and Q. But we do need the facts that for every <p to which we can apply H we can also apply P, Q, PQ and QP , and that (with proper units) we then have

(2) PQ‘-p QP<p = —i(p-

Since P, Q, and H are self-adjoint, for each and xf/ in the domain of H we have (P<p,\p) = ((p,Pxf/), and so on. But since the inner product has the property {i<P,\f) =■ i((p,xf) = {(p,—i\p), we also have

(3) ([P + iQ](p,\J/) = (<p,[P ~ iQ]'!')-

From (1) and (2) we readily compute that for all <p and in the domain of H,

(4) [P - iQ][P + iQ]<p - [2 H + 1 ]<p,

8

The Virginia Journal of Science

[January

(5) [P + iQ][P ~ iQ]+ = 12// - 1]*.

In (5) we choose \p = [P + iQ\<p; with the help of (4), (5) becomes (0) [P + iQ}[2H + l]<p = [2H - 1][P + iQb.

Concerning the spectral resolution of H we need the following well-known facts.1 To each real number u corresponds a projection operator Eu (in another, but not better, terminology Eu<p would be the vector obtained by expanding ip in eigenfunctions and eigendifferentials of H and retaining only that part of the expansion corresponding to characteristic numbers ^u) with the properties

(7) Eu increases with u and is right continuous', as u tends to oo f Eu tends to 1 , and as u tends to oo f Eu tends to 0.

(8) If a S b, then EaEb = EhEa Ea .

(9) J du(Eu<p,\p) = ((p,I) for all <p and f in S.

(Hb For all finite a and b, and all ip, (Eb E„)<p is in the domain of II , and also in the domains of H2, H \ etc.

(11) For all ip in the domain of H and all yp in S, f udu(Evip, yp) = (Hep, yp) .

J —co

Let ip be any vector, and let ip’ (Eh Ea)ip. By (8), we have

Eucpr = Euip Euip = 0 if u S a,

= Eu(p Ea(p = Eb<p Ea(p

if a < u ^ b,

if u > b.

Hence if we replace <p by ip' in the integrals in (9) and (11), on the range of integration oo < u S a the product ( Eutp',\p ) is constant, so this contributes nothing to the integral, and likewise for b < u < oo , so we have

(12)

(13)

J* dJEuv,0 = ([Eb- E.M),

J udu(Eucp,if) = (H[Eb Ea]<p,>f)

If we replace a, b, u by c, d, v, reverse the order in each inner product (which changes it to its complex conjugate) and then interchange letters ip, \ p, these become

(14) dv(ip,Evyp) = (<p,[Ed - Ec]yp),

^See, for example, J. von Neuman Mathematische Grundlapen der Quantenmechanik, Dover Publications, 1943.

1953]

Spectrum of Harmonic Oscillator

9

(15) j vdv(<p,Evif) = ((p,H[Ed E,. ] \p) .

J c

Let <p, $ be any vectors in S, and let R\ a < u ^ b, c < v ^ d lie any rec¬ tangle in the plane. For brevity we write

(1® / = (Ei ; Ea)<p, # (Ed E,.)\p.

Then by (3), (10), (12), (13) and (14) we obtain

(17) ([ P + iQ][2H + l]/,p) i ([2// + 1 ]/, |H - /(?]c/)

= [ (2m + IW„(P„.p,[P - »Q]g)

(2w + l)du([P + f (J ]£/<£>, p)

= £ £ (2m + 1Rd.([P +

Similarly from (3), (10), (14), (15) and (12)

(18) ( [2/7 - I HP + iQ]f,g) = / (2v - \)d,([P + iQ]f,li,t)

= j (2/i - IK(/,[P - iQW.i)

= I [ \2v - JKd,(A>,[P - iQ]E,+).

J a J c

We again apply (3) to the last, and subtract (18) from (17) member by member; by (6) we obtain

(.19) 0 = 2 j £(«-«> + l)dJ,([P + iQ\E&,E.t).

Now let us introduce a factor f(u, v) under the integral sign in (10), producing a symbol

(20)

f f f(M,v)(u

v 1 )dudv([E + iQ]Eu<p,Ee\l/) .

If / is a constant, this is 0 by (10). If/ is a step-function, meaning that R can be cut into finitely many sub-rectangles on each of which / is constant, then by applying (10) to each small rectangle we find that the integral over each is 0, so when we combine them we find that (20) is still 0. If f(u,v) is continuous in R and on its boundary, it can be uniformly approximated by a sequence of step- functions Ji ,jfp , . For each/w the integral (20) (with/B in place of/) vanishes,

so the limit (20) also is 0. Now suppose that u v -j- 1 does not vanish in R or on its boundary. Then we can choose f(u,v) 1/ (u v + 1), and (20) will still vanish. But now the integration is easy; we obtain

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The Virginia Journal of Science

[January

(21) (\P + iQ][Eb - E, ]v, [E„ - EM) = 0

if the rectangle a S u ^ b, c ^ v ^ d does not meet the line u v -j- 1 = 0. By (3) this can be written

&,[E> - E.][P -