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Local Heisenberg inequalities
Volume
LOCAL HEISENBERG
30 September
PHYSICS LXl-l-ERS
111, number 7
1985
INEQUALITIES
J.-M. LEVY-LEBLOND Laboratoire de Physique Thskrique ‘, Uniuersitk de Nice, Parr V&me, Received 23 May 1985; accepted for publication
06034 Nice Cedex, France
15 July 1985
Heisenbarg-like inequalities are derived from commutator identities, linking the quadratic dispersion several functions of the position variable (or conversely). They have various physical and mathematical provide new links between the quadratic widths and the overall and mean peak widths of wave-functions.
Several works [l-4] recently have stressed that the standard Heisenberg inequalities are not the only way, nor, for that matters, the best one, to express the quantum constraints on the dispersions of the canonical variables, position and momentum. Indeed, the usual quadratic defmition of the spectral widths in position and momentum, inspired by the statistical definition of variance, often is too coarse, if it applies at all. Other defmitions of spectral widths have thus been proposed and Heisenberg-like inequalities proved for them [l-S] , In the present work, we stick to the conventional approach in which Heisenberg inequalities are derived from commutator identities, while generalizing it to obtain inequalities between the usual (quadratic) width in one of the canonical variables (position or momentum), and various functions of the other, Such “local Heisenberg inequalities” (according to Faris’ terminology [l] are much stronger than the usual ones. They also provide’a bridge between the new widths (overall width and mean peak width) of Uffink and Hilgevoord [2] , and the standard quadratic widths. We start from the well-known fact that a commutation relation [A,B] =iC,
(1)
linking three physical properties (hermitian operators) A, B, C of a quantum system, leads to the Heisenberg
in momentum to applications, and
inequality AA AB > $l
(2)
where AX is the standard quadratic width of X (AX)2 4 (X2) - (X>2.
(3)
The mean values are quantum expectation values in a given (arbitrary) state. The elementary Heisenberg inequality for a quanton in one dimension is obtained by taking A = P (momentum) and B = X (position). Let us now proceed to a simple generalization; we take A = P and B = F(X) where F is some arbitrary function. From the commutator expression [P, F(X)]
= --S’(X)
(4)
(we put ti = 1 throughout), it follows that APAF>#F?I.
(5)
We will consider three successive choices for the function F. (1) Let us take for the F the Heaviside function: F(x) = 0,
x
= 1,
x >x,.
(6)
Computing now the average values, we obtain: (F2) = (F> = f
Iq(x)12 dx = 1 - 9(x,),
(7)
X0
’ Equipe de Recherche Associ6e au CNRS.
0.375-9601/85/$03.30 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
where 9 is the wavefunction of the quanton, and 353
Volume 111, number 7
PHYSICS LEl’lTRS
3J(xo) is its probability of localization on the halfaxis (--,x0). Furthermore CF’) = (6(X - x0)> = lo?.
(8)
30 September 1985
function (the factor of 2 in (13) may be improved to fl if cpis symmetrical with respect to). (c) The overall width in position at level (Y,defined by PI
We finally obtain the interesting inequality lq(x)l* <2AP{3’(x)[l
-3’(x)]}l’*,
(9)
linking the probability density at some point to the probability itself and the momentum,quadratic width. The algebraic derivation given above must admittedly be supplemented with some analytical considerations in order to achieve full mathematical rigour; this can indeed be done, as shown by Faris [l] who proved a weaker form of (9), immediately deduced from it, due to the fact that 3(1 -9) Q l/4, so that: l~(x)I* < AP.
(10)
The significance of (9) (or (10)) is quite clear; for the
wavefunction to achieve a large value at some point, since it is normalized, it must be rather narrow enough, so that its momentum width, conversely, is necessarily large. Our new inequality goes far beyond Faris’ one (lo), for it shows how cpnecessarily vanishes when x + -00 (+=), since 3 + 0 (1). Furthermore, it proves that the upper bound lq(x)l* = AP may only be achieved for that unique value of x such that 3 = d (the case of the wavefunction q(x) = $ e-klxl which saturates (10) at x = 0, is interesting to study in that respect). Some consequences of the inequality (9) will be given now, without going into the details of the (simple) calculations: (a) by integration from (9), it can be shown that the function arcsin m - AP x is monotonously decreasing. In other words 9(x,) G sin* [arcsindm)
+ AP(x, - xl)] ,
x2 2x1.
(11)
(12)
which stands by itself as an interesting Heisenbergtype inequality for functions with bounded support. (b) From the standard Chebyshev inequality relating p(x) and AX, it follows that l&x)1* < 2AP AXIx - (X>I,
(13)
which is a most interesting local bound on the wave354
dx =a
, (14) I
obeys the following inequality (already given in ref. [41)WJX) > a(AP)-‘,
(15)
complementary to the inequality W,(X) Q 2( 1 +x)-l/* X AX. It may be mentioned, finally, that other inequalities, resulting in lower bounds for llpj*, result from taking F(x) = 1 for x1
F’ = -k sin kx = -kG,
G(x) = sin kx,
G’=kcoskx=kF,
(16)
whence the following pair of inequalities: (AP)* [(cos*kX) - (cos kX)*] > i k*(sin kX)*, (AP)* [(sin*kX) - (sin kX)*] > ik*(cos kX)*.
(17)
Upon addition, and noting that (cos kX)* t (sin kX)* = l(eRx)12,
(18)
we obtain
(AP)* (1 - I(eRX>j2) > $ k* I(eikx)12.
(19)
It has been shown that one may consider
This entails for the support of cp,the bound supp cpAP > a/2,
W,(X)= min Ix2 - x1 l ,j* rmr* ( I Xl
AiX & k-‘(1 - l(eikx)12)‘/*
(20)
as a measure of the position dispersion on scale k-l [5], which after (19) obeys the generalization of the ordinary Heisenberg inequality [(AP)* + 4 k*] (A;X)* > l/4
(21)
(note that AX = lirt~+.~ Ai X). Let us give other consequences of (19), due to the fact that (22)
Volume 111, number 7
where I?,is the Fourier transform of cp. (a) The following inequality holds, relating the autocorrelation of any function $ to its quadratic dispersion A: ISW
30 September 1985
PHYSICS LETTERS
t $(t + s) dt 2 < [l + (s/~A)~] -’
(23)
The inequality, applied to the radial coordinate and momentum in three dimensions, is sufficient - contrary to the ordinary Heisenberg inequality (see ref. [l]) - to show that the hydrogen atom is stable. Indeed, its hamiltonian H = (1 /2m)P2 - e2/R
(29)
(complementary to the well-known lower bound 1 &A)2, where A 1s ’ th e quadratic dispersion of $). (b) The mean peak widh in position at level 0, defined by [2] :
obeys, because of (28),
w,(X) = min I j$~lp(x
In conclusion, let us note that the inequalities (9), (ll), (12), (13), (15), (19), (21) (23) (25) (2% have their counterparts in the exchange P * X. Let us also stress that while the language used here is the one of quantum theory (wavefunctions, position and momentum, etc), the present results are nothing but exercises in the mathematics of Fourier transforms.
+ Z) dx = /3 ,
(I
(24)
I
obeys the following inequality wB(X) G 2p-‘(1 - /32)1’2AX
(25)
(complementary to wb(X) > (1 - p2)lj2 (AP)-l ; note that these inequalities together imply the standard Heisenberg one). (3) Consider fmrdly the case F(x) =x-l.
(26)
Uf> = (1/2m) (1/2m)U?-‘)2
- f&z-I>
> +ne?
(30)
This work was completed at Mishkenot Sha’ananim and the Van Leer Institute, Jerusalem, thanks to the hospitality of Professor Y. Elkana.
The general inequality (2) reads (AP)2 [(x-2> - (x+]
> $(x-l)?
(27)
We suppose in the following that both (X-2) and (x-l) are finite (see ref. El]). Then, from the trivial fact that a2(a - b)-l > 4b for a > b), it follows that AP > (x-l>.
(28)
References [I] W.G. Faris, J. Math. Phys. 19 (1978) 461. [2] J.B.M. Uffmk and J. Hilgevoord, Phys. Lett. 95A (1983) 474; 105A (1984) 216;Found. Phys., to be published. [3] I. Bialynicki-Bimla, Phys. Lett. 103A (1984) 253. [4] J.F. Price, Phys. Lett. 105A (1984) 343. [S] J.-M. Lbvy-Leblond, Ann. Phys. 101 (1976) 319.
355
LOCAL HEISENBERG
30 September
PHYSICS LXl-l-ERS
111, number 7
1985
INEQUALITIES
J.-M. LEVY-LEBLOND Laboratoire de Physique Thskrique ‘, Uniuersitk de Nice, Parr V&me, Received 23 May 1985; accepted for publication
06034 Nice Cedex, France
15 July 1985
Heisenbarg-like inequalities are derived from commutator identities, linking the quadratic dispersion several functions of the position variable (or conversely). They have various physical and mathematical provide new links between the quadratic widths and the overall and mean peak widths of wave-functions.
Several works [l-4] recently have stressed that the standard Heisenberg inequalities are not the only way, nor, for that matters, the best one, to express the quantum constraints on the dispersions of the canonical variables, position and momentum. Indeed, the usual quadratic defmition of the spectral widths in position and momentum, inspired by the statistical definition of variance, often is too coarse, if it applies at all. Other defmitions of spectral widths have thus been proposed and Heisenberg-like inequalities proved for them [l-S] , In the present work, we stick to the conventional approach in which Heisenberg inequalities are derived from commutator identities, while generalizing it to obtain inequalities between the usual (quadratic) width in one of the canonical variables (position or momentum), and various functions of the other, Such “local Heisenberg inequalities” (according to Faris’ terminology [l] are much stronger than the usual ones. They also provide’a bridge between the new widths (overall width and mean peak width) of Uffink and Hilgevoord [2] , and the standard quadratic widths. We start from the well-known fact that a commutation relation [A,B] =iC,
(1)
linking three physical properties (hermitian operators) A, B, C of a quantum system, leads to the Heisenberg
in momentum to applications, and
inequality AA AB > $l
(2)
where AX is the standard quadratic width of X (AX)2 4 (X2) - (X>2.
(3)
The mean values are quantum expectation values in a given (arbitrary) state. The elementary Heisenberg inequality for a quanton in one dimension is obtained by taking A = P (momentum) and B = X (position). Let us now proceed to a simple generalization; we take A = P and B = F(X) where F is some arbitrary function. From the commutator expression [P, F(X)]
= --S’(X)
(4)
(we put ti = 1 throughout), it follows that APAF>#F?I.
(5)
We will consider three successive choices for the function F. (1) Let us take for the F the Heaviside function: F(x) = 0,
x
= 1,
x >x,.
(6)
Computing now the average values, we obtain: (F2) = (F> = f
Iq(x)12 dx = 1 - 9(x,),
(7)
X0
’ Equipe de Recherche Associ6e au CNRS.
0.375-9601/85/$03.30 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
where 9 is the wavefunction of the quanton, and 353
Volume 111, number 7
PHYSICS LEl’lTRS
3J(xo) is its probability of localization on the halfaxis (--,x0). Furthermore CF’) = (6(X - x0)> = lo?.
(8)
30 September 1985
function (the factor of 2 in (13) may be improved to fl if cpis symmetrical with respect to
We finally obtain the interesting inequality lq(x)l* <2AP{3’(x)[l
-3’(x)]}l’*,
(9)
linking the probability density at some point to the probability itself and the momentum,quadratic width. The algebraic derivation given above must admittedly be supplemented with some analytical considerations in order to achieve full mathematical rigour; this can indeed be done, as shown by Faris [l] who proved a weaker form of (9), immediately deduced from it, due to the fact that 3(1 -9) Q l/4, so that: l~(x)I* < AP.
(10)
The significance of (9) (or (10)) is quite clear; for the
wavefunction to achieve a large value at some point, since it is normalized, it must be rather narrow enough, so that its momentum width, conversely, is necessarily large. Our new inequality goes far beyond Faris’ one (lo), for it shows how cpnecessarily vanishes when x + -00 (+=), since 3 + 0 (1). Furthermore, it proves that the upper bound lq(x)l* = AP may only be achieved for that unique value of x such that 3 = d (the case of the wavefunction q(x) = $ e-klxl which saturates (10) at x = 0, is interesting to study in that respect). Some consequences of the inequality (9) will be given now, without going into the details of the (simple) calculations: (a) by integration from (9), it can be shown that the function arcsin m - AP x is monotonously decreasing. In other words 9(x,) G sin* [arcsindm)
+ AP(x, - xl)] ,
x2 2x1.
(11)
(12)
which stands by itself as an interesting Heisenbergtype inequality for functions with bounded support. (b) From the standard Chebyshev inequality relating p(x) and AX, it follows that l&x)1* < 2AP AXIx - (X>I,
(13)
which is a most interesting local bound on the wave354
dx =a
, (14) I
obeys the following inequality (already given in ref. [41)WJX) > a(AP)-‘,
(15)
complementary to the inequality W,(X) Q 2( 1 +x)-l/* X AX. It may be mentioned, finally, that other inequalities, resulting in lower bounds for llpj*, result from taking F(x) = 1 for x1
F’ = -k sin kx = -kG,
G(x) = sin kx,
G’=kcoskx=kF,
(16)
whence the following pair of inequalities: (AP)* [(cos*kX) - (cos kX)*] > i k*(sin kX)*, (AP)* [(sin*kX) - (sin kX)*] > ik*(cos kX)*.
(17)
Upon addition, and noting that (cos kX)* t (sin kX)* = l(eRx)12,
(18)
we obtain
(AP)* (1 - I(eRX>j2) > $ k* I(eikx)12.
(19)
It has been shown that one may consider
This entails for the support of cp,the bound supp cpAP > a/2,
W,(X)= min Ix2 - x1 l ,j* rmr* ( I Xl
AiX & k-‘(1 - l(eikx)12)‘/*
(20)
as a measure of the position dispersion on scale k-l [5], which after (19) obeys the generalization of the ordinary Heisenberg inequality [(AP)* + 4 k*] (A;X)* > l/4
(21)
(note that AX = lirt~+.~ Ai X). Let us give other consequences of (19), due to the fact that (22)
Volume 111, number 7
where I?,is the Fourier transform of cp. (a) The following inequality holds, relating the autocorrelation of any function $ to its quadratic dispersion A: ISW
30 September 1985
PHYSICS LETTERS
t $(t + s) dt 2 < [l + (s/~A)~] -’
(23)
The inequality, applied to the radial coordinate and momentum in three dimensions, is sufficient - contrary to the ordinary Heisenberg inequality (see ref. [l]) - to show that the hydrogen atom is stable. Indeed, its hamiltonian H = (1 /2m)P2 - e2/R
(29)
(complementary to the well-known lower bound 1 &A)2, where A 1s ’ th e quadratic dispersion of $). (b) The mean peak widh in position at level 0, defined by [2] :
obeys, because of (28),
w,(X) = min I j$~lp(x
In conclusion, let us note that the inequalities (9), (ll), (12), (13), (15), (19), (21) (23) (25) (2% have their counterparts in the exchange P * X. Let us also stress that while the language used here is the one of quantum theory (wavefunctions, position and momentum, etc), the present results are nothing but exercises in the mathematics of Fourier transforms.
+ Z) dx = /3 ,
(I
(24)
I
obeys the following inequality wB(X) G 2p-‘(1 - /32)1’2AX
(25)
(complementary to wb(X) > (1 - p2)lj2 (AP)-l ; note that these inequalities together imply the standard Heisenberg one). (3) Consider fmrdly the case F(x) =x-l.
(26)
Uf> = (1/2m)
- f&z-I>
> +ne?
(30)
This work was completed at Mishkenot Sha’ananim and the Van Leer Institute, Jerusalem, thanks to the hospitality of Professor Y. Elkana.
The general inequality (2) reads (AP)2 [(x-2> - (x+]
> $(x-l)?
(27)
We suppose in the following that both (X-2) and (x-l) are finite (see ref. El]). Then, from the trivial fact that a2(a - b)-l > 4b for a > b), it follows that AP > (x-l>.
(28)
References [I] W.G. Faris, J. Math. Phys. 19 (1978) 461. [2] J.B.M. Uffmk and J. Hilgevoord, Phys. Lett. 95A (1983) 474; 105A (1984) 216;Found. Phys., to be published. [3] I. Bialynicki-Bimla, Phys. Lett. 103A (1984) 253. [4] J.F. Price, Phys. Lett. 105A (1984) 343. [S] J.-M. Lbvy-Leblond, Ann. Phys. 101 (1976) 319.
355