Uncertainty relations, one of the fundamental. Heisenberg Uncertainty Relations can be Replaced by Stronger Ones L. SKÁLA, 1,2 V.

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1 Heisenberg Uncertainty Relations can be Replaced by Stronger Ones L. SKÁLA, 1, V. KAPSA 1 1 Faculty of Mathematics Physics, Charles University, Ke Karlovu 3, Prague, Czech Republic Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario NL 3G1, Canada Received 4 July 008; accepted 4 September 008 Published online 3 December 008 in Wiley InterScience ( DOI /qua.1948 ABSTRACT: Two uncertainty relations, one related to the probability density current the other one related to the probability density, are derived discussed. Both relations are stronger than the Heisenberg uncertainty relations. Their generalization to the multidimensional case to the mixed states is also discussed. 008 Wiley Periodicals, Inc. Int J Quantum Chem 109: , 009 Key words: Heisenberg uncertainty relations; probability density current; probability density 1. Introduction Uncertainty relations, one of the fundamental results of quantum mechanics, have been studied in a large number of articles (see e.g., [1 6]; for a detailed review, see [7]). The stard approach to their derivation is based on the wave function ψ. It is shown below that the approach based on the probability density current probability density yields uncertainty relations that are stronger than the corresponding Heisenberg uncertainty relation.. One-Dimensional Case By analogy with continuum mechanics, it is possible to introduce in quantum mechanics not only the probability density ρ but also the probability density current j related to the velocity v j ρv. (1) The quantities ρ 0 v can be expressed in terms of two real functions s 1 s 1 (x, t) s s (x, t) as follows Correspondence to: L. Skála; skala@karlov.mff.cuni.cz Contract grant sponsor: MSMT, Czech Republic. Contract grant number: ρ e s /, () v 1 s 1 m, (3) International Journal of Quantum Chemistry, Vol 109, (009) 008 Wiley Periodicals, Inc.

2 HEISENBERG UNCERTAINTY RELATIONS where x is the coordinate, t denotes time, m is the mass. Then, by introducing the wave function ψ ψ e (is 1 s )/, (4) we can get two basic formulas of quantum mechanics ρ ψ (5) j ( ψ ψ ) mi ψ ψ, (6) where the star denotes the complex conjugate. It is seen that instead of ρ j, the state of the quantum mechanical system can be described by the functions s 1 s or the wave function ψ [8 10]. It is assumed in this article that the probability density ρ has the properties ρdx 1 (7) lim ρ(x) 0. (8) x ± Taking into account that the mean value s ψ s ψdx s ρdx ρ dx (9) equals zero it can easily be shown that ( p) appearing in the Heisenberg uncertainty relation ( x) ( p) /4 (10) can be written as the sum of two quantities ( s1 ) ( p 1 ) ( s ) ( p ) s1 (11) (1) that are greater than or equal to zero. The first quantity ( p 1 ) depends on ( s 1 /) ρ can be different from zero for the nonzero probability density current j only. The second quantity ( p ) depends only on the form of the wave packet given by s is independent of j. Therefore, the separation of ( p) into two parts given above has a good physical meaning. An analogous separation was discussed also in [11] within the framework of one-dimensional stochastic mechanics. Now we can use the Schwarz inequality (u, u)(v, v) (u, v), where the inner product is defined as the integration over all x. For the functions u (x x ) ρ v ( s 1 / s 1 / ) ρ we get the first uncertainty relation ( x) ( p 1 ) ( s1 (x x ) ) s1. (13) Equation (13) was also derived within the framework of Nelson s stochastic mechanics [11]. The second uncertainty relation can be obtained by putting u (x x ) ρ v ( s /) ρ with the result ( x) ( p ) /4. (14) Equation (14) is known for example from [1], see also the stochastic variational approach to the minimum uncertainty states [13]. Another discussion of Eq. (14) can be found in [8 10]. We see that the Heisenberg uncertainty relation (10) can be replaced by two more detailed uncertainty relations (13) (14) for the information carried by the functions s 1 (information related to the probability density current j describing the motion in space) s (information related to the probability density ρ describing the form of the wave packet). For real wave functions ψ, corresponding to s 1 0, Eq. (13) gives trivial result 0 0, Eq. (14) becomes the Heisenberg uncertainty relation (10). In a general case, the uncertainty relations (13) (14) are stronger than the Heisenberg uncertainty relation (10). 3. Multi-Dimensional Case We consider the N-dimensional space with the coordinates x (x 1,..., x N ) the probability density ρ 0 given by the wave function ψ ρ(x, t) ψ(x, t). (15) It is assumed that this probability density fulfills the boundary conditions ρ xm 0, m 1,..., N (16) VOL. 109, NO. 8 DOI /qua INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 167

3 SKÁLA AND KAPSA the stard normalization condition ρ(x, t)dτ 1, dτ dx 1...dx N, (17) where the integration is carried out over the whole space. The mean values of the coordinates are defined as x m x m ρdτ. (18) The N N covariance matrix ( X) is given by the equation ( X) mn (x m x m )(x n x n )ρdτ. (19) Assuming that c m are arbitrary complex numbers it can easily be verified that this matrix is positive semidefinite N c N m ( X) mn c n c m (x m x m ) ρdτ 0. m1 (0) m,n1 Analogously to Section, the wave function ψ is written in the form ψ e (is 1 s )/, (1) where s 1 s 1 (x 1,..., x N, t) s s (x 1,..., x N, t) are real functions. The functions s 1 s give the probability density ρ ρ ψ e s / () the probability density current j k ( ψ ψ ) ψ ψ mi k k 1 s 1 ρ. (3) m k Further, we calculate the mean value of the momentum operator ˆp m i ( / m ), which must be real ˆp m ψ s1 s ˆp m ψdτ ρdτ + i ρdτ m m s1 s1 ρdτ. (4) m m Similarly, we get ˆp m ˆp n (ˆp m ψ) (ˆp n ψ)dτ ( P) mn ( s1 s 1 + s ) s ρdτ (5) m n m n [(ˆp m ˆp m )ψ] (ˆp n ˆp n )ψdτ ˆp m ˆp n ˆp m ˆp n ( s1 s 1 + s s m n m n s1 ρdτ m ) ρdτ s1 n ρdτ. (6) Analogously to the matrix ( X), it can be shown that the matrix ( P) is positive semidefinite. It can be verified that both matrices appearing in Eq. (6) ( P 1 ) mn s1 s 1 s1 s1 ρdτ ρdτ ρdτ m n m n (7) ( P ) mn s m s n ρdτ, (8) ( P 1 ) + ( P ) ( P) (9) are positive semidefinite, too. The matrix ( P 1 ) depends on ( s 1 / m ) ρ (i.e., s ) can be different from zero for the nonzero probability density current j (j 1,..., j N ) only [see Eq. (3)]. The second matrix ( P ) depends only on the form of the wave packet given by s is independent of j. Therefore, the separation of ( P) into two parts given by Eqs. (7) (9) has a good physical meaning. Now, we define a correlation matrix G G mn ( s1 (x m x m ) n s1 n ) ρdτ (30) create a new N N matrix M ( ) ( X) G M T G ( P 1 ), (31) where the superscript T denotes the transposition. Using similar arguments as in case of the matrix 168 INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY DOI /qua VOL. 109, NO. 8

4 HEISENBERG UNCERTAINTY RELATIONS ( X), it can be shown that the matrix M is positive semidefinite. Further, we make use of a general result valid for N N matrices A, B, C, D, where D is a regular matrix having the property det(d) 0 1 denotes the N N unity matrix ( )( ) ( ) 1 BD 1 A B A BD 1 C 0 (3) 0 1 C D C D leading to ( ) A B det det(a BD 1 C) det(d). (33) C D Applying this equation to the matrix M given by Eq. (31), the first multidimensional uncertainty relation for the matrices ( X) ( P 1 ) is obtained det(m) det{( X) ( P 1 ) G T [( P 1 ) ] 1 G( P 1 ) } 0. (34) In the one-dimensional form, Eq. (34) leads to Eq. (13). Further, we replace the function s 1 by s repeat the discussion made in the preceding paragraph. Taking into account the equation s m s ρdτ ρ dτ m m ρ xm dτ 0, (35) where dτ dx 1 dx m 1 dx m+1 dx N, the matrix element G mm can be replaced by G mm G mm (x m x m ) s ρdτ. (36) m Performing here the integration by parts in the variable x m, assuming that [(x m x m )ρ] xm 0 using Eqs. (17), (18) we get G mm [(x m x m )ρ] xm dτ + ρdτ. (37) By using Eqs. (16) (18) we get similarly G mn (x m x m ) ρ dτ n (x m x m )ρ xn dτ 0, m n, (38) where dτ dx 1 dx n 1 dx n+1 dx N. Equation (33) applied to the matrix M ( ) ( X) / / ( P ) (39) then yields the second multidimensional uncertainty relation ] det [( X) ( P ) 0 (40) 4 or, in the one-dimensional form, Eq. (14). The third uncertainty relation can be obtained by replacing ( P ) in the matrix M by ( P) ( P 1 ) + ( P ). The resulting matrix remains positive semidefinite, Eq. (40) then leads to the multidimensional uncertainty relation ] det [( X) ( P) 0 (41) 4 which is known for example from [3 7]. The onedimensional form of this relation is the Heisenberg uncertainty relation (10). There is one important difference between the uncertainty relation (41) the uncertainty relations (34) (40). The uncertainty relation (41) is based on using the wave function ψ (see [3 7]). In contrast to it, the uncertainty relations (34) (40) are based on using the probability density current probability density. Because of Eq. (9), these uncertainty relations are in general stronger than the uncertainty relation (41). Discussion of the multidimensional uncertainty relations (34) (40) can be found also in [14]. 4. Generalization to Mixed States Finally, we note that the uncertainty relations (34), (40), (41) can be generalized to mixed states when the system can be found in the states described with the probabilities by the wave function ψ j (x, t), j 1,, 3,... The density matrix for such a system can be written as j ψ j ψ j. The matrices ( X), ( P 1 ), ( P ), G can be then defined as ( X) mn j ψ j (x m x m )(x n x n )ψ j dτ, (4) VOL. 109, NO. 8 DOI /qua INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 169

5 SKÁLA AND KAPSA where ( P 1 ) mn j j G mn j x m j ψ j ψ j ψ j dτ m k ( P ) mn j ψ j x mψ j dτ, (43) m n ψ j dτ ψ j w k ψ k ( ψ j (x s1j m x m ) n s 1k n ψ k dτ, (44) s j m s j n ψ j dτ (45) s1j n ) ψ j dτ. (46) Here, ψ j e (is 1j s j )/ħ, s 1j s j are real functions j 1. It can easily be shown that the uncertainty relations (34), (40), (41) are valid also in this case. References 1. Heisenberg, W. Z. Phys 197, 43, 17.. Robertson, H. P. Phys Rev 199, 34, Robertson, H. P. Phys Rev 1934, 46, Chistyakov, A. L. Theor Math Phys 1976, 7, Trifonov, D. A. J Phys A 000, 33, L Trifonov, D. A. J Phys A 001, 34, L Dodonov, V. V.; Manko, I. V. Trudy Fiziceskogo Instituta im. P. N. Lebedeva 1987, 183, 5 (in Russian). 8. Skála, L.; Kapsa, V. Phys E 005, 9, Skála, L.; Kapsa, V. Collect Czech Chem Commun 005, 70, Skála, L.; Kapsa, V. Opt Spectrosc 007, 103, De Falco, D.; De Martino, S.; De Siena, S. Phys Rev Lett 198, 49, De la Pena-Auerbach, L.; Cetto, A. M. Phys Lett A 197, 39, Illuminati, F.; Viola, L. J Phys A 1995, 8, Skála, L.; Kapsa, V. J Phys A 008, 41, INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY DOI /qua VOL. 109, NO. 8