- Email: [email protected]
Effective Hamiltonian and Berry phase in a quantum mechanical system with time dependent boundary conditions
Volume
156, number
1,2
PHYSICS
LETTERS
A
3 June 1991
Effective Hamiltonian and Berry phase in a quantum mechanical system with time dependent boundary conditions P. Pereshogin and P. Pronin MoscowState University, Physics Faculty, Department of Theoretical Physics, II 7234 Moscow. USSR Received 19 March 1990; revised manuscript Communicated by J.P. Vigier
received
5 April 199 1; accepted
for publication
8 April 199 1
We discuss the problem of the quantum particle behavior in a one-dimensional infinite square well potential with a moving wall. We show that this system is equivalent to a particle interacting with a vector field dependent on the coordinate and relative velocity of the wall. The effective Hamiltonian is constructed and the extra geometrical phase of the wave function is calculated.
1. In recent years much attention has been paid to the set of problems deeply connected with the nontrivial geometry of the dynamical system state space. Considerable progress in the understanding of these problems has been achieved by Berry [ 11, Simon [ 2 ] and Hannay [ 31. It turns out that state space anholonomy played a central role in the investigation of the evolution dynamics of classical and quantum systems with time dependent Hamiltonians. It has been shown that the form of the ordinary quantum adiabatic theorem originally proposed by Fock and Born [ 41 is not quite valid. Strictly speaking the adiabatic approximation variation of parameters can lead to a “geometrical phase” of the vector state in addition to the dynamic one. Generally published works devoted to investigations of systems with time dependent Hamiltonians have not paid sufftcient attention to the possible role of the boundary conditions. The fact that different geometrical boundary conditions lead to different phase factors and physical effects was discussed in ref. [ 5 1. The very interesting problem of particle movement in a multiply connected space-time was considered by Levy-LeBlond [ 6 1. He regarded the space to consist of three parts one of which is a bounded region. The wave function acquired the phase factor due to the boundary conditions of this bounded region. 12
The main aim of our paper is the investigation of a system with time dependent boundary conditions. A similar problem was first investigated by Doescher and Rice [ 71 and in the last years by Greenberger [ 8 ] and Pinder [ 9 1. But we would like to consider the problem from the geometrical point of view on the state space and on the evolution of the quantum mechanical system. We find the geometrical framework to be very convenient and believe it is adequate to describe this problem. It is pertinent to remind ourselves that the geometry of libre bundles was used to describe the evolution as a parallel transport in ref. [ lo] and it was shown by Simon [ 21 that we can compare the fibre bundle space to the dynamical system state space and the connection of this libre bundle is Hermitian in the adiabatic approximation. There is not sufficient space in this short paper to give the details of this geometrical treatment so we refer those interested in such details to refs. [ 11,121. We will show in this paper that it is necessary to change the original Hamiltonian to the effective one when considering a system with time dependent boundary conditions. We will demonstrate the appearance of gauge structure in such a system and calculate the Berry phase. Finally we will discuss the correspondence between our results and those of Greenberger, and will give short comments to Doescher and Rice’s approach. Elsevier Science Publishers
B.V. (North-Holland)
156, number
1,2
PHYSICS
LETTERS
A
3 June 1991
Effective Hamiltonian and Berry phase in a quantum mechanical system with time dependent boundary conditions P. Pereshogin and P. Pronin MoscowState University, Physics Faculty, Department of Theoretical Physics, II 7234 Moscow. USSR Received 19 March 1990; revised manuscript Communicated by J.P. Vigier
received
5 April 199 1; accepted
for publication
8 April 199 1
We discuss the problem of the quantum particle behavior in a one-dimensional infinite square well potential with a moving wall. We show that this system is equivalent to a particle interacting with a vector field dependent on the coordinate and relative velocity of the wall. The effective Hamiltonian is constructed and the extra geometrical phase of the wave function is calculated.
1. In recent years much attention has been paid to the set of problems deeply connected with the nontrivial geometry of the dynamical system state space. Considerable progress in the understanding of these problems has been achieved by Berry [ 11, Simon [ 2 ] and Hannay [ 31. It turns out that state space anholonomy played a central role in the investigation of the evolution dynamics of classical and quantum systems with time dependent Hamiltonians. It has been shown that the form of the ordinary quantum adiabatic theorem originally proposed by Fock and Born [ 41 is not quite valid. Strictly speaking the adiabatic approximation variation of parameters can lead to a “geometrical phase” of the vector state in addition to the dynamic one. Generally published works devoted to investigations of systems with time dependent Hamiltonians have not paid sufftcient attention to the possible role of the boundary conditions. The fact that different geometrical boundary conditions lead to different phase factors and physical effects was discussed in ref. [ 5 1. The very interesting problem of particle movement in a multiply connected space-time was considered by Levy-LeBlond [ 6 1. He regarded the space to consist of three parts one of which is a bounded region. The wave function acquired the phase factor due to the boundary conditions of this bounded region. 12
The main aim of our paper is the investigation of a system with time dependent boundary conditions. A similar problem was first investigated by Doescher and Rice [ 71 and in the last years by Greenberger [ 8 ] and Pinder [ 9 1. But we would like to consider the problem from the geometrical point of view on the state space and on the evolution of the quantum mechanical system. We find the geometrical framework to be very convenient and believe it is adequate to describe this problem. It is pertinent to remind ourselves that the geometry of libre bundles was used to describe the evolution as a parallel transport in ref. [ lo] and it was shown by Simon [ 21 that we can compare the fibre bundle space to the dynamical system state space and the connection of this libre bundle is Hermitian in the adiabatic approximation. There is not sufficient space in this short paper to give the details of this geometrical treatment so we refer those interested in such details to refs. [ 11,121. We will show in this paper that it is necessary to change the original Hamiltonian to the effective one when considering a system with time dependent boundary conditions. We will demonstrate the appearance of gauge structure in such a system and calculate the Berry phase. Finally we will discuss the correspondence between our results and those of Greenberger, and will give short comments to Doescher and Rice’s approach. Elsevier Science Publishers
B.V. (North-Holland)