- Email: [email protected]
A boundary effect in quantum mechanics
4 October 1999
Physics Letters A 261 Ž1999. 30–33 www.elsevier.nlrlocaterphysleta
A boundary effect in quantum mechanics Yao Qian-Kai
1
Department of Physics, Zhengzhou UniÕersity, Zhengzhou, Henan 450052, People’s Republic of China Received 25 November 1998; received in revised form 28 May 1999; accepted 1 June 1999 Communicated by P.R. Holland
Abstract We show that a quantum system consisting of particles inside an infinite spherical well with a moving wall is equivalent to the particle interacting with an electromagnetic field that dependens on the wall motion. The effective Hamiltonian is constructed and the properties of the electromagnetic field are discussed. q 1999 Published by Elsevier Science B.V. All rights reserved.
1. Introduction In 1983 Berry w1x discovered that a localized change in the spatial boundary conditions of a quantum mechanical particle can lead to a geometrical phase of its vector state in addition to the dynamic one, and therefore this is called the Berry phase. Since then, much attention w2,3x has been paid to the set of problems deeply connected with the geometrical boundary conditions of the quantum system. It turns out that state space anholonomy played a central role in the investigation of the evolution dynamics of the quantum system with time dependent boundary w4,5x. As an analysis of the boundary quantum effect, we here consider a similar problem: the quantum behaviour of the particle bounded by an infinite spherical potential well with a moving wall. The result shows that the change in the geometrical boundary conditions can lead to an effective electro-
1
magnetic field which can exert interaction on the particle. Finally, we will determine this electromagnetic field.
2. Equation and its solutions The actual boundary quantum effect is very simple, it consists of a particle of mass M inside an infinite spherically symmetric square well of range a 0 , for which V Ž r . s 0 as r F a 0 and ` otherwise. So that the particle can be considered to be confined to the inner part of the well, far away from the wall. This situation is just like the quark bounded by a ‘bag’, the so-called ‘bag model’ for discussing the quark confinement. We use the spherical polar coordinates Ž r, u , f . and give the original Hamiltonian of this mechanical system Hˆ0 s y
Fax: q86-0371-7973895; e-mail: [email protected]
"2 2m
"2
2
= sy
2m
0375-9601r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 9 9 . 0 0 6 2 2 - 2
ž
2
=r q
=u ,2f r2
/
.
Ž 1.
Y. Qian-Kair Physics Letters A 261 (1999) 30–33
As usual the quantum state of the particle is described by the wave function C Ž r, u , f ,t ., which is governed by the Schrodinger equation ¨
EC i"
Et
s Hˆ0C ,
Ž 2.
C Ž r s a0 ,t . s 0 .
Ž 3.
It gives the Hamiltonian eigenfunctions
En Ž a 0 . t ,
Ž 4.
(
t
X
H E Ž aŽ t . . dt " 0 n
X
,
Ž 5.
(
xn r l a
,
Ž 6.
xn r l is the n r th solution of the spherical Bessel function of order l, n r the radial quantum number. The normalization constant Cn is expressed as y2 Cn s
a 3 jly1 Ž xn r l . jlq1 Ž xn r l .
E
E Cn
=a s
E lnCn
yi
jl Yl m q Cn
E k E jl
Y . Ž 9. Ea E a E k lm In the spherical polar coordinate, we have operator rˆ ™ i 1k EEk k, and the result of it acting on the Bessel functions, rj ˆ l Ž kr . s rjl Ž kr ., suggests
Ek
Ž rlnj ˆ l. y
E lnk
Ea Ea Ea E ln Ž Cnrk . E lnk s y ir Ž iPk. , Ž 10 . Ea Ea here the relation EEk s 1k Ž EEk k y 1 . is used, i s kr< k < denotes the unit vector in momentum direction. With help of Eqs. Ž6. and Ž7. and kˆ ™ yi= , we can decompose =a through a connection form in the infinitesimal one =a s y
here the wave number k Ž a. s 2 MEn Ž a . r" can be defined by the boundary condition < jl Ž ka.: s 0, that is ks
We now analyze the action of the operator ErE t on
=a
where EnŽ a0 . are the eigenvalues for Hˆ0 , the normalization constant CnŽ a0 . and the wave number k Ž a 0 . Žs 2 MEn Ž a0 . r" . are relative to the potential scale a 0 , jl Ž k Ž a0 . r . the spherical Bessel functions w6x. If now the range of the potential well is variable, so that the confining bag expands or shrinks, namely a 0 ™ a s aŽ t ., we must solve the Schrodinger equa¨ tion subject to the time dependent boundary condition, C Ž r s aŽ t .,t . s 0. The normalized solutions to it take the time dependent form as
=exp y
3. Effective Hamiltonian
i"
endowed with boundary condition
=exp y
31
1
r q
2a
a
iP= .
Ž 11 .
Now, it is possible to consider our original problem as a problem with the effective Hamiltonian Hˆeff s Hˆ0 y i"
a˙ a
1 2
y ri P = .
Ž 12 .
The ordinary time derivative in the new Schrodinger ¨ equation is
EC
s HˆeffC . Ž 13 . Et The effective Hamiltonian is the Hamiltonian for the interaction between the charge particle and the electromagnetic field due to the vector potential that only has a spherical component w5x i"
1r2
.
Ž 7.
A Ž r ,t . s
Mca˙ qa
reu , f ,
Ž 14 .
Y. Qian-Kair Physics Letters A 261 (1999) 30–33
32
where q is the charge of the particle in terms of A, e r and eu , f represent the unit vectors along the radial and spherical directions respectively. So the result for the action of the operator = s 1r Ž EEr r q =u , f . on A is immediately obtained
Ž= P A. s
Mca˙ qa
=u , f eu , f s y
Mca˙ qa
.
Ž 15 .
This relation can help us to express the effective Hamiltonian for the particle of charge q moving in the electromagnetic field as i"q
Hˆeff s Hˆ0 q
2 Mc i"q
s Hˆ0 q
Ž = P A . q i"
ar ˙ a
iP=
w= P A q A P = x 2 Mc ar ˙ q i" Ž i y eu , f . P = . a
Ž 16 .
In the position representation, the momentum operator is pˆ s yi"= , and thus we have approximately Hˆeff f
1 2M
ž
pˆ y
q c
2
A
/
q qf q ,
Ž 17 .
effective Hamiltonian for a particle inside a spherical well with a moving wall. This Hamiltonian reflects the interaction between the charged particle and the electromagnetic field caused by the motion of the geometrical boundary. And therefore, we have the fact that, when the boundary condition becomes stationary Ži.e. a˙ s 0., the field tends to zero and the effective Hamiltonian comes back Eq. Ž1.. Further, Comparing Eq. Ž17. with Eq. Ž1., we find Hˆ0 can be achieved through the following transformation w7,8x Hˆeff s U Ž t . Hˆ0 Uy1 Ž t . , Ž 19 . where UŽ t . s expŽ if Ž t .. and the gauge function f Ž t . s qrŽ "c .HA P dr y qr"Hf q dt. The Hamiltonian eigenfunctions also experience a time-dependent transformation such that
with the scalar potential
f q Ž r ,t . s
a˙ qa
r Ž pu , f y p . s
a˙ qa
a
ž
(
yLln
L y r 2 MEn ,
/
Ž 18 . which represents the work on the system caused by the motion of the wall. L s < r = p < s rpu , f is the orbital angular momentum of the particle, which has the eigenvalue l Ž l q 1 . ". Here, we omit the term proportional to A2 Ž r,t . since only a slow wall motion is considered. Clearly, Eq. Ž18. shows that, when L s 0, f q becomes yarr ˙ Ž qa. 2 MEn . HowŽ . ever, p s pu , f i.e. pr s 0 suggests f q Ž r,t . tends to zero. This is an example of gauge field appearance in a quantum mechanical system with time dependent boundary conditions.
(
(
a0 for
a˙ s a ,
i "
(
ž
Ma r 2
q r 2 MEn
u q sin uf a
at
a y
a0
a0
En t
/
,
Ž 21 .
in the case of a˙ s a s 0, The next step is to determine the vector potential A for a given situation, specified in terms of the magnetic field vector B. We now clarify the determination of the vector potential A by the following expression Mca˙ Mc a L As reu , f s = r , for a˙ s a . qa qa L Ž 22 . Considering the fact that the spherical vector potential can be expressed in terms of the magnetic field B as
4. Non-local effect
A s 12 Ž B = r . ,
By introducing the boundary quantity a into Hamiltonian eigenfunctions Ždifferent from the method adopted in Ref. w5x., we have constructed the
we obtain 2 Mc a Bs L. qaL
Ž 23 . Ž 24 .
Y. Qian-Kair Physics Letters A 261 (1999) 30–33
It shows that, when a ) 0 or a - 0, the vector field B is respectively parallel or anti-parallel to the orbital angular momentum vector L of the particle. According to the model of the orbital angular momentum, the vector L, of length l Ž l q 1 . ", processes about the z-axis, the Ž2 l q 1. allowed projections of L on the z-axis is given by m", with m s yl,y l q 1, . . . ,q l. The vector B of length 2 mc arŽ qa. has the same feature, it also processes about the z-axis and has z-component
(
Bz s
2 Mc a L z qa
L
2 Mc a
m
qa
(l Ž l q 1.
s
,
Ž 25 .
note that the x- and y-components are still not zero although the values of them always vanish. So that, the vector B may be viewed as lying on the surface of a cone with altitude B z which has the z-axis as its symmetry axis, all orientations of B on the surface of the cone being equally likely. And naturally, we have the conclusion that the two components B x and B y cannot in general be assigned precise values simultaneously, just like L x , L y . Up to now, we have shown that a quantum system with time dependent boundary conditions is equivalent to the particle interacting with a magnetic field relative to a confining condition. And with the help of the relation A P p s Ž B = r . P pr2 s B P Lr2, we can rewrite the terms linear in A in Eq. Ž17. as q y Ž pˆ P A . q 2 A P p 2 Mc q sy Ž pˆ P A . q B P L , Ž 26 . 2 Mc so that the effective Hamiltonian becomes q q Hˆeff s Hˆ0 y B P L q qf q . Ž pˆ P A . y 2 Mc 2 Mc
By this example, we analyze a new effect that has been engendered by time dependent boundary conditions. This effect represents the interaction between the charge particle and the electromagnetic field due to the wall motion, and thus the system energy shifts.
5. Summary A discussion of the boundary effect has been presented. We now conclude that a non-stationary confinement of particle can lead to the appearance of an effective electromagnetic field. In our consideration we perform the usual treatment for the particle inside an infinite spherical well, similar to the quark confined by a bag, i.e. the bag model suggested in discussing quark confinement. Therefore, a better application of this work is in connection with quark confinement. If now the bag confining the quark expands or constricts, an effective electromagnetic field will appear accordingly. Perhaps this interpretation can provide part of the explanation for the electromagnetic nature of the particle. Finally, we wish to demonstrate that, besides the spherical well, such a result is probably applicable to enclosures of any shape. And in principle, we can say that the effect certainly appears in any process where one tries to confine a particle to a smaller region. For this effect, one can want ideally an electromagnetic force related to the time dependent boundary conditions acting on the particle to produce.
References
Ž 27 . The second last linear term in expression Ž27. represents the energy of the magnetic moment m s qrŽ2 Mc . L Žcaused by the orbital motion of the particle of charge q . interacting with the magnetic field B, that is a W s ym P B s y l Ž l q 1. " . Ž 28 . a
(
33
w1x w2x w3x w4x w5x w6x
M.V. Berry, Proc. Roy. Soc. ŽLondon. A 392 Ž1984. 45. B. Simon, Phys. Rev. Lett. 51 Ž1983. 2167. J.H. Hannay, J. Phys. A 18 Ž1985. 221. D.M. Greenberger, Physica B 151 Ž1988. 374. P. Pereshogin, P. Pronin, Phys. Lett. A 156 Ž1991. 12. B.H. Bransden, C.J. Joachain, Introduction to Quantum Mechanics, Longman Group UK Ltd, London, 1989. w7x M. Razary, Phys. Rev. A 44 Ž1991. 2384. w8x F.L. Li, S.J. Wang, A. Weiguny, D.L. Lin, J. Phys. A 27 Ž1994. 985. w9x M.V. Berry, G. Klein, J. Phys. A 17 Ž1984. 1805.
Physics Letters A 261 Ž1999. 30–33 www.elsevier.nlrlocaterphysleta
A boundary effect in quantum mechanics Yao Qian-Kai
1
Department of Physics, Zhengzhou UniÕersity, Zhengzhou, Henan 450052, People’s Republic of China Received 25 November 1998; received in revised form 28 May 1999; accepted 1 June 1999 Communicated by P.R. Holland
Abstract We show that a quantum system consisting of particles inside an infinite spherical well with a moving wall is equivalent to the particle interacting with an electromagnetic field that dependens on the wall motion. The effective Hamiltonian is constructed and the properties of the electromagnetic field are discussed. q 1999 Published by Elsevier Science B.V. All rights reserved.
1. Introduction In 1983 Berry w1x discovered that a localized change in the spatial boundary conditions of a quantum mechanical particle can lead to a geometrical phase of its vector state in addition to the dynamic one, and therefore this is called the Berry phase. Since then, much attention w2,3x has been paid to the set of problems deeply connected with the geometrical boundary conditions of the quantum system. It turns out that state space anholonomy played a central role in the investigation of the evolution dynamics of the quantum system with time dependent boundary w4,5x. As an analysis of the boundary quantum effect, we here consider a similar problem: the quantum behaviour of the particle bounded by an infinite spherical potential well with a moving wall. The result shows that the change in the geometrical boundary conditions can lead to an effective electro-
1
magnetic field which can exert interaction on the particle. Finally, we will determine this electromagnetic field.
2. Equation and its solutions The actual boundary quantum effect is very simple, it consists of a particle of mass M inside an infinite spherically symmetric square well of range a 0 , for which V Ž r . s 0 as r F a 0 and ` otherwise. So that the particle can be considered to be confined to the inner part of the well, far away from the wall. This situation is just like the quark bounded by a ‘bag’, the so-called ‘bag model’ for discussing the quark confinement. We use the spherical polar coordinates Ž r, u , f . and give the original Hamiltonian of this mechanical system Hˆ0 s y
Fax: q86-0371-7973895; e-mail: [email protected]
"2 2m
"2
2
= sy
2m
0375-9601r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 9 9 . 0 0 6 2 2 - 2
ž
2
=r q
=u ,2f r2
/
.
Ž 1.
Y. Qian-Kair Physics Letters A 261 (1999) 30–33
As usual the quantum state of the particle is described by the wave function C Ž r, u , f ,t ., which is governed by the Schrodinger equation ¨
EC i"
Et
s Hˆ0C ,
Ž 2.
C Ž r s a0 ,t . s 0 .
Ž 3.
It gives the Hamiltonian eigenfunctions
En Ž a 0 . t ,
Ž 4.
(
t
X
H E Ž aŽ t . . dt " 0 n
X
,
Ž 5.
(
xn r l a
,
Ž 6.
xn r l is the n r th solution of the spherical Bessel function of order l, n r the radial quantum number. The normalization constant Cn is expressed as y2 Cn s
a 3 jly1 Ž xn r l . jlq1 Ž xn r l .
E
E Cn
=a s
E lnCn
yi
jl Yl m q Cn
E k E jl
Y . Ž 9. Ea E a E k lm In the spherical polar coordinate, we have operator rˆ ™ i 1k EEk k, and the result of it acting on the Bessel functions, rj ˆ l Ž kr . s rjl Ž kr ., suggests
Ek
Ž rlnj ˆ l. y
E lnk
Ea Ea Ea E ln Ž Cnrk . E lnk s y ir Ž iPk. , Ž 10 . Ea Ea here the relation EEk s 1k Ž EEk k y 1 . is used, i s kr< k < denotes the unit vector in momentum direction. With help of Eqs. Ž6. and Ž7. and kˆ ™ yi= , we can decompose =a through a connection form in the infinitesimal one =a s y
here the wave number k Ž a. s 2 MEn Ž a . r" can be defined by the boundary condition < jl Ž ka.: s 0, that is ks
We now analyze the action of the operator ErE t on
=a
where EnŽ a0 . are the eigenvalues for Hˆ0 , the normalization constant CnŽ a0 . and the wave number k Ž a 0 . Žs 2 MEn Ž a0 . r" . are relative to the potential scale a 0 , jl Ž k Ž a0 . r . the spherical Bessel functions w6x. If now the range of the potential well is variable, so that the confining bag expands or shrinks, namely a 0 ™ a s aŽ t ., we must solve the Schrodinger equa¨ tion subject to the time dependent boundary condition, C Ž r s aŽ t .,t . s 0. The normalized solutions to it take the time dependent form as
=exp y
3. Effective Hamiltonian
i"
endowed with boundary condition
=exp y
31
1
r q
2a
a
iP= .
Ž 11 .
Now, it is possible to consider our original problem as a problem with the effective Hamiltonian Hˆeff s Hˆ0 y i"
a˙ a
1 2
y ri P = .
Ž 12 .
The ordinary time derivative in the new Schrodinger ¨ equation is
EC
s HˆeffC . Ž 13 . Et The effective Hamiltonian is the Hamiltonian for the interaction between the charge particle and the electromagnetic field due to the vector potential that only has a spherical component w5x i"
1r2
.
Ž 7.
A Ž r ,t . s
Mca˙ qa
reu , f ,
Ž 14 .
Y. Qian-Kair Physics Letters A 261 (1999) 30–33
32
where q is the charge of the particle in terms of A, e r and eu , f represent the unit vectors along the radial and spherical directions respectively. So the result for the action of the operator = s 1r Ž EEr r q =u , f . on A is immediately obtained
Ž= P A. s
Mca˙ qa
=u , f eu , f s y
Mca˙ qa
.
Ž 15 .
This relation can help us to express the effective Hamiltonian for the particle of charge q moving in the electromagnetic field as i"q
Hˆeff s Hˆ0 q
2 Mc i"q
s Hˆ0 q
Ž = P A . q i"
ar ˙ a
iP=
w= P A q A P = x 2 Mc ar ˙ q i" Ž i y eu , f . P = . a
Ž 16 .
In the position representation, the momentum operator is pˆ s yi"= , and thus we have approximately Hˆeff f
1 2M
ž
pˆ y
q c
2
A
/
q qf q ,
Ž 17 .
effective Hamiltonian for a particle inside a spherical well with a moving wall. This Hamiltonian reflects the interaction between the charged particle and the electromagnetic field caused by the motion of the geometrical boundary. And therefore, we have the fact that, when the boundary condition becomes stationary Ži.e. a˙ s 0., the field tends to zero and the effective Hamiltonian comes back Eq. Ž1.. Further, Comparing Eq. Ž17. with Eq. Ž1., we find Hˆ0 can be achieved through the following transformation w7,8x Hˆeff s U Ž t . Hˆ0 Uy1 Ž t . , Ž 19 . where UŽ t . s expŽ if Ž t .. and the gauge function f Ž t . s qrŽ "c .HA P dr y qr"Hf q dt. The Hamiltonian eigenfunctions also experience a time-dependent transformation such that
with the scalar potential
f q Ž r ,t . s
a˙ qa
r Ž pu , f y p . s
a˙ qa
a
ž
(
yLln
L y r 2 MEn ,
/
Ž 18 . which represents the work on the system caused by the motion of the wall. L s < r = p < s rpu , f is the orbital angular momentum of the particle, which has the eigenvalue l Ž l q 1 . ". Here, we omit the term proportional to A2 Ž r,t . since only a slow wall motion is considered. Clearly, Eq. Ž18. shows that, when L s 0, f q becomes yarr ˙ Ž qa. 2 MEn . HowŽ . ever, p s pu , f i.e. pr s 0 suggests f q Ž r,t . tends to zero. This is an example of gauge field appearance in a quantum mechanical system with time dependent boundary conditions.
(
(
a0 for
a˙ s a ,
i "
(
ž
Ma r 2
q r 2 MEn
u q sin uf a
at
a y
a0
a0
En t
/
,
Ž 21 .
in the case of a˙ s a s 0, The next step is to determine the vector potential A for a given situation, specified in terms of the magnetic field vector B. We now clarify the determination of the vector potential A by the following expression Mca˙ Mc a L As reu , f s = r , for a˙ s a . qa qa L Ž 22 . Considering the fact that the spherical vector potential can be expressed in terms of the magnetic field B as
4. Non-local effect
A s 12 Ž B = r . ,
By introducing the boundary quantity a into Hamiltonian eigenfunctions Ždifferent from the method adopted in Ref. w5x., we have constructed the
we obtain 2 Mc a Bs L. qaL
Ž 23 . Ž 24 .
Y. Qian-Kair Physics Letters A 261 (1999) 30–33
It shows that, when a ) 0 or a - 0, the vector field B is respectively parallel or anti-parallel to the orbital angular momentum vector L of the particle. According to the model of the orbital angular momentum, the vector L, of length l Ž l q 1 . ", processes about the z-axis, the Ž2 l q 1. allowed projections of L on the z-axis is given by m", with m s yl,y l q 1, . . . ,q l. The vector B of length 2 mc arŽ qa. has the same feature, it also processes about the z-axis and has z-component
(
Bz s
2 Mc a L z qa
L
2 Mc a
m
qa
(l Ž l q 1.
s
,
Ž 25 .
note that the x- and y-components are still not zero although the values of them always vanish. So that, the vector B may be viewed as lying on the surface of a cone with altitude B z which has the z-axis as its symmetry axis, all orientations of B on the surface of the cone being equally likely. And naturally, we have the conclusion that the two components B x and B y cannot in general be assigned precise values simultaneously, just like L x , L y . Up to now, we have shown that a quantum system with time dependent boundary conditions is equivalent to the particle interacting with a magnetic field relative to a confining condition. And with the help of the relation A P p s Ž B = r . P pr2 s B P Lr2, we can rewrite the terms linear in A in Eq. Ž17. as q y Ž pˆ P A . q 2 A P p 2 Mc q sy Ž pˆ P A . q B P L , Ž 26 . 2 Mc so that the effective Hamiltonian becomes q q Hˆeff s Hˆ0 y B P L q qf q . Ž pˆ P A . y 2 Mc 2 Mc
By this example, we analyze a new effect that has been engendered by time dependent boundary conditions. This effect represents the interaction between the charge particle and the electromagnetic field due to the wall motion, and thus the system energy shifts.
5. Summary A discussion of the boundary effect has been presented. We now conclude that a non-stationary confinement of particle can lead to the appearance of an effective electromagnetic field. In our consideration we perform the usual treatment for the particle inside an infinite spherical well, similar to the quark confined by a bag, i.e. the bag model suggested in discussing quark confinement. Therefore, a better application of this work is in connection with quark confinement. If now the bag confining the quark expands or constricts, an effective electromagnetic field will appear accordingly. Perhaps this interpretation can provide part of the explanation for the electromagnetic nature of the particle. Finally, we wish to demonstrate that, besides the spherical well, such a result is probably applicable to enclosures of any shape. And in principle, we can say that the effect certainly appears in any process where one tries to confine a particle to a smaller region. For this effect, one can want ideally an electromagnetic force related to the time dependent boundary conditions acting on the particle to produce.
References
Ž 27 . The second last linear term in expression Ž27. represents the energy of the magnetic moment m s qrŽ2 Mc . L Žcaused by the orbital motion of the particle of charge q . interacting with the magnetic field B, that is a W s ym P B s y l Ž l q 1. " . Ž 28 . a
(
33
w1x w2x w3x w4x w5x w6x
M.V. Berry, Proc. Roy. Soc. ŽLondon. A 392 Ž1984. 45. B. Simon, Phys. Rev. Lett. 51 Ž1983. 2167. J.H. Hannay, J. Phys. A 18 Ž1985. 221. D.M. Greenberger, Physica B 151 Ž1988. 374. P. Pereshogin, P. Pronin, Phys. Lett. A 156 Ž1991. 12. B.H. Bransden, C.J. Joachain, Introduction to Quantum Mechanics, Longman Group UK Ltd, London, 1989. w7x M. Razary, Phys. Rev. A 44 Ž1991. 2384. w8x F.L. Li, S.J. Wang, A. Weiguny, D.L. Lin, J. Phys. A 27 Ž1994. 985. w9x M.V. Berry, G. Klein, J. Phys. A 17 Ž1984. 1805.