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Phase-space Fisher information
Chemical Physics Letters 437 (2007) 132–137 www.elsevier.com/locate/cplett
Phase-space Fisher information ´ . Nagy I. Hornya´k, A
*
Department of Theoretical Physics, University of Debrecen, and Atomic and Molecular Physics Research Group of the Hungarian Academy of Sciences, H-4010 Debrecen, Hungary Received 3 January 2007; in final form 26 January 2007 Available online 3 February 2007
Abstract The concept of phase-space Fisher information, F(q, p), is introduced and a theorem, stating that this quantity is not less than the sum of the position space and the momentum space Fisher information, is proved. Applications are made to the hydrogen-like atoms and the isotropic harmonic oscillator. An uncertainty-like relation for the Fisher information sum is also presented. 2007 Elsevier B.V. All rights reserved.
1. Introduction Information theoretical concepts have a role of growing importance in studying quantum mechanical systems. Shannon information [1] has been applied in physics for a long time and recently Fisher information [2] has also attracted special attention. Fisher information (Finf) was originally introduced by Fisher as a measure of ‘intrinsic accuracy’ in statistical estimation theory. Sears et al. [3] were the first who noticed the importance of Finf [2] in quantum mechanics and density functional theory (DFT) more than 20 years ago. They argued that the quantum mechanical kinetic energy can be considered a measure of the information distribution. Since then Finf has proved to be a very useful concept, e.g. the equations of non-relativistic quantum mechanics [4] have been derived using the principle of minimum Finf [5]. The time-independent Kohn–Sham equations and the time-dependent Euler equation of DFT were also derived by applying the above principle [6,7]. Other applications include single-particle systems with a central potential [8], two-electron ‘entangled artificial’ atom proposed by Moshinsky [9]. In a recent Letter, atomic Finf [10] has also been investigated.
*
Corresponding author. ´ . Nagy). E-mail address: [email protected] (A
0009-2614/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2007.01.084
Phase-space representation of quantum mechanics enjoys a renewed interest in many branches of physics (e.g. solid-state physics, quantum optics, quantum chaos, quantum chemistry). Here we cite only some important papers [11–26]. In this Letter, the concept of phase-space Finf is introduced. In this respect we prove that the phase-space Finf is larger or equal to the sum of the position space and the momentum space Finf. Elucidating applications are made for the hydrogen-like atoms and the isotropic harmonic oscillator. An uncertainty-like relationship for the Finf sum is presented. The Letter, is organized as follows: in Section 2 phasespace Finf is introduced. An inequality for the marginal configuration and momentum Fisher information is derived in Section 3. The last section is devoted to illustrative examples, discussion and the presentation of uncertainty-like relationship for the Finf sum. 2. Phase-space distribution functions It was Wigner [11] who first realized the possibility of formulating quantum mechanics in the phase-space. He examined a distribution function F(q, p) which yields the proper quantum mechanical marginal distributions when either one of the variables is integrated over: Z F ðq; pÞ dp ¼ jwðqÞj2 ¼ .ðqÞ; ð1Þ
I. Hornya´k, A´. Nagy / Chemical Physics Letters 437 (2007) 132–137
and Z F ðq; pÞ dq ¼ j/ðpÞj2 ¼ cðpÞ;
ð2Þ
where w(q) and /(p) are the configuration and momentum wave functions, respectively and .(q) and c(p) are the configuration-space and momentum-space densities, respectively. Both are normalized to 1. The Wigner function cannot be considered a proper probability distribution, because it may assume negative values. Since Wigner’s original Letter, several other distribution functions have been found which satisfy conditions (1) and (2). However, these may take on not only negative but also imaginary values. Wigner proved [12] that positive distributions which are bilinear in the wave function do not exist. Later, Cohen and Zaparovanny [27] proved that positive distributions satisfying conditions (1) and (2) exist, though these are not bilinear in the wave function. They also showed that the set of all possible functions of q and p consistent with (1) and (2) have the form Z 1 F ðq; pÞ ¼ 2 expðihq isp þ ihuÞf ðh; sÞ 4p
w ðu s=2Þwðu þ s=2Þ dh ds du; where f(h, s) satisfies the conditions f ðh; 0Þ ¼ f ð0; sÞ ¼ 1; By taking f = 1 we arrive at the Wigner distribution. By means of the joint probability density h(v, w) of random variables v, w defined over the range (0, 1) one can define the probability function H(q, p) as follows
1
wðpÞ ¼
The Shannon information [1] can be defined both in the configuration space Z S . ¼ .ðqÞ ln .ðqÞ dq and in the momentum space Z S c ¼ cðpÞ ln cðpÞ dp: The sum of the position and momentum Shannon information has turned to be a very important quantity and one can show that the following uncertainty relation [29] S . þ S c P ð1 þ ln pÞ holds. Using the positive quantum distribution F(q, p) the phase-space Shannon information can be defined as Z S F ¼ F ðq; pÞ ln F ðq; pÞ dq dp: This is the Leipnik entropy [28]. From the subadditivity of the Shannon information one can obtain the inequality [34]
where p(xjh) is a probability density function depending on a parameter h and obeying proper regularity conditions. If h is a parameter of locality then
p
cðp0 Þ dp0 :
1
pðxjhÞ ¼ pðx þ hÞ;
The positive quantum distributions have the form F ðq; pÞ ¼ .ðqÞcðpÞ½1 þ cvðq; pÞ;
3. Phase-space Fisher information
Let us turn to the Finf. The Finf functional [2] is defined as 2 Z Z 0 2 o ln pðxjhÞ ½p ðxjhÞ F I ðhÞ ¼ pðxjhÞ dx; ð4Þ dx ¼ oh pðxjhÞ
The functions v(q) and w(p) are taken to be Z q vðqÞ ¼ .ðq0 Þ dq0 ; Z
writing the distribution function F in terms of h(v, w) is that the marginals will always be satisfied by any h(v, w). The class of positive quantum distributions are determined by the function h(v, w). By selecting different probability function h(v, w) one can get different positive quantum distributions F(q, p).
SF P S. þ Sc:
H ðq; pÞ ¼ .ðqÞcðpÞhðv; wÞ:
and
133
ð3Þ
where vðq; pÞ ¼ hðvðqÞ; wðpÞÞ h1 ðvðqÞÞ h2 ðwðpÞÞ þ 1 and h1, h2 are the marginal distributions of H(q, p) defined by the relations: Z H ðq; pÞ dp ¼ h1 ðvÞ.ðqÞ ¼ H 1 ðqÞ; and Z H ðq; pÞ dq ¼ h2 ðwÞcðpÞ ¼ H 2 ðpÞ: Cohen and Zaparovanny [27] showed that c can be chosen to make F positive. They emphasized that the advantage of
and Finf (Eq. (4)) has the form 2 Z opðx þ hÞ I F ðhÞ ¼ pðx þ hÞ dx: oðx þ hÞ This is Finf per observation with respect to the locality parameter h. As the expression does not depend on h, we may set the locality at zero: Z 0 2 ½p ðxÞ I F ðh ¼ 0Þ ¼ dx: pðxÞ In general x can be vector-valued and in this case Finf has the form Z 2 ½rpðxÞ F dx: I ðh ¼ 0Þ ¼ pðxÞ
I. Hornya´k, A´. Nagy / Chemical Physics Letters 437 (2007) 132–137
134
This locality Finf is called intrinsic accuracy and measures the ‘narrowness’ of a distribution. For the normal distribution it is equal to the inverse variance. When the parameter is one of locality Finf characterizes the behavior of the distribution itself. Let us define now the phase-space Finf as Z 2 ½rF ðp; qÞ I FF ¼ dq dp F ðq; pÞ " 2 2 # Z 1 oF ðp; qÞ oF ðp; qÞ ¼ dq dp: ð5Þ þ F ðq; pÞ oq op The position and momentum space Finf can be written as 2 Z 1 d.ðqÞ I F. ¼ dq .ðqÞ dq
for any q. Consequently Z cðpÞvðq; pÞ dp ¼ 0
ð10Þ
with the exception of points q, where .(q) vanishes. Similarly, we can immediately obtain Z .ðqÞvðq; pÞ dq ¼ 0 ð11Þ with the exception of points p, where c(p) vanishes. Next, we shall show that I0 = 0. After partial integration with respect to variable q, the first term of I0 takes the form 1 Z Z d.ðqÞ d2 .ðqÞ dpcðpÞvðq; pÞ 2c dq 2c dq dq2 1 Z dpcðpÞvðq; pÞ
and I Fc ¼
Z
2 1 dcðpÞ dp: cðpÞ dp
and vanishes because of Eq. (10). A partial integration of the second term of I0 for the variable p, leads also to 0 after using Eq. (11). From Eq. (8) we obtain
Theorem 1. The following inequality holds: I FF P I F. þ I Fc :
ð6Þ
I FF ¼ I F. þ I Fc " 2 2 # Z 2 c .ðqÞcðpÞ d.ðqÞ dcðpÞ þ þ dq dp: 1 þ cvðq; pÞ dq dp
There is equality if F ðq; pÞ ¼ .ðqÞcðpÞ:
ð12Þ ð7Þ
Proof. By substituting the explicit form of the positive quantum distribution F(q, p) (Eq. (3)) into the expression of the phase-space Finf (5) we obtain I FF ¼
2 Z 1 d.ðqÞ 1 þ c cðpÞvðq; pÞ dp dq .ðqÞ dq 2 Z Z 1 dcðpÞ þ 1 þ c .ðqÞvðq; pÞ dq dp þ I 0 cðpÞ dp " 2 2 # Z 2 c .ðqÞcðpÞ d.ðqÞ dcðpÞ dq dp; þ þ 1 þ cvðq; pÞ dq dp Z
As the last term of Eq. (12) is positive, we immediately arrive at the inequality (6). It can also be seen from Eq. (12) that we can obtain equality in (6) if and only if c = 0. The above theorem and its proof can be straightforwardly generalized for systems of any degree of freedom. h 4. Illustrative examples and discussion
ð8Þ where
Z Z Z d.ðqÞ ovðq; pÞ dcðpÞ I 0 ¼ 2c dq dpcðpÞ þ 2c dp dq oq dp Z ovðq; pÞ : ð9Þ dq.ðqÞ op Substituting the form of positive quantum distribution F(q, p) (Eq. (3)) into the left-hand side of Eq. (1) Z Z F ðq; pÞ dp ¼ .ðqÞ cðpÞ½1 þ cvðq; pÞ dp and comparing it with the right-hand side of Eq. (1) we are led to the expression Z c.ðqÞ cðpÞvðq; pÞ dp ¼ 0
Electronic wave functions for atoms and molecules may be quite complicated and difficult to understand or interpret. phase-space information concepts may provide appreciable advantage in this respect. The Shannon information sum has already proved to be useful. Gadre and Bendale [30] argued that the maximization of entropy in one space would lead to an ‘imbalance’ with the conjugate space. Therefore, they proposed the Shannon information sum. For hydrogen atom, Gadre et al. [31] showed that S. is minimum for the ground state and increases monotonically with the excited-state energy, whereas the momentum space Shannon information Sc decreases monotonically. They also found that the Shannon information sum assumes its minimum value for the ground state. For the linear harmonic oscillator, however, they noticed that the Shannon information of both spaces and hence their sum increase with the quantum number n. A recent Letter, [32] presented position space and momentum space Finf of single-particle systems with a central potential. Romera et al. have also found that the product of the position space and the momentum space Finf is
I. Hornya´k, A´. Nagy / Chemical Physics Letters 437 (2007) 132–137
also a useful quantity satisfying important uncertainty relation. In this Letter, we have shown that the sum of the position space and the momentum space Finf is also a meaningful quantity. We have defined a whole class of phase-space Finfs. Characterising the system under study with different probability function h(v, w) leads to different positive quantum distribution F(q, p) and consequently different phasespace Finf. The phase-space Finf can be used to give the information content of a given phase-space distribution function. From the class of the positive distribution functions defined by Cohen and Zaparovanny [27] we can select the one having the desired information. As an illustration we consider the simplest case (Eq. (7)) that corresponds to the minimal phase-space Finf, that is, the sum of the position space and the momentum space Finf. Here we deal with single-particle systems with a central potential. Making use of the position space I F. ¼ 4hp2 i 2ð2l þ 1Þjmjhr2 i and the momentum space Fisher information derived by Romera et al. [32]
135
The second example is the hydrogen atom and the hydrogen-like ions. The potential is V(r) = Z/r. In a recent Letter, Romera et al. [32] studied the Finf for this system. They obtained 4Z 2 jmj F ð16Þ I. ¼ 2 1 n n for the position space Finf and I Fc ¼
2n2 2 5n þ 1 3lðl þ 1Þ ½8n 3ð2l þ 1Þjmj 2 Z
ð17Þ
for the momentum space one, where n = 1, 2, . . .; l = 0, 1, . . . , n 1 and m = l, l + 1, . . . l 1, l. Then, the Finf sum has the form I Fmin ¼ I F. þ I Fc 4Z 2 jmj ¼ 2 1 n n 2 2n þ 2 5n2 þ 1 3lðl þ 1Þ ½8n 3ð2l þ 1Þjmj : Z ð18Þ
I Fc ¼ 4hr2 i 2ð2l þ 1Þjmjhp2 i;
A uniform scaling of coordinates results the scaled N-particle wave function
we can write
Wk ðr1 ; r2 ; . . . ; rN Þ ¼ k3N =2 Wðkr1 ; kr2 ; . . . ; krN Þ
I Fmin
¼
I F.
þ 2
I Fc
ð19Þ
and the scaled density 2
2
2
¼ 4ðhp i þ hr iÞ 2ð2l þ 1Þjmjðhr i þ hp iÞ;
.k ðrÞ ¼ k3 .ðkrÞ
ð20Þ
where l and m are the orbital and magnetic quantum numbers. That is, the Finf sum can be given by a couple of moments. This result clearly shows that the Fisher information sum provides a measure of the spreading of quantum states in phase space. The ground state Finf sum has an especially simple form
in the momentum space. This expression gives rise the scaled density
I Fmin ¼ 4ðhp2 i þ hr2 iÞ:
ck ðpÞ ¼ k3 cðk1 pÞ
It is worth mentioning that the first term is proportional to the kinetic energy. (In systems having more than one or two-electrons it is proportional to the Weizsa¨cker kinetic energy.) As a first example consider the isotropic harmonic oscillator. Atomic units are used in this Letter. The potential is V ðrÞ ¼ 12 x2 r2 . Romera et al. [32] obtained 3 F I . ¼ 4 2n þ l jmj þ x ð13Þ 2
in the momentum space. Gadre et al. [31] showed that S .k ¼ S . 3 ln k and S ck ¼ S c þ 3 ln k in the position and momentum space, respectively while the coordinate scaling does not change the Shannon information sum: S .k þ S ck ¼ S . þ S c . On the other hand, we immediately see that the scaled Finf takes the form
for the position space Finf and 3 I Fc ¼ 4 2n þ l jmj þ x1 2
in the position space. The Fourier transform of the scaled N-particle wave function (19) leads to the wave function Uk ðp1 ; p2 ; . . . ; pN Þ ¼ k3N =2 Uðk1 p1 ; k1 p2 ; . . . ; k1 pN Þ
ð21Þ
ð22Þ
I F.k ¼ k2 I F. ; I Fck ¼ k2 I Fc and
ð14Þ
for the momentum space Finf, where n = 1, 2, . . .; l = 0, 1,. . . and m = l, l + 1, . . . l 1, l. Then, the Finf sum defined in this Letter, takes the form 3 1 F I min ¼ 4 2n þ l jmj þ xþ : ð15Þ 2 x
I F.k þ I Fck ¼ k2 I F. þ k2 I Fc :
ð23Þ
Thus, the coordinate scaling changes not only the position and the momentum space Finf, but their sum, too. For the normal distribution the Finf is equal to the inverse variance. Eq. (20) shows that a narrower distribution has a larger Finf. A wave function and density having a smaller variance in the position space corresponds to a
I. Hornya´k, A´. Nagy / Chemical Physics Letters 437 (2007) 132–137
136
wave function and density of larger variance in the momentum space and consequently, a smaller momentum space Finf. On the other hand, a distribution with a larger variance, hence smaller Finf in the position space, leads to smaller variance and larger Finf in the momentum space. Eq. (23) shows that the coordinate scaling changes the role of the position and momentum space Finf in the sum. It is demonstrated in the examples mentioned above. For the isotropic harmonic oscillator I F. > I Fc if x > 1, F I . < I Fc if x < 1 and I F. ¼ I Fc if x = 1, as can be seen from Eqs. (13) and (14). Consequently, the sum is dominated by the position space Finf if x 1 and by the momentum space Finf if x 1. According to Eq. (15) the sum increases monotonically with the quantum number n for fixed quantum numbers l and m. Gadre et al. [31] showed the same behavior for the Shannon information sum of the linear harmonic oscillator. The hydrogen-like ions show a more complicated behavior. Table 1 presents the position space Finf, the momentum space Finf and the Finf sum for the hydrogen atom (Z = 1) for the ground and some excited states. As it can be seen from Eqs. (16) and (17), the position space Finf depends only on the quantum number n, while the momentum space Finf depends also on the quantum numbers l and m. There is a monotonic decrease in the position space Finf with increasing excitation. On the other hand, the momentum space Finf increases with the excitation. The sum is dominated by the momentum space Finf, while the position space Finf has a negligible contribution to the sum for higher excited states. For fixed quantum numbers l and m the sum increases monotonically with the quantum number n. For non-spherical orbitals the momentum space Finf and hence the sum are smaller than that of the spherical orbitals. Turning to the hydrogen-like ions, consider first the ground state. As it can be seen from Eq. (18) I Fmin ¼ 4ðZ 2 þ 3=Z 2 Þ. Therefore, I F. > I Fc for atomic numbers Z P 2, while the sum is dominated by the position space Finf if Z 2. For excited states the behavior is more complicated. If m = l = 0, for a given atomic number Z, I F. > I Fc for a quantum number n satisfying the condition 2Z4 > (5n2 + 1)n4 and I F. < I Fc for n satisfying the condition Table 1 Position space, momentum space Finf and the Finf sum for the hydrogen atom (in a.u.) Orbital
I Fq
I Fc
I Fq þ I Fc
1s 2s 3s 4s 2p, 2p, 3p, 3p, 3d, 3d, 3d,
4 1 0.444 0.25 1 0.5 0.444 0.296 0.444 0.296 0.148
12 168 828 2592 120 64 720 450 504 342 180
16 169 828.444 2592.25 121 64.5 720.444 450.296 504.444 342.296 180.148
jmj = 0 jmj = 1 jmj = 0 jmj = 1 jmj = 0 jmj = 1 jmj = 2
2Z4 < (5n2 + 1)n4. That also means that for a given Z the sum is dominated by the momentum space Finf for large enough quantum numbers n. Also, the sum increases monotonically with high enough quantum numbers n for fixed quantum numbers l and m. Finally, we turn to the uncertainty-like relationships. Gadre and Bendale [30] derived rigorous upper and lower bounds to atomic Shannon information sums in terms of the second moments Ær2æ and Æp2æ. To obtain a similar inequality for the Finf sum we have to apply Stam’s uncertainty principle [33,34]. The Stam’s uncertainty principle relates the Finf of the Rposition space with the momentum space quantity hp2 i ¼ p2 cðpÞdp: I F. 6 4hp2 i and the Finf ofR the momentum with the position space quantity hr2 i ¼ r2 .ðrÞdr: I Fc 6 4hr2 i: These inequalities lead to the uncertainty-like relationship for Finf sum
I F. þ I Fc 6 4 hp2 i þ hr2 i : Romera et al. [32] derived the Finf of a single-particle system with a central potential: I F. ¼ 4hp2 i 2ð2l þ 1Þjmjhr2 i and I Fc ¼ 4hr2 i 2ð2l þ 1Þjmjhp2 i: From these equalities we arrive at the relationship for Finf sum for a single-particle system with a central potential:
I F. þ I Fc ¼ 4 hp2 i þ hr2 i 2ð2l þ 1Þjmj hp2 i þ hr2 i : It is worth mentioning that there exist another class of positive distribution functions, the Husimi functions [18]. These are essentially Gaussian convolutions in coordinate and momentum variables of the Wigner function and they are everywhere non-negative. However, it is disadvantageous – from the point of view emphasized here – that they do not satisfy the marginal relations (1) and (2). The phasespace Finf can be defined with the Husimi functions, too. Properties of Husimi-based phase-space Finf should be the subject of a further study. Concluding, we emphasize that the phase-space Finf defined here satisfies an inequality; its lower bound is the sum of the position and momentum space Finf and it provides a measure of the spreading of quantum states in phase-space. In the examples above the lower bound of the phase-space Fisher information was studied. Acknowledgement This work was supported by the grant OTKA No. T 042505.
I. Hornya´k, A´. Nagy / Chemical Physics Letters 437 (2007) 132–137
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C.E. Shannon, Bell Syst. Tech. J. 27 (1948) 379. R.A. Fisher, Proc. Cambridge Philos. Soc. 22 (1925) 700. S.B. Sears, R.G. Parr, U. Dinur, Israel J. Chem. 19 (1980) 165. M. Reginatto, Phys. Rev. A 58 (1998) 1775. B.R. Frieden, Physics from Fisher Information. A unification, Cambridge UP, 1998. R. Nalewajski, Chem. Phys. Lett. 372 (2003) 28. ´ . Nagy, J. Chem. Phys. 119 (2003) 9401. A E. Romera, P. Sa´nchez-Morena, J.S. Dehesa, Chem. Phys. Lett. 414 (2005) 468. ´ . Nagy, Chem. Phys. Lett. 425 (2006) 157. A ´ . Nagy, K.D. Sen, Phys. Lett. A 360 (2006) 291. A E. Wigner, Phys. Rev. 40 (1932) 749. E. Wigner, Quantum-mechanical distribution functions revisited, in: W. Yourgrau, A. van der Merwe (Eds.), Perspectives in Quantum Theory, M. I. T., Cambridge, 1971. J.P. Dahl, Theor. Chim. Acta 81 (1982) 329. M. Springborg, J.P. Dahl, in: J.P. Dahl, J. Avery (Eds.), Local Density Approximations in Quantum Chemistry and Solid-State Physics, Plenum, 1984. M. Springborg, J.P. Dahl, Phys. Rev. A 36 (1997) 1050. J.P. Dahl, M. Springborg, J. Chem. Phys. 88 (1988) 4535.
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[17] J.P. Dahl, W.P. Schleich, Phys. Rev. A 65 (2002) 022109. [18] K. Husimi, Proc. Phys. Math. Soc. Japan 22 (1940) 264. [19] M.E. Casida, J.E. Harriman, J.L. Anchell, Int. J. Quantum Chem. Symp. 21 (1987) 435. [20] J.E. Harriman, J. Chem. Phys. 88 (1988) 6399. [21] J.L. Anchell, J.E. Harriman, J. Chem. Phys. 89 (1988) 6860. [22] J.J. Wlodarz, J. Chem. Phys. 100 (1994) 7476. [23] J.J. Wlodarz, Int. J. Quantum Chem. 51 (1994) 123. [24] J.J. Wlodarz, Int. J. Quantum Chem. 56 (1995) 233. [25] Q.S. Li, G.M. Wei, L.Q. Lu, Phys. Rev. A 70 (2004) 022105. [26] J. Dunkel, S.A. Trigger, Phys. Rev. A 71 (2005) 052102. [27] L. Cohen, Y.I. Zaparovanny, J. Math. Phys. 21 (1980) 794. [28] R. Leipnik, Inf. Control 2 (1959) 64. [29] I. Bialynicki-Birula, J. Mycielski, Commun. Math. Phys. 44 (1975) 129. [30] S.R. Gadre, R.D. Bendale, Int. J. Quantum Chem. 28 (1985) 311. [31] S.R. Gadre, S.B. Sears, S.J. Chakravorty, R.D. Bendale, Phys. Rev. A 32 (1985) 2602. [32] E. Romera, P. Sa´nchez-Morena, J.S. Dehesa, Chem. Phys. Lett. 414 (2005) 468. [33] A.J. Stam, Inf. Control 2 (1959) 101. [34] A. Dembo, T.A. Cover, J.A. Thomas, IEEE Trans. Inform. Theor. 37 (1991) 1501.
Phase-space Fisher information ´ . Nagy I. Hornya´k, A
*
Department of Theoretical Physics, University of Debrecen, and Atomic and Molecular Physics Research Group of the Hungarian Academy of Sciences, H-4010 Debrecen, Hungary Received 3 January 2007; in final form 26 January 2007 Available online 3 February 2007
Abstract The concept of phase-space Fisher information, F(q, p), is introduced and a theorem, stating that this quantity is not less than the sum of the position space and the momentum space Fisher information, is proved. Applications are made to the hydrogen-like atoms and the isotropic harmonic oscillator. An uncertainty-like relation for the Fisher information sum is also presented. 2007 Elsevier B.V. All rights reserved.
1. Introduction Information theoretical concepts have a role of growing importance in studying quantum mechanical systems. Shannon information [1] has been applied in physics for a long time and recently Fisher information [2] has also attracted special attention. Fisher information (Finf) was originally introduced by Fisher as a measure of ‘intrinsic accuracy’ in statistical estimation theory. Sears et al. [3] were the first who noticed the importance of Finf [2] in quantum mechanics and density functional theory (DFT) more than 20 years ago. They argued that the quantum mechanical kinetic energy can be considered a measure of the information distribution. Since then Finf has proved to be a very useful concept, e.g. the equations of non-relativistic quantum mechanics [4] have been derived using the principle of minimum Finf [5]. The time-independent Kohn–Sham equations and the time-dependent Euler equation of DFT were also derived by applying the above principle [6,7]. Other applications include single-particle systems with a central potential [8], two-electron ‘entangled artificial’ atom proposed by Moshinsky [9]. In a recent Letter, atomic Finf [10] has also been investigated.
*
Corresponding author. ´ . Nagy). E-mail address: [email protected] (A
0009-2614/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2007.01.084
Phase-space representation of quantum mechanics enjoys a renewed interest in many branches of physics (e.g. solid-state physics, quantum optics, quantum chaos, quantum chemistry). Here we cite only some important papers [11–26]. In this Letter, the concept of phase-space Finf is introduced. In this respect we prove that the phase-space Finf is larger or equal to the sum of the position space and the momentum space Finf. Elucidating applications are made for the hydrogen-like atoms and the isotropic harmonic oscillator. An uncertainty-like relationship for the Finf sum is presented. The Letter, is organized as follows: in Section 2 phasespace Finf is introduced. An inequality for the marginal configuration and momentum Fisher information is derived in Section 3. The last section is devoted to illustrative examples, discussion and the presentation of uncertainty-like relationship for the Finf sum. 2. Phase-space distribution functions It was Wigner [11] who first realized the possibility of formulating quantum mechanics in the phase-space. He examined a distribution function F(q, p) which yields the proper quantum mechanical marginal distributions when either one of the variables is integrated over: Z F ðq; pÞ dp ¼ jwðqÞj2 ¼ .ðqÞ; ð1Þ
I. Hornya´k, A´. Nagy / Chemical Physics Letters 437 (2007) 132–137
and Z F ðq; pÞ dq ¼ j/ðpÞj2 ¼ cðpÞ;
ð2Þ
where w(q) and /(p) are the configuration and momentum wave functions, respectively and .(q) and c(p) are the configuration-space and momentum-space densities, respectively. Both are normalized to 1. The Wigner function cannot be considered a proper probability distribution, because it may assume negative values. Since Wigner’s original Letter, several other distribution functions have been found which satisfy conditions (1) and (2). However, these may take on not only negative but also imaginary values. Wigner proved [12] that positive distributions which are bilinear in the wave function do not exist. Later, Cohen and Zaparovanny [27] proved that positive distributions satisfying conditions (1) and (2) exist, though these are not bilinear in the wave function. They also showed that the set of all possible functions of q and p consistent with (1) and (2) have the form Z 1 F ðq; pÞ ¼ 2 expðihq isp þ ihuÞf ðh; sÞ 4p
w ðu s=2Þwðu þ s=2Þ dh ds du; where f(h, s) satisfies the conditions f ðh; 0Þ ¼ f ð0; sÞ ¼ 1; By taking f = 1 we arrive at the Wigner distribution. By means of the joint probability density h(v, w) of random variables v, w defined over the range (0, 1) one can define the probability function H(q, p) as follows
1
wðpÞ ¼
The Shannon information [1] can be defined both in the configuration space Z S . ¼ .ðqÞ ln .ðqÞ dq and in the momentum space Z S c ¼ cðpÞ ln cðpÞ dp: The sum of the position and momentum Shannon information has turned to be a very important quantity and one can show that the following uncertainty relation [29] S . þ S c P ð1 þ ln pÞ holds. Using the positive quantum distribution F(q, p) the phase-space Shannon information can be defined as Z S F ¼ F ðq; pÞ ln F ðq; pÞ dq dp: This is the Leipnik entropy [28]. From the subadditivity of the Shannon information one can obtain the inequality [34]
where p(xjh) is a probability density function depending on a parameter h and obeying proper regularity conditions. If h is a parameter of locality then
p
cðp0 Þ dp0 :
1
pðxjhÞ ¼ pðx þ hÞ;
The positive quantum distributions have the form F ðq; pÞ ¼ .ðqÞcðpÞ½1 þ cvðq; pÞ;
3. Phase-space Fisher information
Let us turn to the Finf. The Finf functional [2] is defined as 2 Z Z 0 2 o ln pðxjhÞ ½p ðxjhÞ F I ðhÞ ¼ pðxjhÞ dx; ð4Þ dx ¼ oh pðxjhÞ
The functions v(q) and w(p) are taken to be Z q vðqÞ ¼ .ðq0 Þ dq0 ; Z
writing the distribution function F in terms of h(v, w) is that the marginals will always be satisfied by any h(v, w). The class of positive quantum distributions are determined by the function h(v, w). By selecting different probability function h(v, w) one can get different positive quantum distributions F(q, p).
SF P S. þ Sc:
H ðq; pÞ ¼ .ðqÞcðpÞhðv; wÞ:
and
133
ð3Þ
where vðq; pÞ ¼ hðvðqÞ; wðpÞÞ h1 ðvðqÞÞ h2 ðwðpÞÞ þ 1 and h1, h2 are the marginal distributions of H(q, p) defined by the relations: Z H ðq; pÞ dp ¼ h1 ðvÞ.ðqÞ ¼ H 1 ðqÞ; and Z H ðq; pÞ dq ¼ h2 ðwÞcðpÞ ¼ H 2 ðpÞ: Cohen and Zaparovanny [27] showed that c can be chosen to make F positive. They emphasized that the advantage of
and Finf (Eq. (4)) has the form 2 Z opðx þ hÞ I F ðhÞ ¼ pðx þ hÞ dx: oðx þ hÞ This is Finf per observation with respect to the locality parameter h. As the expression does not depend on h, we may set the locality at zero: Z 0 2 ½p ðxÞ I F ðh ¼ 0Þ ¼ dx: pðxÞ In general x can be vector-valued and in this case Finf has the form Z 2 ½rpðxÞ F dx: I ðh ¼ 0Þ ¼ pðxÞ
I. Hornya´k, A´. Nagy / Chemical Physics Letters 437 (2007) 132–137
134
This locality Finf is called intrinsic accuracy and measures the ‘narrowness’ of a distribution. For the normal distribution it is equal to the inverse variance. When the parameter is one of locality Finf characterizes the behavior of the distribution itself. Let us define now the phase-space Finf as Z 2 ½rF ðp; qÞ I FF ¼ dq dp F ðq; pÞ " 2 2 # Z 1 oF ðp; qÞ oF ðp; qÞ ¼ dq dp: ð5Þ þ F ðq; pÞ oq op The position and momentum space Finf can be written as 2 Z 1 d.ðqÞ I F. ¼ dq .ðqÞ dq
for any q. Consequently Z cðpÞvðq; pÞ dp ¼ 0
ð10Þ
with the exception of points q, where .(q) vanishes. Similarly, we can immediately obtain Z .ðqÞvðq; pÞ dq ¼ 0 ð11Þ with the exception of points p, where c(p) vanishes. Next, we shall show that I0 = 0. After partial integration with respect to variable q, the first term of I0 takes the form 1 Z Z d.ðqÞ d2 .ðqÞ dpcðpÞvðq; pÞ 2c dq 2c dq dq2 1 Z dpcðpÞvðq; pÞ
and I Fc ¼
Z
2 1 dcðpÞ dp: cðpÞ dp
and vanishes because of Eq. (10). A partial integration of the second term of I0 for the variable p, leads also to 0 after using Eq. (11). From Eq. (8) we obtain
Theorem 1. The following inequality holds: I FF P I F. þ I Fc :
ð6Þ
I FF ¼ I F. þ I Fc " 2 2 # Z 2 c .ðqÞcðpÞ d.ðqÞ dcðpÞ þ þ dq dp: 1 þ cvðq; pÞ dq dp
There is equality if F ðq; pÞ ¼ .ðqÞcðpÞ:
ð12Þ ð7Þ
Proof. By substituting the explicit form of the positive quantum distribution F(q, p) (Eq. (3)) into the expression of the phase-space Finf (5) we obtain I FF ¼
2 Z 1 d.ðqÞ 1 þ c cðpÞvðq; pÞ dp dq .ðqÞ dq 2 Z Z 1 dcðpÞ þ 1 þ c .ðqÞvðq; pÞ dq dp þ I 0 cðpÞ dp " 2 2 # Z 2 c .ðqÞcðpÞ d.ðqÞ dcðpÞ dq dp; þ þ 1 þ cvðq; pÞ dq dp Z
As the last term of Eq. (12) is positive, we immediately arrive at the inequality (6). It can also be seen from Eq. (12) that we can obtain equality in (6) if and only if c = 0. The above theorem and its proof can be straightforwardly generalized for systems of any degree of freedom. h 4. Illustrative examples and discussion
ð8Þ where
Z Z Z d.ðqÞ ovðq; pÞ dcðpÞ I 0 ¼ 2c dq dpcðpÞ þ 2c dp dq oq dp Z ovðq; pÞ : ð9Þ dq.ðqÞ op Substituting the form of positive quantum distribution F(q, p) (Eq. (3)) into the left-hand side of Eq. (1) Z Z F ðq; pÞ dp ¼ .ðqÞ cðpÞ½1 þ cvðq; pÞ dp and comparing it with the right-hand side of Eq. (1) we are led to the expression Z c.ðqÞ cðpÞvðq; pÞ dp ¼ 0
Electronic wave functions for atoms and molecules may be quite complicated and difficult to understand or interpret. phase-space information concepts may provide appreciable advantage in this respect. The Shannon information sum has already proved to be useful. Gadre and Bendale [30] argued that the maximization of entropy in one space would lead to an ‘imbalance’ with the conjugate space. Therefore, they proposed the Shannon information sum. For hydrogen atom, Gadre et al. [31] showed that S. is minimum for the ground state and increases monotonically with the excited-state energy, whereas the momentum space Shannon information Sc decreases monotonically. They also found that the Shannon information sum assumes its minimum value for the ground state. For the linear harmonic oscillator, however, they noticed that the Shannon information of both spaces and hence their sum increase with the quantum number n. A recent Letter, [32] presented position space and momentum space Finf of single-particle systems with a central potential. Romera et al. have also found that the product of the position space and the momentum space Finf is
I. Hornya´k, A´. Nagy / Chemical Physics Letters 437 (2007) 132–137
also a useful quantity satisfying important uncertainty relation. In this Letter, we have shown that the sum of the position space and the momentum space Finf is also a meaningful quantity. We have defined a whole class of phase-space Finfs. Characterising the system under study with different probability function h(v, w) leads to different positive quantum distribution F(q, p) and consequently different phasespace Finf. The phase-space Finf can be used to give the information content of a given phase-space distribution function. From the class of the positive distribution functions defined by Cohen and Zaparovanny [27] we can select the one having the desired information. As an illustration we consider the simplest case (Eq. (7)) that corresponds to the minimal phase-space Finf, that is, the sum of the position space and the momentum space Finf. Here we deal with single-particle systems with a central potential. Making use of the position space I F. ¼ 4hp2 i 2ð2l þ 1Þjmjhr2 i and the momentum space Fisher information derived by Romera et al. [32]
135
The second example is the hydrogen atom and the hydrogen-like ions. The potential is V(r) = Z/r. In a recent Letter, Romera et al. [32] studied the Finf for this system. They obtained 4Z 2 jmj F ð16Þ I. ¼ 2 1 n n for the position space Finf and I Fc ¼
2n2 2 5n þ 1 3lðl þ 1Þ ½8n 3ð2l þ 1Þjmj 2 Z
ð17Þ
for the momentum space one, where n = 1, 2, . . .; l = 0, 1, . . . , n 1 and m = l, l + 1, . . . l 1, l. Then, the Finf sum has the form I Fmin ¼ I F. þ I Fc 4Z 2 jmj ¼ 2 1 n n 2 2n þ 2 5n2 þ 1 3lðl þ 1Þ ½8n 3ð2l þ 1Þjmj : Z ð18Þ
I Fc ¼ 4hr2 i 2ð2l þ 1Þjmjhp2 i;
A uniform scaling of coordinates results the scaled N-particle wave function
we can write
Wk ðr1 ; r2 ; . . . ; rN Þ ¼ k3N =2 Wðkr1 ; kr2 ; . . . ; krN Þ
I Fmin
¼
I F.
þ 2
I Fc
ð19Þ
and the scaled density 2
2
2
¼ 4ðhp i þ hr iÞ 2ð2l þ 1Þjmjðhr i þ hp iÞ;
.k ðrÞ ¼ k3 .ðkrÞ
ð20Þ
where l and m are the orbital and magnetic quantum numbers. That is, the Finf sum can be given by a couple of moments. This result clearly shows that the Fisher information sum provides a measure of the spreading of quantum states in phase space. The ground state Finf sum has an especially simple form
in the momentum space. This expression gives rise the scaled density
I Fmin ¼ 4ðhp2 i þ hr2 iÞ:
ck ðpÞ ¼ k3 cðk1 pÞ
It is worth mentioning that the first term is proportional to the kinetic energy. (In systems having more than one or two-electrons it is proportional to the Weizsa¨cker kinetic energy.) As a first example consider the isotropic harmonic oscillator. Atomic units are used in this Letter. The potential is V ðrÞ ¼ 12 x2 r2 . Romera et al. [32] obtained 3 F I . ¼ 4 2n þ l jmj þ x ð13Þ 2
in the momentum space. Gadre et al. [31] showed that S .k ¼ S . 3 ln k and S ck ¼ S c þ 3 ln k in the position and momentum space, respectively while the coordinate scaling does not change the Shannon information sum: S .k þ S ck ¼ S . þ S c . On the other hand, we immediately see that the scaled Finf takes the form
for the position space Finf and 3 I Fc ¼ 4 2n þ l jmj þ x1 2
in the position space. The Fourier transform of the scaled N-particle wave function (19) leads to the wave function Uk ðp1 ; p2 ; . . . ; pN Þ ¼ k3N =2 Uðk1 p1 ; k1 p2 ; . . . ; k1 pN Þ
ð21Þ
ð22Þ
I F.k ¼ k2 I F. ; I Fck ¼ k2 I Fc and
ð14Þ
for the momentum space Finf, where n = 1, 2, . . .; l = 0, 1,. . . and m = l, l + 1, . . . l 1, l. Then, the Finf sum defined in this Letter, takes the form 3 1 F I min ¼ 4 2n þ l jmj þ xþ : ð15Þ 2 x
I F.k þ I Fck ¼ k2 I F. þ k2 I Fc :
ð23Þ
Thus, the coordinate scaling changes not only the position and the momentum space Finf, but their sum, too. For the normal distribution the Finf is equal to the inverse variance. Eq. (20) shows that a narrower distribution has a larger Finf. A wave function and density having a smaller variance in the position space corresponds to a
I. Hornya´k, A´. Nagy / Chemical Physics Letters 437 (2007) 132–137
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wave function and density of larger variance in the momentum space and consequently, a smaller momentum space Finf. On the other hand, a distribution with a larger variance, hence smaller Finf in the position space, leads to smaller variance and larger Finf in the momentum space. Eq. (23) shows that the coordinate scaling changes the role of the position and momentum space Finf in the sum. It is demonstrated in the examples mentioned above. For the isotropic harmonic oscillator I F. > I Fc if x > 1, F I . < I Fc if x < 1 and I F. ¼ I Fc if x = 1, as can be seen from Eqs. (13) and (14). Consequently, the sum is dominated by the position space Finf if x 1 and by the momentum space Finf if x 1. According to Eq. (15) the sum increases monotonically with the quantum number n for fixed quantum numbers l and m. Gadre et al. [31] showed the same behavior for the Shannon information sum of the linear harmonic oscillator. The hydrogen-like ions show a more complicated behavior. Table 1 presents the position space Finf, the momentum space Finf and the Finf sum for the hydrogen atom (Z = 1) for the ground and some excited states. As it can be seen from Eqs. (16) and (17), the position space Finf depends only on the quantum number n, while the momentum space Finf depends also on the quantum numbers l and m. There is a monotonic decrease in the position space Finf with increasing excitation. On the other hand, the momentum space Finf increases with the excitation. The sum is dominated by the momentum space Finf, while the position space Finf has a negligible contribution to the sum for higher excited states. For fixed quantum numbers l and m the sum increases monotonically with the quantum number n. For non-spherical orbitals the momentum space Finf and hence the sum are smaller than that of the spherical orbitals. Turning to the hydrogen-like ions, consider first the ground state. As it can be seen from Eq. (18) I Fmin ¼ 4ðZ 2 þ 3=Z 2 Þ. Therefore, I F. > I Fc for atomic numbers Z P 2, while the sum is dominated by the position space Finf if Z 2. For excited states the behavior is more complicated. If m = l = 0, for a given atomic number Z, I F. > I Fc for a quantum number n satisfying the condition 2Z4 > (5n2 + 1)n4 and I F. < I Fc for n satisfying the condition Table 1 Position space, momentum space Finf and the Finf sum for the hydrogen atom (in a.u.) Orbital
I Fq
I Fc
I Fq þ I Fc
1s 2s 3s 4s 2p, 2p, 3p, 3p, 3d, 3d, 3d,
4 1 0.444 0.25 1 0.5 0.444 0.296 0.444 0.296 0.148
12 168 828 2592 120 64 720 450 504 342 180
16 169 828.444 2592.25 121 64.5 720.444 450.296 504.444 342.296 180.148
jmj = 0 jmj = 1 jmj = 0 jmj = 1 jmj = 0 jmj = 1 jmj = 2
2Z4 < (5n2 + 1)n4. That also means that for a given Z the sum is dominated by the momentum space Finf for large enough quantum numbers n. Also, the sum increases monotonically with high enough quantum numbers n for fixed quantum numbers l and m. Finally, we turn to the uncertainty-like relationships. Gadre and Bendale [30] derived rigorous upper and lower bounds to atomic Shannon information sums in terms of the second moments Ær2æ and Æp2æ. To obtain a similar inequality for the Finf sum we have to apply Stam’s uncertainty principle [33,34]. The Stam’s uncertainty principle relates the Finf of the Rposition space with the momentum space quantity hp2 i ¼ p2 cðpÞdp: I F. 6 4hp2 i and the Finf ofR the momentum with the position space quantity hr2 i ¼ r2 .ðrÞdr: I Fc 6 4hr2 i: These inequalities lead to the uncertainty-like relationship for Finf sum
I F. þ I Fc 6 4 hp2 i þ hr2 i : Romera et al. [32] derived the Finf of a single-particle system with a central potential: I F. ¼ 4hp2 i 2ð2l þ 1Þjmjhr2 i and I Fc ¼ 4hr2 i 2ð2l þ 1Þjmjhp2 i: From these equalities we arrive at the relationship for Finf sum for a single-particle system with a central potential:
I F. þ I Fc ¼ 4 hp2 i þ hr2 i 2ð2l þ 1Þjmj hp2 i þ hr2 i : It is worth mentioning that there exist another class of positive distribution functions, the Husimi functions [18]. These are essentially Gaussian convolutions in coordinate and momentum variables of the Wigner function and they are everywhere non-negative. However, it is disadvantageous – from the point of view emphasized here – that they do not satisfy the marginal relations (1) and (2). The phasespace Finf can be defined with the Husimi functions, too. Properties of Husimi-based phase-space Finf should be the subject of a further study. Concluding, we emphasize that the phase-space Finf defined here satisfies an inequality; its lower bound is the sum of the position and momentum space Finf and it provides a measure of the spreading of quantum states in phase-space. In the examples above the lower bound of the phase-space Fisher information was studied. Acknowledgement This work was supported by the grant OTKA No. T 042505.
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