160
6 Dynamics of Quantum Mechanical Macroscopic Systems
Proposition 6.4.7 shows the fulfillment of the ground state condition also for τ := τ
Q .
The validity of the remaining assertions is clear.
6.4.13 Note. A brief version of the here presented theory together with applications
to models of BCS theory and of Josephson junction was published in [40, 41]. Cf.
also the next section.
6.5 An Example: The B.C.S. Model of Superconductivity
6.5.1 We shall illustrate in this section the above developed theory by description
and analysis of a perhaps simplest nontrivial and physically interesting mathematical
model: The strong coupling version of the Bardeen-Cooper-Schrieffer model of the
phenomenon of superconductivity in the quasi spin formulation; it was formulated
and analyzed in [168, 311, 312], in the framework of the traditional QM formalism.
It can be presented, completed, and solved in the framework of the constructions of
the present work as follows:
It is a tensor product type model of Sect. 5.1 with G := SU (2), H := H 0 :=
C
2
, , := Z, the generators of U (G) in C
2 are
X ξ j := i
d
dt
t=0
U (exp(tξ j )) =
1
2
σ j , j = 1, 2, 3,
(6.5.1)
where σ j are the Pauli matrices and the elements ξ j ∈ g of the chosen basis satisfy
the relations
[ξ j , ξ k ] = ε jkm ξ m , j, k, (m) = 1, 2, 3.
(6.5.2)
Let F j := F(ξ j ) be used for the functions f ξ j on g
∗
F as well as for their numerical
values in the points F ∈ g
∗ . The dynamics of the system is specified by the function
Q on g
∗ :
Q(F) = −2εF 3 − λ(F
2
1 + F
2
2 ), ε, λ are some positive numbers.
(6.5.3)
This specifies the model completely.
6.5.2 The Poisson structure on g
∗
= su(2)
∗ is determined by the Poisson brackets
{F j , F k } = −ε jkm F m , j, k, (m) = 1, 2, 3,
(6.5.4)
which are obtained from (6.5.2) according to (5.1.145). The classical dynamics corresponding to the given Hamiltonian function Q ∈ C
∞
(su(2)
∗
, R) is then described
by the flow ϕ
Q on su(2)
∗ which is determined by the Hamilton equations
˙
F j (ϕ
Q
t F) :=
d
dt
F j (ϕ
Q
t F) = {Q, F j }(ϕ
Q
t F), t ∈ R, j = 1, 2, 3.
(6.5.5)
6 Dynamics of Quantum Mechanical Macroscopic Systems
Proposition 6.4.7 shows the fulfillment of the ground state condition also for τ := τ
Q .
The validity of the remaining assertions is clear.
6.4.13 Note. A brief version of the here presented theory together with applications
to models of BCS theory and of Josephson junction was published in [40, 41]. Cf.
also the next section.
6.5 An Example: The B.C.S. Model of Superconductivity
6.5.1 We shall illustrate in this section the above developed theory by description
and analysis of a perhaps simplest nontrivial and physically interesting mathematical
model: The strong coupling version of the Bardeen-Cooper-Schrieffer model of the
phenomenon of superconductivity in the quasi spin formulation; it was formulated
and analyzed in [168, 311, 312], in the framework of the traditional QM formalism.
It can be presented, completed, and solved in the framework of the constructions of
the present work as follows:
It is a tensor product type model of Sect. 5.1 with G := SU (2), H := H 0 :=
C
2
, , := Z, the generators of U (G) in C
2 are
X ξ j := i
d
dt
t=0
U (exp(tξ j )) =
1
2
σ j , j = 1, 2, 3,
(6.5.1)
where σ j are the Pauli matrices and the elements ξ j ∈ g of the chosen basis satisfy
the relations
[ξ j , ξ k ] = ε jkm ξ m , j, k, (m) = 1, 2, 3.
(6.5.2)
Let F j := F(ξ j ) be used for the functions f ξ j on g
∗
F as well as for their numerical
values in the points F ∈ g
∗ . The dynamics of the system is specified by the function
Q on g
∗ :
Q(F) = −2εF 3 − λ(F
2
1 + F
2
2 ), ε, λ are some positive numbers.
(6.5.3)
This specifies the model completely.
6.5.2 The Poisson structure on g
∗
= su(2)
∗ is determined by the Poisson brackets
{F j , F k } = −ε jkm F m , j, k, (m) = 1, 2, 3,
(6.5.4)
which are obtained from (6.5.2) according to (5.1.145). The classical dynamics corresponding to the given Hamiltonian function Q ∈ C
∞
(su(2)
∗
, R) is then described
by the flow ϕ
Q on su(2)
∗ which is determined by the Hamilton equations
˙
F j (ϕ
Q
t F) :=
d
dt
F j (ϕ
Q
t F) = {Q, F j }(ϕ
Q
t F), t ∈ R, j = 1, 2, 3.
(6.5.5)