I
Boris Musulin
Southern Illinois University Carbondale
O~eratorFormalism in Dimensional dnalysis I
-~
Traditionally, dimensional analysis is introduced in many physics and engineering courses and in some chemistry courses in order to familiarize the student with techniques for converting units and/or for avoiding errors in physical equations. The usual approach is similar to the approach developed by Bridgman.' However, we have used another method of teaching this subject which has a third pedagogical advantage. This method has been used for six years a t Southern Illinois University a t the graduate level, the senior level, and the freshman level. It has also been used, in abbreviated form, in a terminal liberal arts course. The purpose of this paper is to briefly outline the method in order that others may use. it The term dimension is used in modern terminology2 to signify the exponents of the units attached to a physical magnitude. By convention the term is restricted to whatever fundamental units have been adopted as a basis set. We adopt these conventions. Further, we create an operator which we symbolize by ',dim
The fourth property shows that the operator d i m does differ from the algebra of usual numbers. Further, the concept of dimensional homogeneity is easily introduced a t this point. Of course, consistent use of units also follows naturally. The Mth property brings in the concept of dimensionless constants as opposed to those which depend on units, since any unit raised to the zero power is simply the pure number, one. Finally, in advanced courses, this is an excellent opportunity to point out that methods which students have learned in advanced mathematics courses, such as the establishment of different sets of postulates for groups, rings, etc., are pertinent to their study of physical sciences. After establishing the fundamental units in mechanics, electricity, etc., the algebra of operators is demonstrated. For example, What are the dimensions of energy? dim E
=
dim ( m l )
=
(dim m)(dim l)(dim l)/(dim 1') mp-9
=
7,
The use of operators is an abstraction which many students find difficult. Yet they are extremely important to teaching quantum mechanics and quantum chemistry. With the increased emphasis on modern physical o hem is try,^ more students will have to master the operator concept. We have found that the early use of the term breeds the sort of familiaritv which causes a student to believe he knows the subject. First, the student is taught the algebra of the new operator. Briefly, these are 1. Multiplication dim (ab)
=
(dim a)(dim b)
dim (a/b)
=
(dim a)/(dim b)
2. Division
3. Exponentiation
Another example, which illustrates use of units not commonly associated with particular physical ,,,agnitude, is given by the following: Show that energy may he expressed in reciprocal centimeters, From the formula E
=
hell
E m l/h dim E = 1-1
The usual tie-in to dimensioned constants is then made. Second, use of the operator dim can he used to "derive" some fundamental relations. For example, the student finds in studying electromagnetic radiation that dim (hv)
=
dim v
From this relation,
dim (a')
=
(dim a)b
4. Addition and subtraction dim (a
+ b)
=
dim a
=
dim b
5. Establishment of the identity dim a
=
0
BRIDGMAN, P. W., "Dimensional Analysis," Rev. ed., Yale Univ. Press, New Haven, Conn., 1931. FOCKEN, C. M., "Dimensional Methods and Their Applications," Edward Arnold & Co., London, 1953. a ACS Committee an Professional Training, "Report," 1963.
622
/ Journal of Chemical Education
The ensuing discussion fixes the value of k as well as the value of u. I t is a t this point the student is taught the relation of a physical equation to experimental measurements and is shown the limitation of the mathematical formalism of dimensional analysis. The operator approach is a useful alternate to the rigorous development of functional relations from the I1 Theorem.?
