A historical derivation of Heisenberg’s uncertainty relation is flawed

John H. Marburger IIIa兲

Office of Science and Technology Policy, Executive Office of the President, New Executive Office Building,
Washington, District of Columbia 20502
共Received 17 August 2007; accepted 26 January 2008兲
Kennard’s 1927 demonstration of the uncertainty relation, cited by Heisenberg as the earliest
derivation using the formalism of quantum theory, invokes a trial function that severely limits the
class of wave functions for which the uncertainty relation is shown to be valid. © 2008 American
Association of Physics Teachers.
关DOI: 10.1119/1.2870628兴

I. HISTORICAL BACKGROUND tion from the mean. This factor multiplies the right-hand side
of the relations by a factor of two so, for example, Eq. 共14兲 in
Heisenberg proposed his famous uncertainty relation 80 Heisenberg’s Chicago lectures,4
years ago in the form ⌬q⌬p ⬃ h and asserted, but did not h
demonstrate, that it could be derived directly from the math- ␦ p␦q 艌 , 共1兲
2␲
ematics of the new matrix mechanics, then all of 18 months
old.1 The quantities ⌬q and ⌬p are uncertainty measures of is correct using the definition

冕
the position q and momentum p, and h is Planck’s constant.
A derivation appeared immediately thereafter in a fascinating 共␦q兲2 = 2 共q − qm兲2兩␺共q兲兩2dq, 共2兲
paper by Kennard, which Heisenberg cited in connection
with his own proof in his 1929 University of Chicago where qm is the expected value of q with the distribution 兩␺兩2.
lectures.3,4 At about the same time Weyl sketched a proof Kennard set qi = ␦q, pi = ␦ p, and I will use this notation and
based on the Schwarz inequality in his book on applications also set the means qm, pm to zero, which amounts to resetting
of group theory in quantum mechanics, citing a remark by the origins of the coordinate and momentum frames. I also
Pauli.5 Historians credit Kennard with the first proof.6 How- reserve ⌬q and ⌬p for the conventional root-mean-square
ever, Kennard’s proof is flawed, and we must look to one of deviations, that is, omitting the factor of two on the right of
the other actors for the first unambiguously correct demon- Eq. 共2兲, and replace h / 2␲ everywhere with ប. In the follow-
stration of this famous inequality. ing ␺共q兲 is the probability amplitude 共wave function兲 in the
Bohr appears to have been the first to point out the path to position representation and ␾共p兲 is its corresponding mo-
a proper derivation by linking it 共in his 1927 Como lecture兲 mentum space representation.
to a theorem of physical optics.7 But he gave no details and
in any case his method of derivation was not what Heisen-
berg had in mind when he said the relation followed directly
from quantum theory. By juxtaposing the uncertainty relation II. KENNARD’S DERIVATION
with the commutator qp− pq= i共h / 2␲兲I Heisenberg gave the
clear impression that the latter directly implies the former. Kennard begins by examining a wave function that has a
Here q and p are matrices representing position and momen- Gaussian amplitude and a quadratic term in its phase. The
position and momentum space forms 共in our notation兲,

冉 冊
tum and I is the identity matrix. The general link between
commutators and uncertainty products like ⌬q⌬p was estab- q2
lished by Robertson in 1929 and extended by Schrödinger ␺共q兲 = C exp − + ib2q2 , 共3a兲
2q2i
the following year,8,9 but most derivations for the position-
momentum uncertainty product do not explicitly invoke the
commutator. One of these is Kennard’s.
Kennard’s paper consists of two long parts—an overview
␾共p兲 = C1 exp − 冉 p2
2p2i
冊
+ ib21 p2 , 共3b兲

of the new quantum mechanical formalism, and several ap- are related by a Fourier transform, which leads to the rela-
plications to “simple motions,” between which the uncer- tions C2 = 1 / 冑␲qi, C21 = 1 / 冑␲ pi, b21 = b2q2i / ប2共1 + 4b4q4i 兲, and
tainty principle derivation appears beginning on p. 337 of
Ref. 3. The paper has not been translated into English or ប2
included in any of the useful compilations of historical pa- p2i = 共1 + 4b4q4i 兲. 共4兲
q2i
pers from this period,2,10,11 but Nieto has remarked that Ken-
nard’s result for the motion of harmonic oscillator wave Here qi and b are arbitrary parameters. For this particular
packets was a discovery of “squeezed states” that was “too class of functions
far ahead of 关its兴 time.”12 Bohr also cited Kennard’s work on qi pi 艌 ប for Gaussian wave packets with quadratic phase,
the motion of wave packets in some simple cases.7 The dif-
ficulty with Kennard’s approach to the uncertainty relation 共5兲
seems to have passed undetected for eight decades. and the minimum uncertainty product for these functions
The uncertainty measure in these early papers often differs is achieved when the quadratic part of the phase vanishes:
by 冑2 from the now-conventional root-mean-square devia- b = 0.

585 Am. J. Phys. 76 共6兲, June 2008 http://aapt.org/ajp © 2008 American Association of Physics Teachers 585
 To establish the lower bound generally, it is necessary to that adds to the non-negative integral in Eq. 共10兲 depends
consider an arbitrary wave packet, which Kennard repre- entirely on Eq. 共9兲 for ␺0共q兲. We know from the proofs based
sented as on the Schwarz inequality that Eq. 共5兲 is generally valid, and

␺共q兲 = f共q兲␺0共q兲 = f共q兲C0 exp − 冉 冊 q2

2q2i
. 共6兲
the minimum is achieved by a Gaussian wave packet. But
Kennard’s derivation is incapable of giving this information.
Is there a way to save this proof? Rather than defining f共q兲
Here C0 is a real normalization factor and ␺共q兲 is an arbitrary in terms of the arbitrary function ␺共q兲, which makes Eq. 共8兲
normalizable wave function whose uncertainty measure is qi an identity, we can imagine choosing f共q兲 freely and defining
determined from Eq. 共2兲 with ␦q = qi. The function f共q兲 is ␺共q兲 through Eq. 共6兲. Then the parameters C0 and qi would
defined by this relation. Kennard substituted this form into have to be determined self-consistently from the normaliza-
the definition of p2i , which he wrote in the position represen- tion condition and Eq. 共2兲. The same mathematical steps
tation in the form 共after an integration by parts兲 would be valid, but now f共q兲 would be an independent arbi-

冕 冋 册 冕
trary function. Unfortunately, the class of functions for which
ប ⳵ 2
this approach can be made to work is limited.
p2i = 2 ␺*共q兲 ␺共q兲dq = 2ប2 ␺*⬘共q兲␺⬘共q兲dq.
i ⳵q The problem appears when we attempt to determine the
共7兲 parameter qi. We insert Eq. 共6兲 for ␺共q兲 into Eq. 共2兲 with
qm = 0, and find
Primes here denote differentiation with respect to q; the
reader will recognize 共ប / i兲共⳵ / ⳵q兲 as the momentum operator
in the position representation. After simple manipulations the
q2i = 2 冕 q2C20兩f共q兲兩2e−q
2/q2
i dq. 共11兲
reader may verify using Eq. 共6兲 that Eq. 共7兲 can be expressed
as The normalization condition may be used to evaluate C20
ប2
冉 冕 冊 whose inverse is a function G共qi兲 of qi:

冕
p2i = 1 + 2q2i f *⬘ f ⬘␺0*共q兲␺0共q兲dq . 共8兲
q2i 2/q2 1
兩f共q兲兩2e−q i dq = ⬅ G共qi兲. 共12兲
Compare Eq. 共8兲 with Eq. 共4兲. In Kennard’s words “since the C20
integral on the right cannot be negative, 关Eq. 共5兲兴 holds in all
We next rewrite Eq. 共11兲 in terms of G共qi兲 to find
generality” 共“gilt ganz allgemein”兲.
dG
q2i G = 2q4i 共13兲
III. ANALYSIS OF KENNARD’S TRIAL FUNCTION d共q2i 兲

This argument, however, is circular. Equation 共8兲 is an or

identity contrived to look like Eq. 共4兲 by the particular choice
of trial function in Eq. 共6兲. Kennard’s derivation can be ex- dG G
= . 共14兲
ecuted with any trial function represented by an amplitude dqi qi
f共q兲 times a factor ␺0共q兲 whose second derivative is propor-
Equation 共14兲 is not a differential equation for G共qi兲, but
tional to itself, with the coefficient of proportionality at most
an implicit equation for qi. Solutions occur whenever a ray
quadratic in q 共a linear term is excluded by our convention
from the origin G = qi = 0 is tangent to the 共everywhere posi-
qm = 0兲. Thus we let
tive兲 graph of G共qi兲. Unfortunately, it does not possess physi-
d2␺0共y兲 cally meaningful solutions for arbitrary choices of f共q兲. It is
= 共␣ + ␤y 2兲␺0共y兲, 共9兲
dy 2 a nonlinear eigenvalue equation that serves not only to de-
termine qi, but also to restrict the class of amplitudes 兩f兩 that
where y = q / qi and ␣ and ␤ are arbitrary real constants. We permit real, positive, nonzero values of qi. For example, if
evaluate the derivatives in Eq. 共7兲 with ␺0共q兲 satisfying Eq. 兩f共q兲兩2 remains finite and smooth for small q, then G共qi兲 for
共9兲 and find small qi becomes

p2i =
ប2
q2i
冉− 共2␣ + ␤兲 + 2q2i 冕 冊
f *⬘ f ⬘␺0*共q兲␺0共q兲dq . 共10兲
G共qi兲 →
qismall
冑␲qi兩f共0兲兩2 . 共15兲
Because ␣ and ␤ are arbitrary, the lower limit achieved by Although G共qi兲 automatically satisfies Eq. 共14兲, it leads to a
setting the non-negative integral on the right to zero can be trial function ␺ proportional to a Dirac delta function with a
adjusted at will. For Kennard’s ␺0共q兲 of Eq. 共6兲, ␣ = −1 and vanishing coefficient:

冉 冊
␤ = 1. But why choose these values? Replacing qi in Eq. 共6兲
by qi / ␥, for example, gives ␣ = −␥2 and ␤ = ␥4. Choosing 1 q2
␺共q兲 = f共q兲 exp −
␥2 = 1 / 2 would then suggest a minimum uncertainty product 冑G共qi兲 2q2i
of 3ប / 4. Other forms for ␺0共q兲 that satisfy Eq. 共9兲, such as qismall
cos共␥q / qi兲 permit similar flexibility in choosing the lower → 共2冑␲兲1/2冑qiei␪共q兲␦共q兲, 共16兲
limit. The point is that this reasoning does not establish a
definite lower limit for the uncertainty product, nor tell us where ␪ is the phase of f. For these functions well-behaved
which function will achieve it. Kennard’s trial function near zero, Eq. 共14兲 may possess solutions other than qi = 0 for
“stacks the deck” in favor of the Gaussian wave function. some f共q兲, but that is not necessary. For example, if f is a
The arbitrariness of ␺共q兲 is irrelevant because the expression simple Gaussian with an arbitrary width q f ,

586 Am. J. Phys., Vol. 76, No. 6, June 2008 John H. Marburger III 586
 f共q兲 = exp −冉 冊 q2
2q2f
, 共17兲
Heisenberg’s.14 Perhaps Pauli was the ultimate source of in-
spiration for the first rigorous derivation of this important
consequence of quantum theory.
we find
ACKNOWLEDGMENT
冑␲qi
G共qi兲 = 共18兲 The author is indebted to Robert P. Crease for stimulating
冑1 + q2i /q2f , discussions about the historical context of the early develop-
ment of quantum mechanics and the uncertainty relation.
for which qi = 0 is the only solution of Eq. 共14兲. The same is
true for f = cos共kq兲. The function whose squared amplitude is
q sinh共kq兲 leads to the imaginary solution qi = 2i / k. All these
a兲
On leave from State University of New York at Stony Brook, Department
of Physics, Stony Brook, New York 11794. Electronic mail:
functional forms are excluded from the class of wave func-
jhmiii@jhmarburger.com
tions for which this approach to Kennard’s derivation can be 1
W. Heisenberg, “Über den anschaulichen inhalt der quanten theoretischen
implemented. Kinematik und Mechanik,” Zeit. f. Phys. 43, 172–198 共1927兲. A transla-
Note that a nonzero solution of Eq. 共14兲 can occur only if tion is reprinted in Ref. 2. In this paper, Heisenberg calculates the width
G共qi兲 possesses an inflection point. As the width qi of the of Gaussian wave packets in position and momentum space and finds that
Gaussian factor in the integral of Eq. 共12兲 increases, spread- their product is h / 2␲.
2
Quantum Theory and Measurement, edited by J. Wheeler and W. Zurek
ing like an umbrella over the amplitude factor 兩f共q兲兩2, the 共Princeton University Press, NJ, 1983兲.
integral can increase abruptly as the Gaussian overlaps a 3
E. H. Kennard, “Zur Quantenmechanik einfacher Bewegungstypen,” Zeit.
peak in the amplitude. Thus, a function f共q兲 whose squared f. Phys. 44, 326–352 共1927兲.
4
amplitude has peaks displaced from the origin can possess W. Heisenberg, The Physical Principles of the Quantum Theory, trans-
real, positive, nonzero values of qi. It is precisely among lated by C. Eckart and F. C. Hoyt 共Dover, NY, 1930兲, p. 18.
5
H. Weyl, Group Theory and Quantum Mechanics, translated by H. P.
wave functions that do possess a single central peak, how- Robertson 共Dover, NY, 1928兲.
ever, that we might expect to find one that reduces the un- 6
M. Jammer, The Conceptual Development of Quantum Mechanics
certainty product below the lower limit already found in Eq. 共McGraw–Hill, NY, 1966兲, p. 333.
共5兲 for a Gaussian, and a correct proof needs to consider 7
N. Bohr, “The quantum postulate and the recent development of atomic
these functions. theory,” Nature 共London兲 121, 580–590 共1928兲. Reprinted in Wheeler
and Zureck, Ref. 2. Bohr wrote that “These relations—well known from
the theory of optical instruments, especially from Rayleigh’s investiga-
IV. CONCLUSION tion of the resolving power of spectral apparatus—express the condition
that the wave-trains extinguish each other by interference at the space-
No equation in Kennard’s proof is incorrect, only the time boundary of the wave field.”
8
statement that the proof is general, or that the minimum un- H. P. Robertson, “The uncertainty principle,” Phys. Rev. 34, 163–164
certainty product has been demonstrated. Depending on 共1929兲.
9
E. Schrödinger, “Zum Heisenbergschen Unschärfeprinzip,” Abh. Preuss.
whether ␺共q兲 or f共q兲 in Eq. 共6兲 is regarded as the given Akad. Wiss., Phys.-Math. Kl. 19, 296–303 共1930兲.
arbitrary function, the argument is either a tautology or en- 10
Sources of Quantum Mechanics, edited by B. L. van der Waerden 共Dover,
tails a condition that restricts the range of functions for NY, 1968兲.
11
which the proof is valid. E. Schrödinger, Collected Papers on Wave Mechanics 共Chelsea, NY,
This conclusion raises the question of who deserves prior- 1982兲, 3rd English ed.
12
ity for a rigorous demonstration of the correct mathematical M. M. Nieto, “The discovery of squeezed states – In 1927,” in Proceed-
ings of the Fifth International Conference on Squeezed States and Uncer-
uncertainty relation. Many renowned physicists and math- tainty Relations, NASA Conference Publication NASA/CP-1998-206855,
ematicians were active during this period, and it is difficult to edited by D. Han, J. Janszky, Y. S. Kim, and V. I. Man’ko 共NASA,
know from their published work who first conceived the Washington, DC, 1998兲, pp. 175–180, ArXiv:quant-ph/9708012.
proof that Heisenberg used in his Chicago lectures.4 My fa- 13
V. A. Fock, Fundamentals of Quantum Mechanics, translated by E. Yank-
vorite statement of the proof that is close to the modern ovsky 共MIR, Moscow, 1978兲, pp. 111–112.
14
version appears in Fock’s 1931 text.13 Fock cites Weyl, W. Pauli, Pauli Lectures on Physics: Wave Mechanics, edited by Charles
P. Enz, translated by H. R. Lewis and S. Margulies 共MIT Press, Cam-
whose derivation is rigorous if not elegant, and Weyl ac- bridge, MA, 1973兲, p. 7. These lectures were delivered in 1956–57, but
knowledges Pauli, with whom Heisenberg had shared his drew on material in Pauli’s famous article, “Die allgemeinen Prinzipien
manuscript for Ref. 1 even before he showed it to Bohr. der Wellenmechanik,” Handbuch der Physik, Band 24/1 共Springer, Ber-
Pauli’s derivation in his lectures is very similar to lin, 1933兲.

587 Am. J. Phys., Vol. 76, No. 6, June 2008 John H. Marburger III 587