Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 7, Issue 3, Article 80, 2006
ON THE SHARPENED HEISENBERG-WEYL INEQUALITY
JOHN MICHAEL RASSIAS
P EDAGOGICAL D EPARTMENT, E. E.
NATIONAL AND C APODISTRIAN U NIVERSITY OF ATHENS
S ECTION OF M ATHEMATICS AND I NFORMATICS
4, AGAMEMNONOS S TR ., AGHIA PARASKEVI
ATHENS 15342, G REECE
jrassias@primedu.uoa.gr
URL: http://www.primedu.uoa.gr/∼jrassias/
Received 21 June, 2005; accepted 21 July, 2006
Communicated by S. Saitoh
A BSTRACT. The well-known second order moment Heisenberg-Weyl inequality (or uncertainty
relation) in Fourier Analysis states: Assume that f : R → C is a complex valued function
of a random real variable x such that f ∈ L2 (R). Then the product of the second moment
2
2
of the random real x for |f | and the second moment of the random real ξ for fˆ is at least
.
R
E|f |2 4π, where fˆ is the Fourier transform of f , such that fˆ (ξ) = R e−2iπξx f (x) dx, f (x) =
R 2iπξx
R
2
e
fˆ (ξ) dξ, and E|f |2 = R |f (x)| dx.
R
This uncertainty relation is well-known in classical quantum mechanics. In 2004, the author
generalized the afore-mentioned result to higher order moments and in 2005, he investigated
a Heisenberg-Weyl type inequality without Fourier transforms. In this paper, a sharpened form
of this generalized Heisenberg-Weyl inequality is established in Fourier analysis. Afterwards,
an open problem is proposed on some pertinent extremum principle.These results are useful in
investigation of quantum mechanics.
Key words and phrases: Sharpened, Heisenberg-Weyl inequality, Gram determinant.
2000 Mathematics Subject Classification. 26, 33, 42, 60, 52.
1. I NTRODUCTION
The serious question of certainty in science was high-lighted by Heisenberg, in 1927, via his
uncertainty principle [1]. He demonstrated the impossibility of specifying simultaneously the
position and the speed (or the momentum) of an electron within an atom. In 1933, according
to Wiener [7] a pair of transforms cannot both be very small. This uncertainty principle was
stated in 1925 by Wiener, according to Wiener’s autobiography [8, p. 105-107], at a lecture in
Göttingen. The following result of the Heisenberg-Weyl Inequality is credited to Pauli according
to Weyl [6, p. 77, p. 393-394]. In 1928, according to Pauli [6] the less the uncertainty in |f |2 ,
ISSN (electronic): 1443-5756
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188-05
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J OHN M ICHAEL R ASSIAS
2
the greater the uncertainty in fˆ , and conversely. This result does not actually appear in
Heisenberg’s seminal paper [1] (in 1927). The following second order moment HeisenbergWeyl inequality provides a precise quantitative formulation of the above-mentioned uncertainty
principle according to W. Pauli.
1.1. Second Order Moment Heisenberg-Weyl Inequality ([3, 4, 5]): For any f ∈ L2 (R), f :
R → C, such that
Z
kf k22 =
|f (x)|2 dx = E|f |2 ,
R
any fixed but arbitrary constants xm , ξm ∈ R, and for the second order moments
Z
2
(µ2 )|f |2 = σ|f |2 =
(x − xm )2 |f (x)|2 dx,
ZR
2
(µ2 ) fˆ 2 = σ 2 ˆ 2 =
(ξ − ξm )2 fˆ(ξ) dξ,
| |
|f |
R
the second order moment Heisenberg-Weyl inequality
kf k42
(H1 )
·
≥
,
16π 2
holds. Equality holds in (H1 ) if and only if the generalized Gaussians
f (x) = c0 exp (2πixξm ) exp −c (x − xm )2
2
σ|f
|2
σ2ˆ 2
|f |
hold for some constants c0 ∈ C and c > 0.
1.2. Fourth Order Moment Heisenberg-Weyl
Inequality ([3, pp. 26-27]): For any f ∈
R
2
2
2
L (R), f : R → C, such that kf k2 = R |f (x)| dx = E|f |2 , any fixed but arbitrary constants
xm , ξm ∈ R, and for the fourth order moments
Z
(x − xm )4 |f (x)|2 dx
and
(µ4 )|f |2 =
R
Z
2
(µ4 ) fˆ 2 =
(ξ − ξm )4 fˆ(ξ) dξ,
| |
R
the fourth order moment Heisenberg-Weyl inequality
1
2
(µ4 )|f |2 · (µ4 ) fˆ 2 ≥
E2,f
,
4
| |
64π
(H2 )
holds, where
E2,f = 2
Z h
2 2
(1−4π 2 ξm
xδ ) |f (x)|2
−
x2δ
0
2
|f (x)| −
i
4πξm x2δ Im(f (x)f 0 (x))
dx,
R
with xδ = x − xm , ξδ = ξ − ξm , Im (·) is the imaginary part of (·), and |E2,f | < ∞.
The “inequality” (H2 ) holds, unless f (x) = 0.
We note that if the ordinary differential equation of second order
(ODE)
fα00 (x) = −2c2 x2δ fα (x)
holds, with α = −2πξm i, fα (x) = eαx f (x), and a constant c2 = 21 k22 > 0, k2 ∈ R and k2 6=
0,then “equality” in (H2 ) seems to occur. However, the solution of this differential equation
(ODE), given by the function
p
1
1
2πixξm
2
2
f (x) = |xδ |e
c20 J−1/4
|k2 | xδ + c21 J1/4
|k2 | xδ
,
2
2
J. Inequal. Pure and Appl. Math., 7(3) Art. 80, 2006
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O N T HE S HARPENED H EISENBERG -W EYL I NEQUALITY
3
in terms of the Bessel functions J±1/4 of the first kind of orders ±1/4, leads to a contradiction,
because this f ∈
/ L2 (R). Furthermore, a limiting argument is required for this problem. For the
proof of this inequality see [3].
It is open to investigate cases, where the integrand on the right-hand side of integral of
E2,f will be nonnegative. For instance, for xm = ξm = 0, this integrand is: = |f (x)|2 −
x2 |f 0 (x)|2 (≥ 0). In 2004, we ([3, 4]) generalized the Heisenberg-Weyl inequality and in 2005
we [5] investigated a Heisenberg-Weyl type inequality without Fourier transforms. In this paper, a sharpened form of this generalized Heisenberg-Weyl inequality is established in Fourier
analysis. We state our following two pertinent propositions. For their proofs see [3].
Proposition 1.1 (Generalized differential
identity, [3]). If f : R → C is a complex valued
k
dj
function of a real variable x, 0 ≤ 2 is the greatest integer ≤ k2 , f (j) = dx
j f , and (·) is the
conjugate of (·), then
[ k2 ]
X
2
dk−2i (i)
f (x) ,
k−2i
dx
i=0
holds for any fixed but arbitrary k ∈ N = {1, 2, . . .}, such that 0 ≤ i ≤ k2 for i ∈ N0 =
{0, 1, 2, . . .}.
(*)
f (x) f
(k)
(x) + f
(k)
(x) f¯ (x) =
k
(−1)
k−i
i
k−i
i
Proposition 1.2 (Lagrange type differential identity, [3]). If f : R →√C is a complex valued
function of a real variable x, and fa = eax f , where a = −βi, with i = −1 and β = 2πξm for
any fixed but arbitrary real constant ξm , as well as if
p 2
Apk =
β 2(p−k) , 0 ≤ k ≤ p,
k
and
p p
β 2p−j−k , 0 ≤ k < j ≤ p,
Bpkj = spk
k
j
where spk = (−1)p−k (0 ≤ k ≤ p), then
(LD)
p
X
2
(p) 2 X
fa =
Apk f (k) + 2
Bpkj Re rpkj f (k) f (j) ,
k=0
0≤khj≤p
holds for any fixed but arbitrary p ∈ N0 , where (·) is the conjugate of (·), and rpkj = (−1)p−
(0 ≤ k < j ≤ p), and Re (·) is the real part of (·).
k+j
2
2. S HARPENED H EISENBERG -W EYL I NEQUALITY
We assume that f : R → C is a complex valued function of a real variable x (or absolutely
continuous in [−a, a], a > 0), and w : R → R a real valued weight function of x, as well as
√ xm ,
ξm any fixed but arbitrary real constants. Denote fa = eax f , where a = −2πξm i with i = −1,
and fˆ the Fourier transform of f , such that
Z
Z
−2iπξx
ˆ
f (ξ) =
e
f (x) dx and f (x) =
e2iπξx fˆ(ξ) dξ.
R
R
Also we denote
Z
w2 (x) (x − xm )2p |f (x)|2 dx,
ZR
2
(µ2p ) fˆ 2 =
(ξ − ξm )2p fˆ(ξ) dξ
| |
R
(µ2p )w,|f |2 =
J. Inequal. Pure and Appl. Math., 7(3) Art. 80, 2006
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4
J OHN M ICHAEL R ASSIAS
the 2pth weighted
moment of x for |f |2 with weight function w : R → R and the 2pth moment
2
of ξ for fˆ , respectively. In addition, we denote
hpi p−q
p
p
q
, if 0 ≤ q ≤
Cq = (−1)
= the greatest integer ≤
,
p−q
q
2
2
Z
hpi
(l)
2
p−2q
(p−2q)
Iql = (−1)
wp
(x) f (x) dx, if 0 ≤ l ≤ q ≤
,
2
ZR
hpi
p−2q
(p−2q)
(k)
(j)
Iqkj = (−1)
wp
(x) Re rqkj f (x) f (x) dx, if 0 ≤ k < j ≤ q ≤
,
2
R
k+j
where rqkj = (−1)q− 2 ∈ {±1, ±i} and wp = (x − xm )p w. We assume that all these integrals
exist. Finally we denote
Dq =
q
X
Aql Iql + 2
l=0
X
Bqkj Iqkj ,
0≤khj≤q
if |Dq | < ∞ holds for 0 ≤ q ≤ p2 , where
q 2
q q 2(q−l)
Aql =
β
,
Bqkj = sqk
β 2q−j−k ,
l
k
j
P
with β = 2πξm , and sqk = (−1)q−k , and Ep,f = [p/2]
q=0 Cq Dq , if |Ep,f | < ∞ holds for p ∈ N.
In addition, we assume the two conditions:
p−2q−1
X
(2.1)
2 (p−2q−r−1)
(−1)r lim wp(r) (x) f (l) (x)
= 0,
|x|→∞
r=0
for 0 ≤ l ≤ q ≤
p
, and
2
p−2q−1
(2.2)
X
(p−2q−r−1)
= 0,
(−1)r lim wp(r) (x) Re rqkj f (k) (x) f (j) (x)
|x|→∞
r=0
for 0 ≤ k < j ≤ q ≤
p
. Also,
2
∗
|Ep,f
|=
q
2
Ep,f
+ 4A2 (≥ |Ep,f |),
R
R
where A = kuk x0 − kvk y0 , with L2 −norm k·k2 = R |·|2 , inner product (|u| , |v|) = R |u| |v|,
and
u = w(x)xpδ fα (x),
v = fα(p) (x);
Z
Z
x0 =
|ν(x)| |h(x)| dx,
y0 =
|u(x)| |h(x)| dx,
R
R
as well as
2
1
− 1 x−µ
√ e 4( σ ) ,
2π σ
where µ is the mean and σ the standard deviation, or
s
Γ n+1
1
1
2
,
·
h(x) = √
n+1
n
4
nπ
Γ 2
x2
4
1+ n
h(x) = √
4
J. Inequal. Pure and Appl. Math., 7(3) Art. 80, 2006
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O N T HE S HARPENED H EISENBERG -W EYL I NEQUALITY
5
where n ∈ N, and
Z
2
kh(x)k =
|h(x)|2 dx = 1.
R
Theorem 2.1. If f ∈ L2 (R) (or absolutely continuous in [−a, a], a > 0), then
q
q
1 q
p ∗
∗ 2p
2p
√
(µ2p )w,|f |2 (µ2p ) fˆ 2 ≥
E
(Hp )
p,f ,
| |
2π p 2
holds for any fixed but arbitrary p ∈ N.
Equality holds in (Hp∗ ) iff v(x) = −2cp u(x) holds for constants cp > 0, and any fixed but
arbitrary p ∈ N; cp = kp2 /2 > 0, kp ∈ R and kp 6= 0, p ∈ N, and A = 0, or
h(x) = c1p u(x) + c2p v(x)
and x0 = 0, or y0 = 0,where cip (i = 1, 2) are constants and A2 > 0.
Proof. In fact, from the generalized Plancherel-Parseval-Rayleigh identity [3, (GPP)], and the
fact that |eax | = 1 as a = −2πξm i, one gets
(2.3) Mp∗ = Mp −
(2.4)
1
A2
2p
(2π)
1
A2
= (µ2p )w,|f |2 · (µ2p ) fˆ 2 −
| |
(2π)2p
Z
Z
2 1
2p
2
2p ˆ
2
=
w (x) (x − xm ) |f (x)| dx ·
(ξ − ξm ) f (ξ) dξ −
A2
2p
(2π)
R
R
Z
Z
1
2
2p
2
2
(p)
2
=
w (x) (x − xm ) |fa (x)| dx ·
fa (x) dx − A
(2π)2p
R
R
1 =
kuk2 kvk2 − A2
2p
(2π)
(p)
with u = w(x)xpδ fα (x), v = fα (x).
From (2.3) – (2.4), the Cauchy-Schwarz inequality (|u| , |v|) ≤ kuk kvk and the non-negativeness
of the following Gram determinant [2]
kuk2
(|u| , |v|) y0 (2.5)
0 ≤ (|v| , |u|) kvk2
x0 y0
x0
1 = kuk2 kvk2 − (|u| , |v|)2 − kuk2 x20 − 2(|u| , |v|)x0 y0 + kvk2 y02 ,
0 ≤ kuk2 kvk2 − (|u| , |v|)2 − A2
with
Z
A = kuk x0 − kvk y0 , x0 =
Z
|ν(x)| |h(x)| dx, y0 =
R
|u(x)| |h(x)| dx,
R
and
2
Z
kh(x)k =
|h(x)|2 dx = 1,
R
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6
J OHN M ICHAEL R ASSIAS
we find
1
(|u| , |v|)2
(2π)2p
2
Z
1
|u| |v|
=
(2π)2p
R
Z
2
1
(p)
=
wp (x) fa (x) fa (x) dx ,
(2π)2p
R
Mp∗ ≥
(2.6)
where wp = (x − xm )p w, and fa = eax f . In general, if khk =
6 0, then one gets
(u, v)2 ≤ kuk2 kvk2 − R2 ,
where R = A/ khk = kuk x − kvk y, such that x = x0 / khk , y = y0 / khk.
In this case, A has to be replaced by R in all the pertinent relations
of this paper.
1
From (2.6) and the complex inequality, |ab| ≥ 2 ab + ab with a = wp (x) fa (x), b =
(p)
fa (x), we get
Mp∗
(2.7)
Z
2
1
1
(p)
(p)
=
wp (x) fα (x)fα (x) + fα (x)fα (x) dx .
(2π)2p 2 R
From (2.7) and the generalized differential identity (*), one finds
Mp∗ ≥
(2.8)
1
22(p+1) π 2p


 2
Z
[p/2]
p−2q X
d
2
 wp (x) 
Cq p−2q fa(q) (x)  dx .
dx
q=0
R
From (2.8) and the Lagrange type differential identity (LD), we find
Mp∗
≥
1
22(p+1) π 2p


Z
[p/2]
X
dp−2q
 wp (x) 
Cq p−2q
dx
R
q=0
+2
q
X
2
Aql f (l) (x)
l=0
X
 2
Bqkj Re rqkj f (k) (x) f (j) (x)  dx .
0≤khj≤q
From the generalized integral identity [3], the two conditions (2.1) – (2.2), and that all the
integrals exist, one gets
Z
Z
2
2
dp−2q (l)
p−2q
wp (x) p−2q f (x) dx = (−1)
wp(p−2q) (x) f (l) (x) dx = Iql ,
dx
R
R
as well as
Z
dp−2q
wp (x) p−2q Re rqkj f (k) (x) f (j) (x)
dx
R
Z
p−2q
= (−1)
wp(p−2q) (x)Re rqkj f (k) (x) f (j) (x) = Iqkj .
R
J. Inequal. Pure and Appl. Math., 7(3) Art. 80, 2006
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O N T HE S HARPENED H EISENBERG -W EYL I NEQUALITY
7
Thus we find

Mp∗ ≥
=
where Ep,f =
P[p/2]
q=0
[p/2]
1
X
22(p+1) π 2p
1
22(p+1) π 2p


Cq 
q=0
q
X
2
X
Aql Iql + 2
l=0
Bqkj Iqkj 
0≤khj≤q
2
,
Ep,f
Cq Dq , if |Ep,f | < ∞ holds, or the sharpened moment uncertainty formula
q
p
1 q
1 p
p 2p
∗
√
√
Mp ≥
|Ep,f | ,
Ep,f
≥
2π p 2
2π p 2
1
2
where Mp = Mp∗ + (2π)
2p A .
We note that the corresponding Gram matrix to the above Gram determinant is positive definite if and only if the above Gram determinant is positive if and only if u, v, h are linearly
independent. Besides, the equality in (2.5) holds if and only if h is a linear combination of linearly independent u and v and u = 0 or v = 0, completing the proof of the above theorem. Let
Z
(m2p )|f |2 =
x2p |f (x)|2 dx
R
th
2
be the 2p moment of x for |f | about the origin xm = 0, and
Z
2
2p ˆ
(m2p ) fˆ 2 =
ξ f (ξ) dξ
| |
R
2
the 2pth moment of ξ for fˆ about the origin ξm = 0. Denote
p
p!
p−q
p−q
εp,q = (−1)
·
,
p − q (2q)!
q
if p ∈ N and 0 ≤ q ≤ p2 .
Corollary 2.2. Assume that f : R → C is a complex valued function of a real variable x,
w = 1, xm = ξm = 0, and fˆ is the Fourier transform of f , described in our theorem. If
f : R → C (or absolutely continuous in [−a, a], a > 0), then the following inequality
v
2
u
u[p/2]
q
X
q
1 u
p
2p
2
t
√
2 + 4A ,
(Sp )
(m2p )|f |2 2p (m2p ) fˆ 2 ≥
ε
(m
)
p,q
2q
|f (q) | | |
2π p 2 q=0
holds for any fixed but arbitrary p ∈ N and 0 ≤ q ≤ p2 , where
Z
2
(m2q ) f (q) 2 =
x2q f (q) (x) dx
| |
R
and A is analogous to the one in the above theorem.
We consider the extremum principle (via (9.33) on p. 51 of [3]):
1
,
p∈N
2π
for the corresponding “inequality” (Hp ) [3, p. 22], p ∈ N.
(R)
J. Inequal. Pure and Appl. Math., 7(3) Art. 80, 2006
R(p) ≥
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8
J OHN M ICHAEL R ASSIAS
Problem 2.1. Employing our Theorem 8.1 on p. 20 of [3], the Gaussian function, the Euler
gamma function Γ, and other related special functions, we established and explicitly proved the
above extremum principle (R), where
Γ p + 21
,
R(p) = P
[p/2]
p−q
p
p!
· (2q)!
Γq q=0 (−1)p−q p−q
q
with
[q/2]
X
q 2 1
1
2
2
Γ k+
Γ 2q − 2k +
Γq =
2
2
2k
k=0
q q X
k+j k+j
+2
(−1) 2
2k
2j
0≤k≤j≤[q/2]
1
1
1
×Γ k+
Γ j+
Γ 2q − k − j +
,
2
2
2
q
q
p!
0 ≤ 2 is the greatest integer ≤ 2 for q ∈ N ∪ {0} = N0 , pq = q!(p−q)!
for p ∈ N, q ∈ N0
and 0 ≤ q ≤ p, p! = 1 · 2 · 3 · · · · · (p − 1) · p and 0! = 1, as well as
√
1 (2p)! √
1
1
Γ p+
= 2p ·
π, p ∈ N and Γ
= π.
2
2
p!
2
2k
Furthermore, by employing computer techniques, this principle was verified for p = 1, 2, 3, . . .,
32, 33, as well. It now remains open to give a second explicit proof of verification for the extremum principle (R) using only special functions techniques and without applying our HeisenbergPauli-Weyl inequality [3].
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J. Inequal. Pure and Appl. Math., 7(3) Art. 80, 2006
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