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Uncertainty features of microscopic particles described by nonlinear Schrüdinger equation
ARTICLE IN PRESS Physica B 404 (2009) 4327–4331
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Uncertainty features of microscopic particles described by nonlinear ¨ Schrodinger equation Pang Xiao-fenga,b, a b
Institute of Life Science and Technology, University of Electronic Science and Technology of China, Chengdu 610054, China International Centre for Materials Physics, Chinese Academy of Science, Shenyang 110015, China
a r t i c l e in fo
abstract
Article history: Received 6 October 2008 Received in revised form 15 July 2009 Accepted 3 August 2009
The uncertainty relationship between position and momentum of the microscopic particles is calculated by nonlinear quantum theory in which the states of the particles are described by a nonlinear ¨ Schrodinger equation. The results show that the uncertainty relation differs from that in the quantum mechanics and has a minimum value in this case. This means that the position and momentum of the particles could be determined simultaneously to a certain degree, which could be caused by the wave– ¨ corpuscle duality of the microscopic particles described by the nonlinear Schrodinger equation. & 2009 Published by Elsevier B.V.
PACS: 02.60.Cb 03.65.w Keywords: Microscopic particle Uncertainty relationship ¨ Nonlinear Schrodinger equation Quantum mechanics Nonlinear systems
1. Introduction As it is known, the states and properties of microscopic particles are depicted by the quantum mechanics, which was ¨ established by Bohr, Born, Schrodinger and Heisenberg in the early 1900s [1–7], and is also the pillar and foundation of modern science. In the quantum mechanics, the state of microscopic ¨ particles is described by the Schrodinger equation: i‘
2
@c ‘ ¼ r2 c þ Vð~ r ; tÞc @t 2m
ð1Þ
2 2 where ‘ r =2m is the kinetic energy operator, Vð~ r ; tÞ is the externally applied potential operator, m is the mass of particles, cð~ r ; tÞ is a wave function describing the states of particles, ~ r is its r ; tÞ is coordinate, and t is the time. Since no nonlinear terms of cð~ ¨ included in Eq. (1), hence we here refer to it as linear Schrodinger equation. If only the externally applied potentials are known, we can find the solutions of the equation. However, Eq. (1) is a wave equation and has only a wave solution. Then the microscopic
Correspondence address: Institute of Life Science and Technology, University of Electronic Science and Technology of China, Chengdu 610054, China. Tel.: +86 2883202595; fax: +86 2883208238. E-mail address: [email protected]
0921-4526/$ - see front matter & 2009 Published by Elsevier B.V. doi:10.1016/j.physb.2009.08.027
particles described by the quantum mechanics, which always disperse over the space–time in the form of a wave, do not have a r ; tÞj2 to represent the corpuscle nature. Thus we have to use jcð~ ! probability of the particles occurring in position r at time t 2 ~ according to Born’s idea. On account of jcðr ; tÞj which has a certain value at every point in the space–time, the particle can occur at each point at the same time. In this case, a fraction of the particle must appear in the systems, which is a very strange phenomenon and is quite difficult to understand. However, in physical experiments, the particles are always captured as a whole and not as a fractional one by a detector placed at an exact position. Therefore, the concept of probability representing the corpuscle behavior of the particles cannot be accepted. Because of the above nature of the quantum mechanics, some novel results, such as the uncertainty relationship between the position and the momentum and the mechanical quantities that are denoted by some average values in any state, also occur. These results are considerably incompatible not only with de k, of wave–corpuscle p ¼ ‘~ Broglie relation, E ¼ hu ¼ ‘ o and ~ duality for microscopic particles and Davisson and Germer’s experimental result of electron diffraction on double seam in 1927 [6–8], but also contradictory to the traditional concept of particles [7–9]. These problems of the quantum mechanics inevitably evoked the physicists’ discussion and contentions, where many
ARTICLE IN PRESS 4328
X.-f. Pang / Physica B 404 (2009) 4327–4331
scientists were involved in the debate. It was quite exceptional in the history of physics that so many prominent physicists from different institutions were involved and the scope of the debate has been so wide. Unfortunately, after being debated for almost a century, there remain no definite answers to most of these questions. In such a case, we want to know whether the uncertainty relationship is an intrinsic feature of microscopic particles or an artifact of the quantum mechanics or measuring instruments? This is worth studying deeply [7–9]. Therefore, to clarify the essence of the uncertainty relationship is very necessary in physics. This paper focuses on just this problem. Obviously, the uncertainty relationship is related to the natures ¨ of microscopic particles. Thus instead of the linear Schrodinger equation in Eq. (1) in quantum mechanics, we now use the ¨ following nonlinear Schrodinger equation [10–14] @f ‘ ¼ r2 f7bjfj2 f þ Vð~ r ; tÞf i‘ @t 2m
ð2Þ
to describe microscopic particles, where fð~ r ; tÞ is the wave function of the microscopic particle in this condition, b is a nonlinear interaction coefficient, the bjfj2 f is a nonlinear term, and represents a nonlinear interaction among the microscopic particles. In such a case, it is worth to study the concrete form and features that are the uncertainty relationship of microscopic particles. In this paper we calculate the uncertainty relationship of the position and the momentum of microscopic particles depicted in Eq. (2). We expect that the uncertainty relationship is different from that in the linear quantum mechanics. Then the significance of the uncertainty relationship can be revealed by comparing them in the two theories.
2. Correct form of uncertainty relationship in quantum mechanics and its roots We first calculate the uncertainty relationship of microscopic ¨ particles described by the linear Schrodinger equation in quantum mechanics. It is well known that the uncertainty relationship can be, in the coordinate representation, obtained from [1–7] Z ^ cð~ jðxDA^ þ iDBÞ r ; tÞj2 d~ r Z0 ð3Þ IðxÞ ¼
F^ ðxÞ ¼
Z
^ ~ ^ ~ c ð~ r ; tÞ~ F ½Að r ; tÞ; Bð r ; tÞcð~ r ; tÞ d~ r
where ~ A and ~ B are operators of two physical quantities, for example, position and momentum, or energy and time, and ^ B ^ ¼ iC^ , cðx; tÞ and c ðx; tÞ are satisfy the commutation relation ½A; wave functions of the microscopic particle satisfying the linear ¨ Schrodinger equation (1) and its conjugate equation, respectively, F^ ¼ ðDAx þ DBÞ2 , (DA^ ¼ A^ A, DB^ ¼ B^ B, A and B are the average values of the physical quantities in the state denoted by cðx; tÞ), is an operator of physical quantity related to A and B, x is a real variational parameter. After some simplifications, we can get 2 2 2 I ¼ F ¼ DA^ x þ 2DA^ DB^ x þ DB^ Z0
or 2
2
DA^ x2 þ C^ x þ DB^ Z0
ð4Þ
Using mathematical identities, this can be written as 2
2
DA^ DB^ Z
ðC^ Þ2 4
or
DA^
2
!2
C^
xþ
2DA^
2
2
þDB^
ðC^ Þ2 4DA^
2
ð6Þ
Z0
2
2 This equation shows that DA^ a0, if ðDA^ DB^ Þ2 or C^ =4 is not zero. 2
2
Or else, we cannot obtain Eq. (5) and DA^ DB^ 4ðDADBÞ2 because 2 2 when DA^ ¼ 0, Eq. (6) does not hold. Thus, ðDA^ Þa0 is a necessary
2
or
This is the uncertainty relationship often used in linear quantum mechanics. Since Eq. (4) can be written in the following form: !2 ^ ^ ^ ^ 2 ^F ¼ DA^ 2 x þ DA DB þDB^ 2 ðDA DB Þ Z0 2 2 DA^ DA^
ð5Þ
2 condition for the uncertainty relationship in Eq. (5). DA^ can approach zero, but cannot be equal to zero. This means that Eq. (6) should be denoted as !2 2 2 ðC^ Þ2 C^ ^ DA x þ þDB^ 40 2 2 2DA^ 4DA^
Then a right uncertainty relationship in quantum mechanics should take the form: 2
2
DA^ DB^ 4
ðC^ Þ2 4
ð7Þ
The uncertainty relationship in quantum mechanics is an important representation and indicates that the values of conjugate mechanical quantities, such as the position and the momentum, or energy and time cannot be, permanently accurately and simultaneously determined in quantum mechanics. How does this uncertainty relationship interpret? What is its essence? From the above calculation we know clearly that the uncertainty relation in Eq. (7) was obtained based on fundamental hypotheses of the linear quantum mechanics, including properties of operators of the mechanical quantities, the state of microscopic particle represented by the wave function, which satisfies the € linear Schrodinger equation (1), the concept of average values of mechanical quantities and the commutation relations and eigenequation of operators. Therefore, we can conclude that the uncertainty relationship in Eq. (7) is a necessary result of quantum mechanics, not related to the measurement at all. Thus the basic reason for this relationship is just that the microscopic particles have only a wave feature, not a corpuscle nature in quantum mechanics, and thus the position and momentum of the particle simultaneously are not determinant. Then the uncertainty relation also exists necessarily. This analysis can be also verified from the solutions of Eq. (1). As it is known, the solution of Eq. (1) at Vð~ r ; tÞ ¼ 0 is represented by cð~ r ; tÞ ¼ A0 exp½ið~ k ~ r otÞ ð8Þ it is a plane wave, where k is its wavevector, o is the frequency r ; tÞa0, the solutions are a de and A0 is the amplitude. When Vð~ Broglie wave or a Bloch wave, etc. Therefore, the microscopic particle is represented by a wave in the quantum mechanics, it disperses over whole space and cannot be localized, thus the r ; tÞj2 , of the microscopic particle is not equal to probability, jcð~ zero at all points. This means that the particle does not have a determinant position in this space and thus the uncertainty relationship of its position and momentum occurs inevitably in the quantum mechanics.
ARTICLE IN PRESS X.-f. Pang / Physica B 404 (2009) 4327–4331
Why does this phenomenon occur in the quantum mechanics? As it is known, the Hamiltonian operator of the systems’ corresponding Eq. (1) in the quantum mechanics is [1–7] 2 b r ; tÞ HðtÞ ¼ ‘ r2 =2m þ Vð~
ð9Þ
Obviously, it consists only of kinetic and potential operator of particles, the potential operator is not related to the wave function of state of the particle, and thus can only change the states of microscopic particles, and cannot change its nature. Then the nature of the particles is only determined by the kinetic term, thus it disperses over the whole space and cannot be localized because there is no interaction to suppress the dispersed effect of kinetic energy on the particles in the system, after which the uncertainty relationship occurs. The above results clearly show that the uncertainty relationship is caused by the quantum mechanics; the Hamiltonian in Eq. (9) does not represent truly the features of microscopic particles. In practice, all real systems are composed of many particles, for e.g. the simplest system, hydrogen atom, also contains two particles, this means that there is no system containing only one particle in nature. In such a case, the energy of particles is always related to their states, and the interactions including the nonlinear interaction among the particles or between the particle and background always exist. Therefore, it is not reasonable that the nonlinear interactions and Hamiltonian related to the wave function of state of the particle are replaced by an average field and Hamiltonian unrelated to the wave function of state of particle in the quantum mechanics, respectively [10– 11]. This indicates that the quantum mechanics is an approximate and linear theory and cannot describe truly the properties of motion of microscopic particles.
pffiffiffi
fs ðx0 ; t0 Þ ¼ 2 2Zsechf2Zðx0 x00 Þ 8Zxt0 g ð10Þ
pffiffiffiffiffiffiffi pffiffiffi pffiffiffi where x0 ¼ x 2m=‘ ; t 0 ¼ t=‘ , 2 2Z is the amplitude, 2 2x 0 0 denotes the velocity, y is a constant. The function fs ðx ; t Þ is a square integrable function localized at x00 a0 in the coordinate space. If the microscopic particle is localized at x00 a0, the Fourier transform of this function is 1 fs ðp; t Þ ¼ pffiffiffiffiffiffi 2p 0
Z
1 0
0
ipx0
fs ðx ; t Þe
dx
0
1
1
We can thus find that pffiffiffi pffiffiffi B2 p2 0 0 /x0 S ¼ 4 2ZB20 x00 ; /x 2 S ¼ 0pffiffiffi þ 4 2B20 Zx02 12 2Z
ð14Þ
respectively. Similarly, the momentum of the center of mass of the microscopic particle, /pS, and its square, /p2 S, are given by Z 1 Z 1 b ðpÞj2 dp; /p2 S ¼ b ðpÞj2 dp /pS ¼ pjf p2 jf ð15Þ s s 1
1
which yield
pffiffiffi pffiffiffi 32 2 2 3 3 B0 Z þ 32 2B20 Zx ð16Þ 3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The standard deviations of position Dx0 ¼ /x0 2 S /x0 S2 and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi momentum Dp ¼ /p2 S /pS2 are given by 2 pffiffiffi p p2 0 ðDx0 Þ2 ¼ B20 þ 4Zx02 ð1 4 2ZB20 Þ ¼ 12Z 96Z2
/pS ¼ 16B20 Zx; /p2 S ¼
pffiffiffi 1 pffiffiffi 8 3 ðDpÞ2 ¼ 32 2B20 Z3 þ Zx ð1 4 2ZB20 Þ ¼ Z2 3 3
Dx0 Dp ¼
If the nonlinear interactions in the systems are considered, then the nature and states of the microscopic particles should be ¨ described by the nonlinear Schrodinger equation (2). Then the uncertainty relationship of microscopic particles could be changed, due to the variations of natures of the microscopic particles. At V ¼ 0 and b ¼ 1, the solution, fs , of Eq. (2) is represented by [10,13,14]
2
The results in Eqs. (11) and (12) show that the microscopic particle is localized not only in position space in the shape of soliton, but also in the momentum space in a soliton. For convenience, we introduce the normalization coefficient B0 in 1 ffiffi Z, the position of the certain mass Eqs. (10) and (12), then B20 ¼ 4p 2 0 of the microscopic particle, /x0 S, and its square, /x 2 S, at t0 ¼ 0 given by Z 1 Z 1 0 0 /x0 S ¼ x0 jfs ðx0 Þj2 dx0 ; /x 2 S ¼ x 2 jfs ðx0 Þj2 dx0 ð13Þ
ð17Þ
respectively. Thus we obtain the uncertainty relation between the position and the momentum for the microscopic particle depicted ¨ by nonlinear Schrodinger equation in Eq. (2) at V(x0 ) ¼ 0 as
3. The uncertainty relationship of microscopic particles ¨ equation described by nonlinear Schrodinger
expf2ixx0 i4ðx Z2 Þt 0 þ iyg
4329
ð11Þ
1
It shows that fs ðp; t 0 Þ is localized at p in momentum space. For Eq. (10), the Fourier transform is explicitly given by " # rffiffiffiffi pffiffiffi p p sech pffiffiffi ðp 2 2xÞ fs ðp; t0 Þ ¼ 2 4 2Z pffiffiffi pffiffiffi 2 expfi4ðZ2 þ x px=2 2Þt 0 iðp 2 2xÞx00 þ iyg ð12Þ
p
ð18Þ
6
This result is not related to the features of the microscopic ¨ particle depicted by the nonlinear Schrodinger equation because Eq. (18) has nothing to do with the characteristic parameters of ¨ p in Eq. (18) comes from of the nonlinear Schrodinger equation. pffiffiffiffiffiffi the integral coefficient 1= 2p. For a quantized microscopic particle, p in Eq. (17) should be replaced by p‘ , because Eq. (11) is replaced by Z 1 1 0 fs ðp; t0 Þ ¼ pffiffiffiffiffiffiffiffiffiffi dx0 fs ðx0 ; t 0 Þeipx =‘ ð19Þ 2p‘ 1 The corresponding uncertainty relation of the quantum microscopic particle is given by
DxDp ¼
p‘ 6
¼
h 12
ð20Þ
The uncertainty relation in Eq. (20) or (18) differs from the DxDp4h=2 in quantum mechanics equation (7). However, the minimum value DxDp ¼ h=2 has not been obtained from the ¨ solutions of linear Schrodinger equation and observed in practical quantum mechanical systems up to now except for the coherent and squeezed states of microscopic particles. As a matter of fact, we can represent one-quantum coherent state of harmonic oscillator by [13–15] þ
b a bÞj0S b ¼ ea jaS ¼ expðab
2 =2
1 X
an
b pffiffiffiffiffiffiffiffiffiffiffiffib n1 n¼0
þn
j0S
ð21Þ
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X.-f. Pang / Physica B 404 (2009) 4327–4331
in the number picture, which is a coherent superposition of a large number of microscopic particles (quanta). Eq. (21) represents a state of collective excitation of particles. Thus rffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi ‘ b jaS ¼ i ‘ moða a Þ ða þ a Þ; /ajp x jaS ¼ /ajb 2om
which, respectively, are [15–20] Z 1 pffiffiffi Ns ¼ jfj2 dx0 ¼ 2 2A0 1
p ¼ i
Z
1 1
and
‘
2
x jaS ¼ /ajb
2om
¼
‘ om
where rffiffiffiffiffiffiffiffiffiffiffiffi b x¼
‘
2om
2
2
ða
ða
2
2
E¼
2
b jaS þ a þ 2aa þ 1Þ; /ajp 2
rffiffiffiffiffiffiffiffiffiffiffiffi ‘ om b þ b ðb bÞ; 2
þ
b ðbÞ b is the creation (annihilation) operator of microscopic and b particle (quantum), a and a* are some unknown functions, o is the frequency of the particle, m is its mass. Thus we can get ðDxÞ2 ¼
‘ 2om
; ðDpÞ2 ¼
‘ om 2
; /DxS2 /DpS2 ¼
h2 4
ð22Þ
For the squeezed state of the microscopic particle: jbS ¼ exp½bðbþ2 b2 Þj0S, which is a two-microscopic particle (quanta) coherent state, we can find that [13–16] /bjDx2 jbS ¼
‘ 2mo
e4b ; /bjDp2 jbS ¼
‘ mo 2
e4b
using a similar approach as the above . Here b is the squeezed coefficient and jbjo1. Thus, h Dx 1 8b ¼ e or Dp ¼ DxðomÞe8b 2 Dp mo
DxDp ¼ ;
1
jfx0 j2
1
þ a 2aa 1Þ
bþb bþ Þ; p b¼i ðb
Z
pffiffiffi ðf fx0 ffx0 Þ dx0 ¼ 2 2A0 ve ¼ Ns ve ¼ const
ð23Þ
This shows that the momentum of the microscopic particle (quantum) is squeezed in the two-quanta coherent state compared to that in the one-quantum coherent state. Why do the position and the momentum of microscopic ¨ particles depicted by nonlinear Schrodinger equation have a minimum uncertainty relationship? This is due to the fact that the particles themselves have the wave–corpuscle duality and determinant positions and sizes in such a case. For example, the solution in Eq. (10) of Eq. (2) at V(x,t) ¼ 0 can now be represented by [10,13–20]: ( pffiffiffi ) pffiffiffiffiffi A b pffiffiffiffiffiffiffi fðx; tÞ ¼ A0 sech p0ffiffiffi ½ 2mðx x0 Þ ve t eivc ½ 2mðxx0 Þve t=2‘ 2‘
1 1 jfj4 dx0 ¼ E0 þ Msol v2e 2 2
ð25Þ
pffiffiffi where Msol ¼ Ns ¼ 2 2A0 is a constant and effective mass of microscopic particle. According to the soliton theory [10–13], the soliton in Eq. (4) can move freely over macroscopic distances in a uniform velocity ve in space–time retaining its form, energy, momentum and other quasi-particle properties. Therefore the ¨ microscopic particles depicted by the nonlinear Schrodinger equation in Eq. (2) at Vðx0 Þ ¼ 0 have obviously a corpuscle feature and determinant positions and sizes. Thus they no longer satisfy the rules of motion and uncertainty relationship of particles in quantum mechanics. The above minimum uncertainty relationship arises from its wave-corpuscle duality because it is in essence a solitary wave and goes on propagation in time–space through a carrier wave. Therefore this investigation is helpful to understand the essence of uncertainty relationship as well as the features of microscopic particles. Therefore we can draw the conclusion that the minimum uncertainty relationship is a result of nonlinear effect, instead of linear effect, between the particles or a result of wave–corpuscle duality. From this result we see that when the features of the particles satisfy DxDp4h=2, the particles obey laws of motion in quantum mechanics, and the particles are some waves. When the features of the particles satisfy DxDp ¼ h=12 or p=6, the states of ¨ particles should be described by the nonlinear Schrodinger equation in Eq. (2) at Vðx0 Þ ¼ 0, and the particles have wave– corpuscle duality. If the features of the particles satisfy DxDp ¼ 0, then the particles can be treated as classical particles, with only corpuscle features. Therefore, the theory established by the ¨ nonlinear Schrodinger equation bridges the gap between the classical and quantum mechanics. Therefore to study the properties of microscopic particles described by the nonlinear ¨ Schrodinger equation has important significations in physics.
Acknowledgments The author would like to acknowledge the Major State Basic Research Development Program (973 program) of China for the financial support (Grant no. 2007CB936103).
ð24Þ where rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi v2e 2vc ve A0 ¼ 2 2Z ¼ ; jðx; tÞ 2b ( pffiffiffiffiffiffiffi ) A0 ½ 2mðx x0 Þ ve t pffiffiffi ¼ A0 sech 2‘ pffiffiffiffiffiffiffi is the envelop of the solution, pffiffiffi and expfivc ½ 2mðx x0 Þ ve t=2‘ g is its carrier wave, ve ¼ 2 2x is the group velocity of the envelop, vc is phase velocity of carrier wave. This solution is completely different from Eq. (8) and a bell-type soliton. The envelop j(x,t) is a slow varying function and the mass center of the particle, the position of the mass center ispjust ffiffiffiffiffiffiffi at x0, A0 is its amplitude, and its width is given by pW ffiffiffiffiffiffiffi¼ 2p‘ =ð mbA0 Þ. Thus, the size of the particle is A0 W ¼ 2p‘ = mb, and a constant. In the meanwhile, the particle also has determinant mass, momentum and energy,
References [1] D. Bohr, J. Bub, Phys. Rev. 48 (1935) 169. ¨ [2] E. Schrodinger, Collected Paper on Wave Mechanics, Blackie and Son, London, 1928. ¨ [3] E. Schrodinger, Proc. Cambridge Philos. Soc. 31 (1935) 555. [4] W. Heisenberg, Z. Phys. 33 (1925) 879; W. Heisenberg, H. Euler, Z. Phys. 98 (1936) 714. [5] M. Born, L. Infeld, Proc. Roy. Soc. A 144 (1934) 425. [6] P.A. Dirac, The Principles of Quantum Mechanics, Clarendon Press, Oxford, 1967. [7] S. Diner, D. Farque, G. Lochak, F. Selleri, The Wave-Particle Dualism, Riedel, Dordrecht, 1984. [8] M. Ferrero, A. Van der Merwe, New Developments on Fundamental Problems in Quantum Physics, Kluwer, Dordrecht, 1997. [9] M. Ferrero, A. Van der Merwe, Fundamental Problems in Quantum Physics, Kluwer, Dordrecht, 1995. [10] V.E. Zakharov, A.B. Shabat, Sov. Phys. JETP 34 (1972) 62; V.E. Zakharov, A.B. Shabat, Sov. Phys. JETP 37 (1973) 823. [11] A.S. Davydov, Solitons in Molecular Systems, second ed., Reidel Publishing Comp., Dordrecht, 1985 1991.
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¨ [12] C. Sulem, P.L. Sulem, The Nonlinrar Schrodinger Equation: Self-focusing and Wave Collapse, Springer, Berlin, 1999. [13] X.-f. Pang, Soliton Physics, Sichuan Science and Technology Press, Chengdu, 2003. [14] G. Bai-Lie, X.-f. Pang, Solitons, Chinese Science Press, Beijing, 1987. [15] X.-f. Pang, Phys. Status Solidi (B) 236 (2003) 34. [16] X.-f. Pang, J. Low. Temp. Phys. 58 (1985) 334.
4331
[17] X.-f. Pang, The Theory of Nonlinear Quantum Mechanics, Chinese Chongqing Press, Chongqing, 1994. [18] X.-f. Pang, Research and Development and of World Science and Technology (Chinese) 28 (2006) 11. [19] X.-f. Pang, Research and Development and of World Science and Technology (Chinese) 24 (2003) 54. [20] X.-f. Pang, Phys. Rev. E 62 (2000) 6989.
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Uncertainty features of microscopic particles described by nonlinear ¨ Schrodinger equation Pang Xiao-fenga,b, a b
Institute of Life Science and Technology, University of Electronic Science and Technology of China, Chengdu 610054, China International Centre for Materials Physics, Chinese Academy of Science, Shenyang 110015, China
a r t i c l e in fo
abstract
Article history: Received 6 October 2008 Received in revised form 15 July 2009 Accepted 3 August 2009
The uncertainty relationship between position and momentum of the microscopic particles is calculated by nonlinear quantum theory in which the states of the particles are described by a nonlinear ¨ Schrodinger equation. The results show that the uncertainty relation differs from that in the quantum mechanics and has a minimum value in this case. This means that the position and momentum of the particles could be determined simultaneously to a certain degree, which could be caused by the wave– ¨ corpuscle duality of the microscopic particles described by the nonlinear Schrodinger equation. & 2009 Published by Elsevier B.V.
PACS: 02.60.Cb 03.65.w Keywords: Microscopic particle Uncertainty relationship ¨ Nonlinear Schrodinger equation Quantum mechanics Nonlinear systems
1. Introduction As it is known, the states and properties of microscopic particles are depicted by the quantum mechanics, which was ¨ established by Bohr, Born, Schrodinger and Heisenberg in the early 1900s [1–7], and is also the pillar and foundation of modern science. In the quantum mechanics, the state of microscopic ¨ particles is described by the Schrodinger equation: i‘
2
@c ‘ ¼ r2 c þ Vð~ r ; tÞc @t 2m
ð1Þ
2 2 where ‘ r =2m is the kinetic energy operator, Vð~ r ; tÞ is the externally applied potential operator, m is the mass of particles, cð~ r ; tÞ is a wave function describing the states of particles, ~ r is its r ; tÞ is coordinate, and t is the time. Since no nonlinear terms of cð~ ¨ included in Eq. (1), hence we here refer to it as linear Schrodinger equation. If only the externally applied potentials are known, we can find the solutions of the equation. However, Eq. (1) is a wave equation and has only a wave solution. Then the microscopic
Correspondence address: Institute of Life Science and Technology, University of Electronic Science and Technology of China, Chengdu 610054, China. Tel.: +86 2883202595; fax: +86 2883208238. E-mail address: [email protected]
0921-4526/$ - see front matter & 2009 Published by Elsevier B.V. doi:10.1016/j.physb.2009.08.027
particles described by the quantum mechanics, which always disperse over the space–time in the form of a wave, do not have a r ; tÞj2 to represent the corpuscle nature. Thus we have to use jcð~ ! probability of the particles occurring in position r at time t 2 ~ according to Born’s idea. On account of jcðr ; tÞj which has a certain value at every point in the space–time, the particle can occur at each point at the same time. In this case, a fraction of the particle must appear in the systems, which is a very strange phenomenon and is quite difficult to understand. However, in physical experiments, the particles are always captured as a whole and not as a fractional one by a detector placed at an exact position. Therefore, the concept of probability representing the corpuscle behavior of the particles cannot be accepted. Because of the above nature of the quantum mechanics, some novel results, such as the uncertainty relationship between the position and the momentum and the mechanical quantities that are denoted by some average values in any state, also occur. These results are considerably incompatible not only with de k, of wave–corpuscle p ¼ ‘~ Broglie relation, E ¼ hu ¼ ‘ o and ~ duality for microscopic particles and Davisson and Germer’s experimental result of electron diffraction on double seam in 1927 [6–8], but also contradictory to the traditional concept of particles [7–9]. These problems of the quantum mechanics inevitably evoked the physicists’ discussion and contentions, where many
ARTICLE IN PRESS 4328
X.-f. Pang / Physica B 404 (2009) 4327–4331
scientists were involved in the debate. It was quite exceptional in the history of physics that so many prominent physicists from different institutions were involved and the scope of the debate has been so wide. Unfortunately, after being debated for almost a century, there remain no definite answers to most of these questions. In such a case, we want to know whether the uncertainty relationship is an intrinsic feature of microscopic particles or an artifact of the quantum mechanics or measuring instruments? This is worth studying deeply [7–9]. Therefore, to clarify the essence of the uncertainty relationship is very necessary in physics. This paper focuses on just this problem. Obviously, the uncertainty relationship is related to the natures ¨ of microscopic particles. Thus instead of the linear Schrodinger equation in Eq. (1) in quantum mechanics, we now use the ¨ following nonlinear Schrodinger equation [10–14] @f ‘ ¼ r2 f7bjfj2 f þ Vð~ r ; tÞf i‘ @t 2m
ð2Þ
to describe microscopic particles, where fð~ r ; tÞ is the wave function of the microscopic particle in this condition, b is a nonlinear interaction coefficient, the bjfj2 f is a nonlinear term, and represents a nonlinear interaction among the microscopic particles. In such a case, it is worth to study the concrete form and features that are the uncertainty relationship of microscopic particles. In this paper we calculate the uncertainty relationship of the position and the momentum of microscopic particles depicted in Eq. (2). We expect that the uncertainty relationship is different from that in the linear quantum mechanics. Then the significance of the uncertainty relationship can be revealed by comparing them in the two theories.
2. Correct form of uncertainty relationship in quantum mechanics and its roots We first calculate the uncertainty relationship of microscopic ¨ particles described by the linear Schrodinger equation in quantum mechanics. It is well known that the uncertainty relationship can be, in the coordinate representation, obtained from [1–7] Z ^ cð~ jðxDA^ þ iDBÞ r ; tÞj2 d~ r Z0 ð3Þ IðxÞ ¼
F^ ðxÞ ¼
Z
^ ~ ^ ~ c ð~ r ; tÞ~ F ½Að r ; tÞ; Bð r ; tÞcð~ r ; tÞ d~ r
where ~ A and ~ B are operators of two physical quantities, for example, position and momentum, or energy and time, and ^ B ^ ¼ iC^ , cðx; tÞ and c ðx; tÞ are satisfy the commutation relation ½A; wave functions of the microscopic particle satisfying the linear ¨ Schrodinger equation (1) and its conjugate equation, respectively, F^ ¼ ðDAx þ DBÞ2 , (DA^ ¼ A^ A, DB^ ¼ B^ B, A and B are the average values of the physical quantities in the state denoted by cðx; tÞ), is an operator of physical quantity related to A and B, x is a real variational parameter. After some simplifications, we can get 2 2 2 I ¼ F ¼ DA^ x þ 2DA^ DB^ x þ DB^ Z0
or 2
2
DA^ x2 þ C^ x þ DB^ Z0
ð4Þ
Using mathematical identities, this can be written as 2
2
DA^ DB^ Z
ðC^ Þ2 4
or
DA^
2
!2
C^
xþ
2DA^
2
2
þDB^
ðC^ Þ2 4DA^
2
ð6Þ
Z0
2
2 This equation shows that DA^ a0, if ðDA^ DB^ Þ2 or C^ =4 is not zero. 2
2
Or else, we cannot obtain Eq. (5) and DA^ DB^ 4ðDADBÞ2 because 2 2 when DA^ ¼ 0, Eq. (6) does not hold. Thus, ðDA^ Þa0 is a necessary
2
or
This is the uncertainty relationship often used in linear quantum mechanics. Since Eq. (4) can be written in the following form: !2 ^ ^ ^ ^ 2 ^F ¼ DA^ 2 x þ DA DB þDB^ 2 ðDA DB Þ Z0 2 2 DA^ DA^
ð5Þ
2 condition for the uncertainty relationship in Eq. (5). DA^ can approach zero, but cannot be equal to zero. This means that Eq. (6) should be denoted as !2 2 2 ðC^ Þ2 C^ ^ DA x þ þDB^ 40 2 2 2DA^ 4DA^
Then a right uncertainty relationship in quantum mechanics should take the form: 2
2
DA^ DB^ 4
ðC^ Þ2 4
ð7Þ
The uncertainty relationship in quantum mechanics is an important representation and indicates that the values of conjugate mechanical quantities, such as the position and the momentum, or energy and time cannot be, permanently accurately and simultaneously determined in quantum mechanics. How does this uncertainty relationship interpret? What is its essence? From the above calculation we know clearly that the uncertainty relation in Eq. (7) was obtained based on fundamental hypotheses of the linear quantum mechanics, including properties of operators of the mechanical quantities, the state of microscopic particle represented by the wave function, which satisfies the € linear Schrodinger equation (1), the concept of average values of mechanical quantities and the commutation relations and eigenequation of operators. Therefore, we can conclude that the uncertainty relationship in Eq. (7) is a necessary result of quantum mechanics, not related to the measurement at all. Thus the basic reason for this relationship is just that the microscopic particles have only a wave feature, not a corpuscle nature in quantum mechanics, and thus the position and momentum of the particle simultaneously are not determinant. Then the uncertainty relation also exists necessarily. This analysis can be also verified from the solutions of Eq. (1). As it is known, the solution of Eq. (1) at Vð~ r ; tÞ ¼ 0 is represented by cð~ r ; tÞ ¼ A0 exp½ið~ k ~ r otÞ ð8Þ it is a plane wave, where k is its wavevector, o is the frequency r ; tÞa0, the solutions are a de and A0 is the amplitude. When Vð~ Broglie wave or a Bloch wave, etc. Therefore, the microscopic particle is represented by a wave in the quantum mechanics, it disperses over whole space and cannot be localized, thus the r ; tÞj2 , of the microscopic particle is not equal to probability, jcð~ zero at all points. This means that the particle does not have a determinant position in this space and thus the uncertainty relationship of its position and momentum occurs inevitably in the quantum mechanics.
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Why does this phenomenon occur in the quantum mechanics? As it is known, the Hamiltonian operator of the systems’ corresponding Eq. (1) in the quantum mechanics is [1–7] 2 b r ; tÞ HðtÞ ¼ ‘ r2 =2m þ Vð~
ð9Þ
Obviously, it consists only of kinetic and potential operator of particles, the potential operator is not related to the wave function of state of the particle, and thus can only change the states of microscopic particles, and cannot change its nature. Then the nature of the particles is only determined by the kinetic term, thus it disperses over the whole space and cannot be localized because there is no interaction to suppress the dispersed effect of kinetic energy on the particles in the system, after which the uncertainty relationship occurs. The above results clearly show that the uncertainty relationship is caused by the quantum mechanics; the Hamiltonian in Eq. (9) does not represent truly the features of microscopic particles. In practice, all real systems are composed of many particles, for e.g. the simplest system, hydrogen atom, also contains two particles, this means that there is no system containing only one particle in nature. In such a case, the energy of particles is always related to their states, and the interactions including the nonlinear interaction among the particles or between the particle and background always exist. Therefore, it is not reasonable that the nonlinear interactions and Hamiltonian related to the wave function of state of the particle are replaced by an average field and Hamiltonian unrelated to the wave function of state of particle in the quantum mechanics, respectively [10– 11]. This indicates that the quantum mechanics is an approximate and linear theory and cannot describe truly the properties of motion of microscopic particles.
pffiffiffi
fs ðx0 ; t0 Þ ¼ 2 2Zsechf2Zðx0 x00 Þ 8Zxt0 g ð10Þ
pffiffiffiffiffiffiffi pffiffiffi pffiffiffi where x0 ¼ x 2m=‘ ; t 0 ¼ t=‘ , 2 2Z is the amplitude, 2 2x 0 0 denotes the velocity, y is a constant. The function fs ðx ; t Þ is a square integrable function localized at x00 a0 in the coordinate space. If the microscopic particle is localized at x00 a0, the Fourier transform of this function is 1 fs ðp; t Þ ¼ pffiffiffiffiffiffi 2p 0
Z
1 0
0
ipx0
fs ðx ; t Þe
dx
0
1
1
We can thus find that pffiffiffi pffiffiffi B2 p2 0 0 /x0 S ¼ 4 2ZB20 x00 ; /x 2 S ¼ 0pffiffiffi þ 4 2B20 Zx02 12 2Z
ð14Þ
respectively. Similarly, the momentum of the center of mass of the microscopic particle, /pS, and its square, /p2 S, are given by Z 1 Z 1 b ðpÞj2 dp; /p2 S ¼ b ðpÞj2 dp /pS ¼ pjf p2 jf ð15Þ s s 1
1
which yield
pffiffiffi pffiffiffi 32 2 2 3 3 B0 Z þ 32 2B20 Zx ð16Þ 3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The standard deviations of position Dx0 ¼ /x0 2 S /x0 S2 and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi momentum Dp ¼ /p2 S /pS2 are given by 2 pffiffiffi p p2 0 ðDx0 Þ2 ¼ B20 þ 4Zx02 ð1 4 2ZB20 Þ ¼ 12Z 96Z2
/pS ¼ 16B20 Zx; /p2 S ¼
pffiffiffi 1 pffiffiffi 8 3 ðDpÞ2 ¼ 32 2B20 Z3 þ Zx ð1 4 2ZB20 Þ ¼ Z2 3 3
Dx0 Dp ¼
If the nonlinear interactions in the systems are considered, then the nature and states of the microscopic particles should be ¨ described by the nonlinear Schrodinger equation (2). Then the uncertainty relationship of microscopic particles could be changed, due to the variations of natures of the microscopic particles. At V ¼ 0 and b ¼ 1, the solution, fs , of Eq. (2) is represented by [10,13,14]
2
The results in Eqs. (11) and (12) show that the microscopic particle is localized not only in position space in the shape of soliton, but also in the momentum space in a soliton. For convenience, we introduce the normalization coefficient B0 in 1 ffiffi Z, the position of the certain mass Eqs. (10) and (12), then B20 ¼ 4p 2 0 of the microscopic particle, /x0 S, and its square, /x 2 S, at t0 ¼ 0 given by Z 1 Z 1 0 0 /x0 S ¼ x0 jfs ðx0 Þj2 dx0 ; /x 2 S ¼ x 2 jfs ðx0 Þj2 dx0 ð13Þ
ð17Þ
respectively. Thus we obtain the uncertainty relation between the position and the momentum for the microscopic particle depicted ¨ by nonlinear Schrodinger equation in Eq. (2) at V(x0 ) ¼ 0 as
3. The uncertainty relationship of microscopic particles ¨ equation described by nonlinear Schrodinger
expf2ixx0 i4ðx Z2 Þt 0 þ iyg
4329
ð11Þ
1
It shows that fs ðp; t 0 Þ is localized at p in momentum space. For Eq. (10), the Fourier transform is explicitly given by " # rffiffiffiffi pffiffiffi p p sech pffiffiffi ðp 2 2xÞ fs ðp; t0 Þ ¼ 2 4 2Z pffiffiffi pffiffiffi 2 expfi4ðZ2 þ x px=2 2Þt 0 iðp 2 2xÞx00 þ iyg ð12Þ
p
ð18Þ
6
This result is not related to the features of the microscopic ¨ particle depicted by the nonlinear Schrodinger equation because Eq. (18) has nothing to do with the characteristic parameters of ¨ p in Eq. (18) comes from of the nonlinear Schrodinger equation. pffiffiffiffiffiffi the integral coefficient 1= 2p. For a quantized microscopic particle, p in Eq. (17) should be replaced by p‘ , because Eq. (11) is replaced by Z 1 1 0 fs ðp; t0 Þ ¼ pffiffiffiffiffiffiffiffiffiffi dx0 fs ðx0 ; t 0 Þeipx =‘ ð19Þ 2p‘ 1 The corresponding uncertainty relation of the quantum microscopic particle is given by
DxDp ¼
p‘ 6
¼
h 12
ð20Þ
The uncertainty relation in Eq. (20) or (18) differs from the DxDp4h=2 in quantum mechanics equation (7). However, the minimum value DxDp ¼ h=2 has not been obtained from the ¨ solutions of linear Schrodinger equation and observed in practical quantum mechanical systems up to now except for the coherent and squeezed states of microscopic particles. As a matter of fact, we can represent one-quantum coherent state of harmonic oscillator by [13–15] þ
b a bÞj0S b ¼ ea jaS ¼ expðab
2 =2
1 X
an
b pffiffiffiffiffiffiffiffiffiffiffiffib n1 n¼0
þn
j0S
ð21Þ
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X.-f. Pang / Physica B 404 (2009) 4327–4331
in the number picture, which is a coherent superposition of a large number of microscopic particles (quanta). Eq. (21) represents a state of collective excitation of particles. Thus rffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi ‘ b jaS ¼ i ‘ moða a Þ ða þ a Þ; /ajp x jaS ¼ /ajb 2om
which, respectively, are [15–20] Z 1 pffiffiffi Ns ¼ jfj2 dx0 ¼ 2 2A0 1
p ¼ i
Z
1 1
and
‘
2
x jaS ¼ /ajb
2om
¼
‘ om
where rffiffiffiffiffiffiffiffiffiffiffiffi b x¼
‘
2om
2
2
ða
ða
2
2
E¼
2
b jaS þ a þ 2aa þ 1Þ; /ajp 2
rffiffiffiffiffiffiffiffiffiffiffiffi ‘ om b þ b ðb bÞ; 2
þ
b ðbÞ b is the creation (annihilation) operator of microscopic and b particle (quantum), a and a* are some unknown functions, o is the frequency of the particle, m is its mass. Thus we can get ðDxÞ2 ¼
‘ 2om
; ðDpÞ2 ¼
‘ om 2
; /DxS2 /DpS2 ¼
h2 4
ð22Þ
For the squeezed state of the microscopic particle: jbS ¼ exp½bðbþ2 b2 Þj0S, which is a two-microscopic particle (quanta) coherent state, we can find that [13–16] /bjDx2 jbS ¼
‘ 2mo
e4b ; /bjDp2 jbS ¼
‘ mo 2
e4b
using a similar approach as the above . Here b is the squeezed coefficient and jbjo1. Thus, h Dx 1 8b ¼ e or Dp ¼ DxðomÞe8b 2 Dp mo
DxDp ¼ ;
1
jfx0 j2
1
þ a 2aa 1Þ
bþb bþ Þ; p b¼i ðb
Z
pffiffiffi ðf fx0 ffx0 Þ dx0 ¼ 2 2A0 ve ¼ Ns ve ¼ const
ð23Þ
This shows that the momentum of the microscopic particle (quantum) is squeezed in the two-quanta coherent state compared to that in the one-quantum coherent state. Why do the position and the momentum of microscopic ¨ particles depicted by nonlinear Schrodinger equation have a minimum uncertainty relationship? This is due to the fact that the particles themselves have the wave–corpuscle duality and determinant positions and sizes in such a case. For example, the solution in Eq. (10) of Eq. (2) at V(x,t) ¼ 0 can now be represented by [10,13–20]: ( pffiffiffi ) pffiffiffiffiffi A b pffiffiffiffiffiffiffi fðx; tÞ ¼ A0 sech p0ffiffiffi ½ 2mðx x0 Þ ve t eivc ½ 2mðxx0 Þve t=2‘ 2‘
1 1 jfj4 dx0 ¼ E0 þ Msol v2e 2 2
ð25Þ
pffiffiffi where Msol ¼ Ns ¼ 2 2A0 is a constant and effective mass of microscopic particle. According to the soliton theory [10–13], the soliton in Eq. (4) can move freely over macroscopic distances in a uniform velocity ve in space–time retaining its form, energy, momentum and other quasi-particle properties. Therefore the ¨ microscopic particles depicted by the nonlinear Schrodinger equation in Eq. (2) at Vðx0 Þ ¼ 0 have obviously a corpuscle feature and determinant positions and sizes. Thus they no longer satisfy the rules of motion and uncertainty relationship of particles in quantum mechanics. The above minimum uncertainty relationship arises from its wave-corpuscle duality because it is in essence a solitary wave and goes on propagation in time–space through a carrier wave. Therefore this investigation is helpful to understand the essence of uncertainty relationship as well as the features of microscopic particles. Therefore we can draw the conclusion that the minimum uncertainty relationship is a result of nonlinear effect, instead of linear effect, between the particles or a result of wave–corpuscle duality. From this result we see that when the features of the particles satisfy DxDp4h=2, the particles obey laws of motion in quantum mechanics, and the particles are some waves. When the features of the particles satisfy DxDp ¼ h=12 or p=6, the states of ¨ particles should be described by the nonlinear Schrodinger equation in Eq. (2) at Vðx0 Þ ¼ 0, and the particles have wave– corpuscle duality. If the features of the particles satisfy DxDp ¼ 0, then the particles can be treated as classical particles, with only corpuscle features. Therefore, the theory established by the ¨ nonlinear Schrodinger equation bridges the gap between the classical and quantum mechanics. Therefore to study the properties of microscopic particles described by the nonlinear ¨ Schrodinger equation has important significations in physics.
Acknowledgments The author would like to acknowledge the Major State Basic Research Development Program (973 program) of China for the financial support (Grant no. 2007CB936103).
ð24Þ where rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi v2e 2vc ve A0 ¼ 2 2Z ¼ ; jðx; tÞ 2b ( pffiffiffiffiffiffiffi ) A0 ½ 2mðx x0 Þ ve t pffiffiffi ¼ A0 sech 2‘ pffiffiffiffiffiffiffi is the envelop of the solution, pffiffiffi and expfivc ½ 2mðx x0 Þ ve t=2‘ g is its carrier wave, ve ¼ 2 2x is the group velocity of the envelop, vc is phase velocity of carrier wave. This solution is completely different from Eq. (8) and a bell-type soliton. The envelop j(x,t) is a slow varying function and the mass center of the particle, the position of the mass center ispjust ffiffiffiffiffiffiffi at x0, A0 is its amplitude, and its width is given by pW ffiffiffiffiffiffiffi¼ 2p‘ =ð mbA0 Þ. Thus, the size of the particle is A0 W ¼ 2p‘ = mb, and a constant. In the meanwhile, the particle also has determinant mass, momentum and energy,
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