^{1}

^{2}

^{*}

The pairon field operator
ψ
(r,t) evolves, following Heisenberg’s equation of motion. If the Hamiltonian H contains a condensation energy
α
_{0}
(<0) and a repulsive point-like interparticle interaction
,
, the evolution equation for
ψ
is non-linear, from which we derive the Ginzburg-Landau (GL) equation:
for the GL wave function
where
σ
denotes the state of the condensed Cooper pairs (pairons), and n the pairon density operator (u and
are kind of square root density operators). The GL equation with
holds for all temperatures (T) below the critical temperature T_{c}, where
ε_{g}
(T) is the T-dependent pairon energy gap. Its solution yields the condensed pairon density
. The T-dependence of the expansion parameters near T_{c} obtained by GL:
constant is confirmed.

In 1950 Ginzburg and Landau (GL) [_{c} a superconductor has a complex order parameter (also known as a GL wave function)

where

where A is a vector potential generating_{j}, GL obtained two equations:

with the density condition:

where j is the current density.

The superelectron model has difficulties. Since electrons are fermions, no two electrons can occupy the same particle state by Pauli’s exclusion principle. Cooper [

Equation (3) is the celebrated Ginzburg-Landau equation, which is quantum mechanical and nonlinear. Since the smallness of _{c},_{c} there is a supercondensate whose motion generates a supercurrent and whose presence generates gaps in the elementary excitation energy spectra. The GL wave function

In the present work, we derive the GL wave equation microscopically and show that the GL equation is valid for all temperature below T_{c}.

Fujita, Ito and Godoy in their book [

which is Cooper’s equation, Equation (1) of his 1956 Physical Review [

Equation (7) can be solved as follows. We assume that the energy

Then,

Here

is k-independent. Introducing Equation (9) in Equation (7), and dropping the common factor

We now assume a free electron model. The Fermi surface is a sphere of the radius (momentum):

where m_{1} represents the effective mass of an electron. The energy

The prime on the k-integral in Equation (11) means the restriction:

We may choose the polar axis along q as shown in

where k_{D} is given by

After performing the integration and taking the small-q and small-

where the pairon ground-state energy w_{0} is given by

As expected, the zero-momentum pairon has the lowest energy. The excitation energy is continuous with no energy gap. Equation (17) was first obtained by Cooper (unpublished) and it is recorded in Schrieffer’s book (Ref. [_{q} increases linearly with momentum ^{2}-increase of the kinetic energy. This linear dispersion relation means that a pairon moves like a massless particle with a common speed

The center-of-mass (CM) of pairons move as bosons. We can show this as follows. Second-quantized operators for a pair of “electrons” (i.e., “electron” pairons) are defined by

where c’s and

Using Equation (20) we compute commutation relations for pair operators and obtain

where

are number operators for electrons. Using Equations (19)-(24), we obtain

Hence

We now introduce

and calculate

from which it follows straightforwardly that the eigenvalues

with the eigenstates

Let us take a three dimensional (3D) superconductor such as tin (Sn) and lead (Pb). Both metals form face-cen- tered cubic (fcc) crystals. They are in superconducting states at 0 K.

The system ground-state wave function

We may assume a periodic rectangular-box with side-lengths

where a is the lattice constant.

We introduce a one-body density operator n and the density matrix n_{ab} for the treatment of a many-particle system. The density operator n can be expanded in the form:

where

where the symbol “tr” means a one-body trace and N is the particle number. The density operator n is Hermitean:

Let us introduce kind of a square root density operator u such that

This u is not Hermitean but

where

where L is the ring circumference. The

Let us introduce boson field operators

where

We take a system characterized by many-boson Hamiltonian H:

where

We note that the field equation is nonlinear in the presence of a pair potential.

We can exprress

where

In the Heisenberg picture (HP) the boson states are time-independent and boson operators

The single particle Hamiltonian h contains the kinetic energy_{e} which arises from the phonon exchange attraction. The ground state wave function

where the superscript (0) means the ground state average.

The ground-state energy of the pairon is negative and is given by w_{0} in Equation (18). Hence we may choose

The pairon has charge magnitude 2e and a size. For Pb the pairon linear size is about 10^{3} Å. Because of the Colomb repulsion and Pauli’s exclusion principle two pairons repel each other at short distances. We may represent this repulsion by a point-like pair potential:

Using this and the random phase (factorization) approximation we obtain

Gathering the results (43), (44) and (46), we obtain

For the steady state the time derivative vanishes, yielding

This is precicely, the GL equation, Equation (3) with

In our derivation we assumed that pairons move as bosons, which is essential. Bosonic pairons can multiply occupy the condensed momentum state while fermionic superelectrons cannot. The correct density condition (6) instead of (5) must therefore be used.

We derived the GL equation from first principles. In the derivation we found that the particles that are described by the GL wave function

The nonlinearity of the GL equation arises from the point-like repulsive inter-pairon interaction. In 1950 when Ginzburg and Landau published their work, the Cooper pair (pairon) was not known. They simply assumed the superelectron model.

Our microscopic derivation allows us to interpret the expansion parameters as follows. The

BCS showed [

where

In the original work [_{c}, where _{c}. We derived the original GL equation by examining the superconductor at 0 K from the condensed pairons point of view. The transport property of a superconductor below T_{c} is dominated by the Bose-condensed pairons. Since there is no distribution, the qualitative property of the condensed pairons cannot change with temperature. The pairon size (the minimum of the coherence length derivable directly from the GL equation) naturally exists. There is only one supercondensate whose behavior is similar for all temperatures below T_{c}; only the density of condensed pairons can change. Thus, there is a quantum nonlinear equation (48) for _{c}. The pairon energy spectrum below T_{c} has a discrete ground-state energy, which is separated from the energy continuum of moving pairons [

Solving Equation (48) with Equation (6), we obtain

indicating that the condensed density

We now consider an ellipsoidal macroscopic sample of a type I superconductor below T_{c} subject to a weak magnetic field H applied along its major axis. Because of the Meissner effect, the magnetic fluxes are expelled

from the body, and the magnetic energy is higher by

field is sufficiently raised, the sample reverts to the normal state at a critical field H_{c}, which can be computed in terms of the free-energy expression (1) with the magnetic field included. We obtain after using Equations (6) and (50)

indicating that the measurements of H_{c} give the T-dependent

We stress that the pairon energy gap

In the presence of a supercondensate the energy-momentum relation for an unpaired (quasi) electron changes:

Since the density of condensed pairons changes with the temperature T, the gap Δ is T-dependent and is determined from Equation (54) (originated in the BCS energy gap equation). Two unpaired electrons can be bound by the phonon-exchange attraction to form a moving pairon whose energy

Note that _{c}, Δ = 0 and the lower band edge _{0}. We may then write

We call _{c} and they both grow monotonically as temperature is lowered. The rhs of Equation (54) is a function of_{c} is a regular point such that a small variation ^{2}. Hence we obtain

Using similar arguments we get from Equations (57)-(59)

As noted earlier, moving pairons have finite (zero) energy gaps in the super (normal) states, which makes Equation (53) approximate. But the gaps disappear at T_{c}, and hence the linear-in-_{c}:

which is supported by experimental data. Tunneling and photo absorption data [

In the original GL theory [_{c} were assumed and tested:

all of which are reestablished by our microscopic calculations.

In summary we reached a significant conclusion that the GL equation is valid for all temperatures below T_{c}. The most important results in the GL theory include GL’s introduction of a coherent length [_{c} but for all temperatures below T_{c}.

In the present work the time evolution of the system is described through the field equation for the boson operators

may be useful in deriving the Josephson-Feynman equation and describing dynamical Josephson effect [

GL treated the effect of the magnetic field applied based on the super electron model. We shall treat this effect based on the moving pairons model separately.