14量子力学多体问题基础
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14 Elementary Aspects of the Quantum-Mechanical Many-Body Problem If we consider a system of more than one particle, we derive its Hamiltonian, describing it quantum-mechanically in the usual manner from the Hamiltonian function of the system in classical mechanics. The Hamiltonian function
mined at the location of the particle i , i.e., i only acts on the coordinates ri of
the ith particle. Consequently the momentum operators of different particles com-
mute, i.e., [ pˆi , pˆ j ] 0 for all i, j . Thus the many-particle Hamiltonian reads
Hˆ
N i 1
2
2mi
i
Vi
(ri
,
t
)
Vik (ri , rk ) .
ik
(14.3)
This is obviously a generalization of the Hamiltonian for one particle. We can now
(14.5)
w(r1,, rN , t)dV (r1,, rN , t) (r1,, rN , t)dV
(14.6)
is the probability that the system can be found at time t in the volume element dV
tion) of all particles in three-dimensional space. Therefore a point in configuration
space with 3N coordinates (x1, y1, z1,, xN , yN , zN ) is also called the configuration
H
N i 1
pi2 2mi
Vi (ri , t)
Vik (ri , rk ) ,
ik
ቤተ መጻሕፍቲ ባይዱ
(14.1)
describes a system of N particles with mass mi . Here, Vi (ri ,t) is the externally
given potential (the so-called one-particle potential), in which the ith particle moves;
the fact that the specification of the coordinates of a special point in this space means
the specification of the three-dimensional coordinates of the position rk (xk , yk , zk ) for all particles of the system ( k 1,, N ), and thus determines the state (configura-
(14.4)
The treatment of this many-body problem confronts the same difficulties in quantum
mechanics as in classical physics because of the complexity compared to the one-
formulate a many-particle Schrödinger equation
Hˆ i , t
where the wave function now depends on the 3N coordinates of all particles
and on time:
(r1,, rN , t) , rk (xk , yk , zk ) .
particle problem.
The wave equation is defined in a space with 3N dimensions, in the so-called
configuration space of the system. The name of this fictitious space originates from
mutual Coulomb interaction. To get the Hamiltonian, we replace the momenta by the
corresponding differential operators
pi
pˆ i
i
ri
i
i
,
(14.2)
where the index i of the nabla operator specifies that the gradient has to be deter-
it can, for example, mean the external electric potential. Vik (ri , rk ) stands for the
interaction potentials between two particles i and k ; it can, for example, be their
point of the system. We denote an infinitesimally small volume element in the configuration space by
dV dV1dVN with dVk d3rk dxkdykdzk . Then the quantity
mined at the location of the particle i , i.e., i only acts on the coordinates ri of
the ith particle. Consequently the momentum operators of different particles com-
mute, i.e., [ pˆi , pˆ j ] 0 for all i, j . Thus the many-particle Hamiltonian reads
Hˆ
N i 1
2
2mi
i
Vi
(ri
,
t
)
Vik (ri , rk ) .
ik
(14.3)
This is obviously a generalization of the Hamiltonian for one particle. We can now
(14.5)
w(r1,, rN , t)dV (r1,, rN , t) (r1,, rN , t)dV
(14.6)
is the probability that the system can be found at time t in the volume element dV
tion) of all particles in three-dimensional space. Therefore a point in configuration
space with 3N coordinates (x1, y1, z1,, xN , yN , zN ) is also called the configuration
H
N i 1
pi2 2mi
Vi (ri , t)
Vik (ri , rk ) ,
ik
ቤተ መጻሕፍቲ ባይዱ
(14.1)
describes a system of N particles with mass mi . Here, Vi (ri ,t) is the externally
given potential (the so-called one-particle potential), in which the ith particle moves;
the fact that the specification of the coordinates of a special point in this space means
the specification of the three-dimensional coordinates of the position rk (xk , yk , zk ) for all particles of the system ( k 1,, N ), and thus determines the state (configura-
(14.4)
The treatment of this many-body problem confronts the same difficulties in quantum
mechanics as in classical physics because of the complexity compared to the one-
formulate a many-particle Schrödinger equation
Hˆ i , t
where the wave function now depends on the 3N coordinates of all particles
and on time:
(r1,, rN , t) , rk (xk , yk , zk ) .
particle problem.
The wave equation is defined in a space with 3N dimensions, in the so-called
configuration space of the system. The name of this fictitious space originates from
mutual Coulomb interaction. To get the Hamiltonian, we replace the momenta by the
corresponding differential operators
pi
pˆ i
i
ri
i
i
,
(14.2)
where the index i of the nabla operator specifies that the gradient has to be deter-
it can, for example, mean the external electric potential. Vik (ri , rk ) stands for the
interaction potentials between two particles i and k ; it can, for example, be their
point of the system. We denote an infinitesimally small volume element in the configuration space by
dV dV1dVN with dVk d3rk dxkdykdzk . Then the quantity