Dimensional analysis as the other language of physicsR. W. Robinett Citation: American Journal of Physics 83, 353 (2015); doi: 10.1119/1.4902882 View online: http://dx.doi.org/10.1119/1.4902882 View Table of Contents: http://scitation.aip.org/content/aapt/journal/ajp/83/4?ver=pdfcov Published by the American Association of Physics Teachers Articles you may be interested in ACER: A framework on the use of mathematics in upper-division physics AIP Conf. Proc. 1513, 90 (2013); 10.1063/1.4789659 Online University Physics Lectures from MIT indexed for AP Physics instructors Phys. Teach. 46, 253 (2008); 10.1119/1.2895685 Computation in undergraduate physics: The Lawrence approach Am. J. Phys. 76, 321 (2008); 10.1119/1.2834739 Avatars of Hollywood in Physical Science Phys. Teach. 44, 28 (2006); 10.1119/1.2150756 Gestures as evidence of student thinking about physics AIP Conf. Proc. 720, 61 (2004); 10.1063/1.1807254

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Dimensional analysis as the other language of physics

R. W. Robinetta)

Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802

(Received 24 June 2012; accepted 14 November 2014)

We review the use of dimensional analysis as a tool for the systematic study and analysis of

physical concepts and phenomena at multiple levels in the physics curriculum. After reviewing the

methodology of its use and citing examples from classical physics, we illustrate how it can be

applied to problems in quantum mechanics, including research-level problems, noting both its

power and its limitations. VC 2015 American Association of Physics Teachers.

[http://dx.doi.org/10.1119/1.4902882]

I. INTRODUCTION

Physics majors are expected to have a diverse skill set or“tool box” after completing an undergraduate degree pro-gram. Their experiences generally include gaining a broadphysical intuition, being able to perform “back-of-the-enve-lope” numerical calculations, having experimental skillsacross different technical areas (optics, electronics, etc.),being adept at the use of graphical and statistical data analy-sis and visualization tools, and possessing substantial pro-gramming abilities. But above all, students majoring inphysics must typically meet the most formal requirements interms of mathematics coursework of any program, asidefrom mathematics itself.

The central role played by advanced mathematics inphysics is perhaps most famously described in a quote byEugene Wigner:1

“The miracle of the appropriateness of thelanguage of mathematics for the formulation of thelaws of physics is a wonderful gift which weneither understand nor deserve. We should begrateful for it and hope that it will remain valid infuture research and that it will extend, for better orfor worse, to our pleasure, even though perhapsalso to our bafflement, to wide branches oflearning.”

On the other hand, abstract, advanced, or very specializedmathematical techniques can just as often confuse, confound,and even discourage students at many stages of their physicscareers. Furthermore, such methods are often well beyondthe level of students early in their physics education, andmay never be amenable for use in a “concepts”-type courseor in service classes for non-physics majors.

It can be useful, then, to have other unifying descriptionsor themes that cut across all subfields of physics, ones thatstill allow students to explore, in a serious and nontrivialway, both classical and more contemporary physics topics,but with a minimal amount of abstract mathematical machin-ery. We will argue here that dimensional analysis can besuch a powerful tool.

Dimensional analysis should be familiar, in one sense, toall students in physics, even if only from the admonition to“make sure that the units work in your final answer” or simi-lar a posteriori checks. We have in mind here, however, amore proactive and a priori methodology to confront (if notsolve completely) many problems, by asking “up front” ifone can extract the necessary functional dependence of someintermediate or final answer on the physical parameters

relevant to the problem. Having identified such variables, itcan be the case that the physical dimensions of the desiredfinal result can, in fact, completely determine (or at leastheavily constrain) the dependence on those values, either asbeing rational powers of combinations of the parametersand/or as functions of dimensionless ratios. This type ofapproach can be useful to physics students at every stage oftheir careers, but seems to be increasingly unfamiliar tomany.

Dimensional analysis in this form is familiar in engineer-ing, and many of the standard textbooks on the subject2 havesuch an audience in mind. Even when this topic is discussedin Physics Today, the emphasis can still be on more appliedareas.3 One intent of this review is to encourage the use ofsuch methods in the undergraduate physics curriculum, at alllevels of instruction, and especially beyond standard prob-lems in classical physics, in part by providing a number ofuseful examples.

More fundamentally (and perhaps more metaphorically),we want to stress the fact that dimensional analysis can beused, almost literally, as another language to describe physi-cal phenomena. The fact that almost all physical quantitieshave dimensions that can be written in terms of rationalpowers of a handful (quite literally, five) of basic dimensionsis a powerful unifying theme in physics. We will explorehow a basic “alphabet” of dimensions, including length (L),time (T), mass (M), charge (Q), and temperature (H), canbe used to “encode” the description of the vast majority ofphysical quantities that students will encounter. Thesedimensional quantities mesh well with five of the sevenMKSA-Kelvin units used in the current International Systemof Units (SI); the other two SI units, a measure of quantity(Avogadro’s number) and a unit of luminous intensity(candela), are used far less frequently in problems arising inphysics and so will not be discussed here.4

In Sec. II, we provide a lightning review of introductoryphysics and the “alphabet” of dimensional constructs neces-sary for describing most physical quantities. Along the way,we will also collect some fundamental constants of nature,which we will think of as “basic words” that appear to beessential constructs. We focus especially on the constants c,e, �h, and kB, as the SI will begin to make use of them as basequantities in “A more fundamental International System ofUnits” starting in 2018.5

In Sec. III, we then review the standard methods of dimen-sional analysis as used to predict the dependences (or con-straints) on physical systems in an a priori way. Then, inSec. IV, we provide a number of more modern examples ofthe use of dimensional analysis as applied to problems in

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quantum physics, even approaching recent research prob-lems. We conclude in Sec. V with some general comments.

II. REVIEW OF DIMENSIONAL ANALYSIS:

ESTABLISHING THE “ALPHABET” AND SOME

UNIFYING “WORDS”

In a typical introductory physics text (calculus-based ornot), one usually encounters classical mechanics first, begin-ning with the study of one-dimensional kinematics. In thiscontext, we need only two dimensional quantities, length andtime, which we will denote by L and T. We will use the(seemingly) standard notation that [v] is to be interpreted (orread) as “the dimensions of v,” so that we have

x tð Þ½ � ¼ L; v tð Þ½ � ¼ L

T; and a tð Þ½ � ¼ L

T2; (1)

for position, velocity, and acceleration, respectively. Whilewe might advocate the use of specific units such as metersand seconds in most problem-based applications, one could,in principle, just as logically use furlongs and fortnights.

Extending classical mechanics to include Newton’s laws,we need the concept of inertial mass, with a new dimension(M), where some familiar dimensionful quantities are givenby

F½ � ¼ ML

T2; E½ � ¼ KE½ � ¼ PE½ � ¼ W½ � ¼ ML2

T2;

P½ � ¼ M

LT2; and q½ � ¼

M

L3; (2)

for force, energy (kinetic, potential, etc., including work),pressure, and density, respectively. Much of classicalmechanics, fluid mechanics, many engineering applications,and even the formalism of quantum mechanics require onlythese three dimensions, L, T, and M.

It is often in the context of classical mechanics that stu-dents encounter one of the first true “unifications” in physics,namely, the law of universal gravitation

FG ¼Gm1m2

r2; (3)

with the same force applying to apples and planets, and withNewton’s constant G as a truly fundamental constant of na-ture. For future reference, we note that

G½ � ¼ L3

MT2: (4)

We also note in passing that the principle of equivalencestates that inertial and gravitational masses are one and thesame, so the same basic M dimensionality is used for both.

Later in an introductory course, when one treats electricityand magnetism (EM), a useful “bumper sticker” or “fortunecookie” description might be

“Electric fields are caused by static charges andmagnetic fields are caused by moving charges”

and one can choose the required new dimensional quantity tobe charge, denoted here by [q]¼Q. The situation regardingunits for EM problems has a much more storied history thanthat for mechanical dimensions, with extensive discussions6,7

in the very early pages of this journal. We note that in arecent edition of the most widely used graduate textbook onelectrodynamics,8 the author not only includes an exhaustivediscussion of the relative merits of MKS versus CGS units,but even chooses to “switch gears” between the two, halfwaythrough the book. We also recall that for metrological rea-sons the basic SI dimensionful quantity currently related tocharge is actually the Ampere, hence the MKSA label.

Imagining, as we do, that our discussion will be mostly usedfor undergraduates (majors and non-majors alike, at all levels),we choose to focus on charge (Q) as the new basic dimensionunderlying EM, and are happy to admit that we will alwayshave in the back of our mind an MKSA-like system, but withthe Coulomb as the explicit realization of that new dimension,not the Ampere. In that context, some of the basic quantitiesencountered in EM problems have dimensions

I½ � ¼ Q

T; V½ � ¼ E½ � ¼ ML2

QT2; C½ � ¼ Q2T2

ML2;

R½ � ¼ ML2

Q2T; and B½ � ¼ M

QT; (5)

for current, voltage/EMF, capacitance, resistance, and mag-netic field, respectively.

If we continue to imagine an MKSA-type description ofelectric and magnetic fields, we can use the forms of Gauss’law and Ampere’s law in this language (in differential form,for static fields), namely,

r � E ¼ q�0

and r� B ¼ l0J; (6)

or similarly the electric and magnetic force laws of Coulomband Biot–Savart

E ¼ qr̂

4p�0r2and B ¼ l0

4p

ðI� r̂

r2dl0; (7)

to describe two new fundamental constants governing the lawsof electricity (�0) and magnetism (l0). The extension of theseresults by Maxwell also led to another of the great early unifi-cations of physics, bringing together electricity, magnetism,and optics, with their fundamental constants being related byc ¼ 1=

ffiffiffiffiffiffiffiffiffi�0l0

p. Finally, in contrast to the situation for mechan-

ics, there is a fundamental quantum of charge, namely, e,which will be important for many physical processes.

Thermal physics, covering both thermodynamics and sta-tistical mechanics, introduces a new physical dimensionality,and a new fundamental constant. The concept of absolutetemperature (hereafter denoted as TK, to avoid confusionwith the T used here to represent the time dimension) asdescribed by the ideal gas law, PV / TK, requires a newdimension, which is conventionally written as [TK]¼H. Thefundamental constant associated with this field of study isBoltzmann’s constant kB, appearing in the ideal gas law aswell as in the definition of entropy, S ¼ kB logðXÞ; we notefor future reference that

kB½ � ¼ML2

HT2: (8)

The “unification” sometimes cited here is the connectionbetween the microstates of a physical system and its macro-scopic properties.

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The final topic considered in most introductory textbooksis “modern physics,” which includes both special relativityand quantum mechanics, along with their applications orrealizations in nature. While these approaches are indeedradically new ways of thinking of the way nature works(especially in the realm of the very fast and/or very small)they do not require any new dimensional “letters” in thealphabet: the set L, T, M, Q, and H suffices. A discussion ofmodern physics does, however, introduce two fundamentalconstants of nature, namely, Planck’s constant �h and thespeed of light c, which have dimensions

�h½ � ¼ ML2

Tand; of course; c½ � ¼ L

T: (9)

Of the five most foundational fundamental constantsincluded here, G, e, c, �h, and kB, we note again that the SIsystem of units will begin using the last four as base quanti-ties within the next decade. In a review describing thatchange, Newell5 notes that “…the gravitational constant,G—which might seem a reasonable choice for a fundamentalconstant more directly linked to the traditional base mechani-cal units—is inherently difficult to measure,” citing a recentreview article.9 The important (and dynamic) connectionbetween dimensional analysis and metrology10 is one that isuseful to recall in this context, as are popular treatments fo-cusing on describing these fundamental constants.11

III. REVIEWING CLASSICAL METHODS AND

APPROACHES

We begin our review of some of the standard results ofdimensional analysis with a classic example. Imagine a mass(m) oscillating in one dimension on the end of a spring(spring constant k), and subject to no other forces. Besides mand k, the only other dimensional parameter is the initial dis-placement x0 of the spring from equilibrium. What candimensional analysis tell us about the period (s), amplitudeof motion (A), and total energy (E) of this system, in termsof these parameters? The dimensions of the various quanti-ties in question are

½m� ¼ M; ½k� ¼ F

x

� �¼ M

T2; and ½x0� ¼ L;

so if we assume a power-law dependence, we can writes ¼ Csm

akbxc0, where a, b, and c are arbitrary powers to be

determined, and Cs is a dimensionless constant. Taking thedimensions of both sides, we have

s½ � ¼ Cs½ � m½ �a k½ �b x0½ �c ! T ¼ Ma M

T2

� �b

Lc; (10)

since [Cs] is dimensionless. Then, matching basic dimen-sions, we require that

M : 0 ¼ aþ bþ 0; (11)

L : 0 ¼ 0þ 0þ c; (12)

T : 1 ¼ 0� 2bþ 0; (13)

which is satisfied by a¼ 1/2, b¼�1/2, and c¼ 0, so thats /

ffiffiffiffiffiffiffiffiffim=k

p. Recall that the exact answer one obtains from

solving the differential equation is s ¼ 2pffiffiffiffiffiffiffiffiffim=k

p.

This is the type of result that is the most straightforward,where dimensional analysis alone determines how the resultdepends on the physical parameters, along with an undeter-mined dimensionless constant of Oð1Þ, hopefully within anorder of magnitude of unity, say 1061. If instead we ask howthe amplitude A of the motion could depend on m, k, and x0,we naturally find that A / m0 k0 x1

0 / x0. If we try the sameapproach for the total energy, we write E ¼ CE ma kb xc

0, andmatching dimensions, we find that a¼ 0, b¼ 1, and c¼ 2, sothat E / kx2

0 which is also correct, with CE¼ 1/2.If instead of an initial x0, we give the mass an initial veloc-

ity v0, the analysis proceeds as above, also givings /

ffiffiffiffiffiffiffiffiffik=m

p, but now A / v0

ffiffiffiffiffiffiffiffiffim=k

p, and E / mv2

0, again withthe number of dimensionful parameters matching exactly thenumber of dimensional constraints.

However, considering the more general problem with bothan initial displacement and an initial speed, we find that thesystem is under-constrained, since assuming

E/ ma kb xc0 vd

0!ML2

T2¼ Mð Þa M

T2

� �b

Lð Þc L

T

� �d

(14)

gives

M : þ1 ¼ aþ bþ 0þ 0; (15)

L : þ2 ¼ 0þ 0þ cþ d; (16)

T : �2 ¼ 0� 2bþ 0� d: (17)

We can solve for three of the exponents in terms of thefourth, so that we might write a¼ 1 � b, d¼ 2 � 2b, andc¼ 2b. Thus, the energy can be written in the form

E ¼ Cb m1�b kb x2b0 v2�2b

0 ¼ Cb mv20

kx20

mv20

!b

¼ Cb mv20 Pb; (18)

where the analysis itself has suggested a new dimensionlessratio, defined by

P � kx20

mv20

: (19)

This result does, of course, reproduce the two earlier specialcases, since for b¼ 1 we have the pure-displacement case ofE / kx2

0, while for b¼ 0 we have E / mv20.

Since the dependence in Eq. (18) is possible for any valueof b, and any integral value of b is allowed, along with an ar-bitrary dimensionless constant Cb, we can write even moregenerally that

E ¼ ðmv20ÞðC0 P0 þ C1 P1 þ C2 P2 þ � � �Þ

¼ ðmv20ÞX1b¼0

CbPb

" #¼ mv2

0 FðPÞ; (20)

so that in fact any function of the dimensionless ratio P isallowed. In this case, of course, the appropriate function is avery simple one, since

E¼ mv20

2þ kx2

0

2¼ mv2

0

21þ kx2

0

mv20

!! F Pð Þ ¼ 1

21þPð Þ:

(21)

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This example is a model for the more general results madefamous by Buckingham,12 who was one of the first to system-atically describe methods of dimensional analysis, leading tothe so-called Buckingham Pi theorem.13 If the system ofdimensional equations is under-constrained, having M dimen-sionful variables but only N constraints, there can beI¼M�N different dimensionless ratios, so the general resultmight depend on F(P1, P2,…,PI). Even in cases where oneobtains such a general result, depending on an unknown func-tion, considering limiting cases can still provide informationon the functional form of F(P). We will see explicit cases ofnontrivial forms for F(P) in examples below.

Dimensional analysis methods have been applied to otherclassical mechanics systems14 and to dispersion in waves,15

but is perhaps more familiarly used in fluid dynamics.Jensen16 notes that the drag force (FD) on a spherical object(radius R), moving at constant speed v through a medium ofdensity q and viscosity g (where [g]¼M/LT), can be writtenin terms of these parameters as

FD / qagbRcvd ¼ qag1�aR1þav1þa / gRvqRv

g

� �a

: (22)

As expected, this form correctly reproduces both the Stokes’law limit for small, slow particles in a highly viscous material(F¼ 6pgRv) for a¼ 0, and the limit appropriate for large,fast-moving objects, where F / qR2v2, for a¼ 1. NotingBuckingham’s theorem, we expect more generally that theforce law would be given by F¼ gRvG(P), where in this casethe dimensionless ratio is

P ¼ qRv

g� Re; (23)

where Re is the famous Reynolds number and G(P) is theundetermined function. In contrast to the straightforward caseof the harmonic oscillator, where the general form of theenergy is known, the complex nonlinear problem of fluid flowadmits no such simple, closed-form solution for G(P). Butthis approach does single-out the correct combination of vari-ables as an important dimensionless ratio of direct relevanceto categorizing different kinds of fluid flow. There are literallydozens of dimensionless numbers (ratios of physical quanti-ties) used in fluid dynamics alone,17 so this situation is exem-plary of many other physical and/or engineering systems.

Perhaps even more importantly, such dimensionless ratioscan be used in scaling arguments, so that systems with differ-ent parameters, but identical dimensionless ratios, can exhibitsimilar behavior—an idea used in applications such as windtunnels. The outcome of identifying scaling behavior can beone of the most important aspects of determining the dimen-sional “content” of a physical result through such methods.

Going beyond mechanics, consider next the problem offinding the time-dependent current in a capacitor (C) beingcharged through a resistance (R) by an external battery (E).We can proceed by identifying IðtÞ / EaRbCctd, and match-ing dimensions (using Eq. (5)) to find that

I tð Þ / Eþ1R�1�dC�dtd ¼ ER

t

RC

� �d

! ER

F Pð Þ; (24)

where P� t/RC, implicitly identifying the time constant forthe problem. The standard analysis of the time-dependentdifferential equation of course gives F(P)¼ e�P, so we see

that non-power-law functional dependences of dimensionlessratios are consistent with Buckingham’s Pi theorem. Aninteresting, and far more open-ended, application of dimen-sional analysis to a Lenz’s law problem has been presentedelsewhere.18

As a more advanced example in electrodynamics, considerthe power (P) radiated by an accelerating point charge. Thecharge (q) and acceleration (a) have obvious places (anddimensions), and since this problem involves both electricand magnetic phenomena, we might expect both �0 and l0 toplay a role. Thus, we write

P ¼ CP �a0 lb

0 qc ad ! ML2

T3

¼ Q2T2

ML3

� �aML

Q2

� �b

Qð Þc L

T2

� �d

: (25)

This is easily solved to find a¼ 1/2, b¼ 3/2, c¼ 2, andd¼ 2, giving

P ¼ CP �1=20 l3=2

0 q2 a2 ¼ CPl0q2a2

c; (26)

if we make use of the connection that c ¼ 1=ffiffiffiffiffiffiffiffiffi�0l0

p; this

expression is consistent with the standard result,19 whereCP¼ 1/6p. The derivation of the power radiated by an accel-erating charge is a challenging one in advanced electrody-namics, but can be studied in this way using much simplertechniques.

If we ask instead whether a particle moving at constantspeed v0 can radiate, replacing a ! v0 in this analysis, wefind that no solution of the resulting system of equations ispossible. This is a very useful result, reinforcing the fact thatcharges in uniform motion do not radiate, simply on dimen-sional grounds. Dimensional analysis has also been appliedto the problem of the gravitational power radiated by a bodyin a circular orbit,20 but because there are fewer constraintequations (just three since only M, L, and T are involved inmechanics problems, and not Q as in EM) and more dimen-sional variables, more assumptions are required to connectwith the answer as derived in general relativity.

As an example where dimensional analysis can highlyconstrain, but not necessarily pinpoint, the dependence onphysical quantities, consider the problem of the time depend-ence of the thickness x(t) of an ice sheet floating on water,maintained at a constant (externally imposed) temperaturedifference DText during the winter. One presumes that thephysics of the problem must depend on the thermal proper-ties of the ice, especially the thermal conductivity (j) andlatent heat of fusion (LF, via the heat released through freez-ing one layer of ice as it flows through the existing ice sheet),and the density q relates the volume of ice frozen to its mass.Thus, the simplest likely power-law dependence on thesequantities would be xðtÞ / jaDTb

exttcLd

Fq�, or, in terms oftheir dimensions

L ¼ ML

T3H

� �a

Hð Þb Tð Þc L2

T2

� �dM

L3

� ��: (27)

Since the system is under-constrained, we can solve for fourpowers in terms of the fifth, and we choose to write �¼�a,d ¼ ð1=2Þ � 2a, c¼ 1 � a, and b¼ a, so that

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x tð Þ / jaDTaextt

1�aL1=2�2aF q�a ¼ jDText

tL2Fq

� �a

tffiffiffiffiffiLF

p: (28)

In this case, the fact that there is a dimensionless quantityP � jDText=tL2

Fq, is not as important as the specific power-law dependences of the individual physical parameters.

For example, since the ice presumably grows thicker withtime, we expect that 1 � a> 0 (hence a< 1) on physicalgrounds. We would also expect, all other things being equal,that the ice thickness will be smaller if either DText or j issmaller (less heat flow) or if LF is larger (harder to freeze theice), which suggest that a> 0 and 1/2 � 2a< 0 (or a> 1/4),respectively. Combining these conditions gives 1/4< a< 1,so that the choices a¼ 0, 1 are immediately disallowed onphysical grounds; the dependence L

1=2�2aF itself might then

suggest (but certainly not prove) that a¼ 1/2 is a naturalchoice, giving

x tð Þ ¼ CjDText

qLFt

� �1=2

; (29)

where C is a dimensionless constant. A classic “firstprinciples” calculation21 (from the late 19th century) equat-ing the heat flow through a slab of ice (already of thicknessx), namely,

dQ

dt¼ jA

DText

x; (30)

with the heat given off in freezing the next infinitesimal layerof ice, via

dQ ¼ LF dm ¼ Lf ½dV q� ¼ LF½ðA dxÞq�; (31)

gives the relation

x dx ¼ jDText

LFq

� �dt; (32)

which can be integrated to obtain Eq. (29), with C ¼ffiffiffi2p

.

IV. EXPLORING THE QUANTUM FRONTIER

If we associate the birth of quantum mechanics with theextraction of Planck’s constant (h or �h) from studies of thenature of blackbody radiation, then the quantum era is wellover a century old. Some of the earliest (now historical)results are well known, but can still provide an illustration ofthe use of dimensional analysis in a more “modern physics”setting.

For example, in discussions of blackbody radiation one of-ten starts with the notion of the frequency-dependent inten-sity, namely, the power (energy per unit time) per unit areaper unit frequency interval, sometimes denoted as dRT(f)/df.The total intensity, integrated over all frequencies, is

RT �ð1

0

dRT fð Þdf

df ; (33)

and both quantities can be directly related to experimentalmeasurements. Theoretical derivations related to blackbodyradiation are most easily done, however, using the notion ofthe frequency-dependent energy density (energy per unit

volume, per unit frequency interval), often written as q(f),and we will apply the methods described above to see howq(f) and RT depend on the fundamental physical constants ofthe problem. We first note that the dimensions of these twoquantities are

RT½ � ¼energy

area � time¼ M

T3and

q fð Þ½ � ¼ energy

volume � frequency¼ M

TL: (34)

Since the problem clearly involves quantum mechanics, pho-tons (hence relativity), and thermal physics, we can expect �h,c, TK, and kB all to appear, and for frequency-dependentresults, also f. In the spirit of the original derivation, we willactually use h ¼ 2p�h, which of course has the samedimensions.

Starting with q(f), for example, we try qðf Þ / haf bccTdKk�B,

or in terms of dimensions

M

TL¼ ML2

T

� �a1

T

� �bL

T

� �c

Hð Þd ML2

HT2

� ��: (35)

This can be solved (say in terms of a) to give �¼ 1 � a,d¼ 1 � a, c¼�3, and b¼ 2þ a, or

q fð Þ ¼ C kBTKð Þf 2c�3� � hf

kBTK

� �a

¼ Cf 2

c3hfð Þ

� �hf

kBTK

� �a�1

! f 2

c3hfð ÞF P � hf

kBTK

� �; (36)

where the analysis has correctly identified the appropriatedimensionless ratio P, and F(P) is an arbitrary functionthereof. This result does, of course, agree with the standardform

q fð Þ ¼ f 2

c3hfð Þ 8p

ehf =kBTK � 1

� �: (37)

For the total intensity, since we’ve integrated over fre-quency, we have only four dimensional parameters andmight therefore expect a more determined result (four equa-tions in four unknowns). If we write RT / hacbTc

KkdB, we find

by matching dimensions that

RT ¼ Ch�3c�2k4BT4

K ¼ Ck4

B

h3c2

� �T4

K; (38)

which is indeed consistent with the Stefan–Boltzmann equa-tion RT ¼ rT4

K , where r � ð2p5=15Þðk4B=h3c2Þ.

One of the benchmark quantum mechanical systems is thehydrogen atom, for which one can easily obtain order-of-magnitude estimates of most relevant quantities, includingenergies and spatial extension, using dimensional analysis.With the understanding that this is a problem involving theelectrostatic attraction of two fundamental charges (e), thatthe electron sets the relevant mass scale (me, just as thelighter mass sets the scale in two-body classical orbit prob-lems), and that it involves quantum mechanics, we can pro-pose for the energy scale that

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EH ¼ CHea�b0mce�h

d ! ML2

T2

� �

¼ ðQÞa Q2T2

ML3

� �b

ðMÞc ML2

T

� �d

:

Matching dimensions, one finds that a¼ 4, b¼�2, c¼ 1,and d¼�2, so that

EH ¼ CHe4

�20

m

�h2: (39)

In this case, comparing to the standard expression for the Hatom energy levels, one finds that CH ¼ �1=32p2 for theground state; if one assumes that the appropriate fundamen-tal constant for electrostatics that should be used for dimen-sional analysis problems is actually 4p�0, then the estimateabove would be much closer to the ground-state energyscale, with CH¼�1/2. Using the same dimensional argu-ments, one can find that the associated length scale hasdimensions ð�h2=meÞð�0=e2Þ, which is indeed proportional tothe Bohr radius. In any dimensional-analysis approach to thephysical parameters for a given quantum state wn, we can of-ten find the dimensionful dependences correctly, but weshould keep in mind that each one may depend on an unde-termined function of the quantum number(s) and in the caseof the H-atom energies this is F(n, l, m)¼F(n)¼ 1/n2, pro-viding a different example of an undetermined dimensionlesspre-factor.

Students can certainly hone their dimensional analysisskills against such problems with historical and well-knownanswers, but one of the reasons for pursuing this method isto be able to approach new physics in new contexts. As amore recent example, consider the study of the interactionsof neutrons with Earth’s gravitational field. It was shownlong ago22 that one could measure the local value of g bywatching neutrons “drop” in the Earth’s gravitational field, apurely classical effect. The effect of terrestrial gravity on thephase of the wave function of an otherwise free neutron—amuch more foundational topic—was also measured morethan 35 years ago.23 Far more recently, however, evidencefor quantized states of a neutron24 in the gravitational poten-tial of the earth has been published.

We can think of this last system as a textbook problem ofa particle in a potential described by

VðzÞ ¼ þ1 z < 0;mngz z � 0;

(40)

which can be treated in position space,25 in momentumspace,26 or by WKB methods.27 One can repeat our now-standard dimensional analysis for the energy scales associ-ated with the quantum states of the neutron in this potential,starting with E / ma

ngb�hc, to find that E / ðmng2�h2Þ1=3. If we

substitute numbers, we find that these energies are of order10�36 J� 10�12 eV; the wavelength of radiation associatedwith transitions between such quantum states would be oforder kc ¼ 2p�hc=E � 106 m, which is not remotely experi-mentally accessible. But if we instead ask for the typical spa-tial extension l of the quantum states, which gives anestimate of how high above the flat surface the neutron willhover, we find that l � ð�h2=m2

ngÞ1=3 10 lm, which is smallbut still measurable. The experimental evidence for thequantum structure of the neutron’s bound states was made by

actually probing the spatial extent of the neutron wave func-tion on such table-top length scales.24

In approaching the “full-blown” versions of many quantumproblems (such as the H atom or the neutron in a gravitationalfield) by actually solving the Schr€odinger equation, one ofthe first steps is often to rewrite the differential equation indimensionless form, which often automatically identifies theappropriate length, time, and energy scales. But for studentswho never get that far in the physics curriculum, dimensionalanalysis can provide a way to extract some of the truly inter-esting science in the problem, in a quantitative manner, with-out focusing on the details of the mathematical derivations.

At an even more fundamental level, the problem of how tocombine quantum mechanics (�h) and special relativity (c)with gravitation (G) is still an open question, withapproaches such as string theory and loop quantum gravitybeing widely debated and studied. But presumably in anysuch theory, these three fundamental constants of nature willbe combined to set the time, length, and energy scales forwhich such a description would be necessary.28 Thus, if wetake combinations of the form �hacbGc and match dimen-sions, we find

LP ¼ffiffiffiffiffiffi�hG

c3

r� 10�35 m; TP ¼

ffiffiffiffiffiffi�hG

c5

r� 3� 10�44 s;

and EP ¼ffiffiffiffiffiffiffi�hc5

G

r� 1019 GeV (41)

for the Planck length, time, and energy, respectively. Thepossibility of using such quantities as the basis for a systemof “absolute units” has often been discussed.29

A problem related to the quantum Hall effect (discussedfurther below) is that of the discrete energy levels of acharged particle in a uniform magnetic field, giving rise toso-called Landau levels. The quantized energies EL coulddepend on �h, e, me, and the value of the applied magneticfield B0. Imposing the dimensional matching conditions, wefind that

EL / �h1e1B10m�1 ¼ �h

eB0

m

� �¼ �hxC; (42)

where xC is the classical cyclotron frequency. Even if stu-dents are not expecting to see the classical quantity xC, itautomatically comes out of the analysis.

One of the first experiments that gave information on themechanism of electrical conduction was the (classical) Halleffect, which can probe the sign and density of the actualcharge carriers. In the standard rectangular-bar geometry(long bar of width d and thickness t), there is a current Ialong the long direction, with a magnetic field B in the t orthickness direction, perpendicular to the current. The ratio ofthe voltage induced across the bar to the current along thebar is often called the Hall resistance and can be shown viaan elementary analysis to be RH¼B/(etne), so that measure-ments of RH plotted versus applied field (B) give a straightline with a slope determined by the material (electronic den-sity ne) and geometric (t) properties of the sample.

For some samples (MOSFETs, for example), at suffi-ciently low temperature and high magnetic fields, it wasshown by von Klitzing et al.30 that there were well-definedplateaus in the RH versus B plots, corresponding to quantizedvalues of resistance, given by RH¼RK/i (i an integer), where

358 Am. J. Phys., Vol. 83, No. 4, April 2015 R. W. Robinett 358

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the value of RK was found to be independent of the materialproperties (ne) and the geometry (t, d, etc.). Because thisseems to be a quantum phenomenon, we might guess thatthis new fundamental unit of resistance RK (now known asthe Klitzing constant) might depend on �h, e, perhaps the car-rier mass me, and let’s even imagine a possible dependenceon �0. If we then write RK / �haebmc

e�do and match dimen-

sions, we find that a¼ 1, b¼�2, and c¼ d¼ 0, so thatRK / �h=e2. The conventional definition is actually thatRK¼ h/e2 25.8 kX, and this value is now used extensivelyin the field of metrology.31 To students, at any level, whohave only been introduced to quantization in the context ofdiscrete energy levels or angular momenta, the fact that thereare other important manifestations of quantum mechanicsthat give rise to quantized physical properties, such as resist-ance, can be enlightening.

As two more such examples, we recall that in supercon-ducting materials, both the circulation32 of the superfluid ve-locity vs and the magnetic flux33 are quantized. If we writethese quantities in terms of their definitions, namely,

C �þ

vs � dl and UB �ð

A

B � dA; (43)

then dimensional analysis alone can be used to derive theirdependence on fundamental parameters, using as input thedimensions of the factors in the integrands. For circulation,we might imagine the relevant parameters being �h and m(only mechanics seems to be involved, classical or quantum),while for UB we might guess �h, m, e, and perhaps l0, since itis a magnetic effect. Assuming power-law dependences inboth cases, we find that

C / �h

m! h

2mand Ub /

�h

e! h

2e; (44)

where the last expression in each case is the actual“textbook” result.

As our final example, we consider a neutron star—a quan-tum system that shows just how important dimensionless pa-rameters can be. The stability of a neutron star arises fromthe interplay between the gravitational self-energy VG(r) ofthe star’s mass M* and the quantum mechanical zero-pointenergy E0(r) of the confined neutrons, both of which dependon the star’s radius r. If we write each of these energies as afunction of its expected physical parameters, we have

VGðrÞ / Ga1 Mb1 rc1 and E0ðrÞ / �ha2 mb2

n rc2 : (45)

Matching dimensions in each, we then find that

VG rð Þ / �GM2

rand E0 rð Þ / �h2

mnr2: (46)

The first result is consistent with standard gravitationaltheory (though we have inserted the physically relevantminus sign by hand!), while the second is consistent with“particle in a box” quantum mechanics. Of course there aremany neutrons (Nn�M*/mn� 1057 � 1) so E0(r) will nodoubt also depend on a yet-to-be-determined function F(Nn)of that dimensionless parameter.

If we combine the expressions in Eq. (46) and then mini-mize the total energy as a function of r, we find for the neu-tron star radius

rNS ¼�h2

GM2mn

F Nnð Þ � 10�92 mð ÞF Nnð Þ: (47)

Note that if we had attempted to use dimensional analysisto predict rNS / �haGbMc

mdn directly, we would not have

found this unambiguous prediction—only a dependence ofthe form ð�h2=GÞMc

m�3�cn —so focusing first on the two sepa-

rate energy expressions in this case was a more powerfulapproach.

If F(Nn) were only linear in the number of neutrons (eachneutron having the same zero-point energy), then the predic-tion for the physical size of this system would be roughly10�35 m, which is (accidentally!) of the same magnitude asthe Planck length in Eq. (41) and would be physically irrele-vant. However, this approach based only on dimensionalquantities misses an extremely important aspect of quantummechanics, namely, the Pauli exclusion principle. Becauseneutrons are spin-1/2 fermions, only two of them can be inthe same spatial wave function, and therefore the averageenergy of each increases with Nn. (The estimate based onF(Nn) / Nn would be appropriate for bosons.) A simplifiedcalculation34 based on filling energy levels of a three-dimensional well shows that FðNnÞ � N5=3

n , leading to amore realistic estimate of rNS� 10 km, much closer to theactual answer.

It is hard to think of any other example where dimen-sional analysis does, in fact, find the correct dependenceson the appropriate physical constants, but then proceeds toget the actual numerical answer more wrong! One shouldnever forget that there are important physical constraints(such as the spin-statistics theorem) that can be completelymissed by dimensional analysis; quantum physics is farmore than just �h!

V. COMMENTS AND CONCLUSIONS

In this note, we have reviewed the basic assumptions andnotation underlying dimensional analysis, provided examplesof its use (and limitations) in the realm of classical physics,and illustrated how it can be successfully applied to moremodern scenarios in the quantum regime, including somethat come close to current research problems. This discussionhas been in the spirit of a remark made by Lord Rayleigh35

exactly one century ago, regarding this method (which hedescribes as the principle of similitude):

“I have often been impressed by the scantyattention paid even by original workers in physicsto the great principle of similitude…I have thoughtthat a few examples, chosen almost at randomfrom various fields, may help to direct theattention of workers and teachers to the greatimportance of the principle.”

The examples considered here have been taken from a va-riety of resources, from elementary discussions inintroductory-level texts, to more advanced texts for physicsmajors, and from recent (and not-so-recent) research papers.Large numbers of further examples are readily obtainablewith a minimum of effort. The author maintains a collectionof examples on a public web site,36 and encourages readersto submit additional examples.

The examples in this article have been chosen to be as rep-resentative as possible of the different types of “outcomes”

359 Am. J. Phys., Vol. 83, No. 4, April 2015 R. W. Robinett 359

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that can be obtained using dimensional analysis, includingcases where:

• The dependence on all relevant parameters is completelydetermined;

• No solution using the relevant parameters is found to bepossible;

• New dimensionless combinations are found, which can beuseful for scaling arguments, either phenomenological orfundamental in nature;

• Constraints are placed on the power-law dependences,which can be used to find “likely” solutions; and

• Dimensional analysis predicts the dimensional dependen-ces correctly, but the dimensionless factors are numeri-cally important, sometimes indicating that a significantphysical effect has been ignored.

Dimensional analysis is not the only way to approach anew problem, but it can often be used to gain significantground along the path towards understanding the physicsbehind the mathematics. There are many powerful analytictechniques in physics, such as the use of symmetry, whichcan be brought to bear on some, but not all, problems. Onemeasure of a professional is not only the ability to use a vari-ety of tools, but also the knack for knowing which tool(s) touse, and when.

In deciding whether to make use of this particular tool inone’s courses or curricula, we might turn to discussions aris-ing from physics education research related to the questionof the nature of effective instruction. For example, Wiemanand Perkins37 ask (and answer) an important question:

“But what specifically do we mean by effectivephysics instruction? It is instruction that changesthe way students think about physics and physicsproblem solving and causes them to think morelike experts—practicing physicists. Experts see thecontent of physics as a coherent structure ofgeneral concepts that describe nature and areestablished by experiments, and they usesystematic concept-based problem-solvingapproaches that are applicable to a wide variety ofsituations.”

Whether or not one considers the fact that the dimensionsof almost all important physical quantities can be “spelled”with a minimal five-letter alphabet to be miraculous (in theWigner sense1), dimensional analysis provides a methodol-ogy that allows students to work towards mastery of all ofthe skills mentioned above. We advocate here that dimen-sional analysis be considered as another important pedagogi-cal component, at many levels, complementing higher-levelmathematics, and that it be used in the study of physics asanother language.

a)Electronic mail: rick@phys.psu.edu1E. Wigner, “The unreasonable effectiveness of mathematics in the natural

sciences,” (Richard Courant Lecture in Mathematical Sciences) Commun.

Pure Appl. Math. 13, 1–14 (1960).2P. W. Bridgeman, Dimensional Analysis (Yale U. P., New Haven, 1922),

available at <https://archive.org/details/dimensionalanaly00bridrich>; N.

F. Murphy, “Dimensional Analysis,” Bull. Virginia Polytech. Inst.

XLII(6), 1–41 (1949); H. L. Langhaar, Dimensional Analysis and Theoryof Models (Wiley, New York, 1951); H. E. Huntley, Dimensional Analysis(Rinehart and Company, New York, 1955), reprinted by Dover, New

York, 1967; P. LeCorbeiller, Dimensional Analysis (Appleton-Century-

Crofts, New York, 1966); L. I. Sedov, Similarity and Dimensional

Methods in Mechanics, 10th ed. (CRC Press, Boca Raton, FL, 1993); T.

Szirtes and P. R�ozsa, Applied Dimensional Analysis and Modeling, 2nd

ed. (Elsevier, Amsterdam, 1997); H. G. Hornung, Dimensional Analysis:Examples of the Use of Symmetry (Dover, New York, 2006); A. C. Palmer,

Dimensional Analysis and Intelligent Experimentation (World Scientific,

Hackensack, NJ, 2008).3D. Bolster, R. E. Hershberger, and R. J. Donnelly, “Dynamic similarity,

the dimensionless science,” Phys. Today 42(9), 42–47 (2011). This review

also provides a brief history of dimensional analysis, citing contributions

from Newton, Euler, Fourier, and Rayleigh.4In somewhat the same way that the Q and X tiles in Scrabble are not

played as often as others.5D. B. Newell, “A more fundamental international system of units,” Phys.

Today 67(7), 35–41 (2014).6R. T. Birge, “On electric and magnetic units and dimensions,” Am. J.

Phys. 2, 41–48 (1934); “On the establishment of fundamental and derived

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1998).9C. Speake and T. Quinn, “The search for Newton’s constant,” Phys. Today

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System of Measurement (Norton, New York, 2011).11M. Rees, Just Six Numbers: The Deep Forces That Shape the Universe

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dimensional equations,” Phys. Rev. 4, 345–376 (1914).13See also S. Corrsin, “A simple geometrical proof of Buckingham’s p-theo-

rem,” Am. J. Phys. 19, 180–181 (1951); J. H. Evans, “Dimensional analy-

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en.wikipedia.org/wiki/Reynolds_number>, and in similar online sources.18J. A. Pelesko, M. Cesky, and S. Huertas, “Lenz’s law and dimensional ana-

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360 Am. J. Phys., Vol. 83, No. 4, April 2015 R. W. Robinett 360

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27J. J. Sakurai, Modern Quantum Mechanics, revised edition, edited by S. F.

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tized flux in superconducting cylinders,” Phys. Rev. Lett. 7, 43–46 (1961);

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tion in a superconducting ring,” ibid. 7, 51–52 (1961).34R. W. Robinett, Quantum Mechanics: Classical Results, Modern Systems,

and Visualized Examples, 2nd ed. (Oxford U. P., Oxford, 2006), pp.

200–206.35L. Rayleigh, “The principle of similitude,” Nature 95, 66–68 (18 March

1915).36A short “dictionary” compiling the L, T, M, Q, H spelling of many basic

dimensionful quantities, as well as a collection of problems similar to ones

considered here, is provided as an online supplement to this article at E-

AJPIAS-83-015501, and is also available at <http://users.phys.psu.edu/

~rick/DimensionalAnalysis.html>.37C. Wieman and K. Perkins, “Transforming physics education,” Phys.

Today 58(11), 36–37 (2005).

361 Am. J. Phys., Vol. 83, No. 4, April 2015 R. W. Robinett 361

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