Superconductivity (SC) was discovered in 1911 in the Leiden laboratory of Kamerlingh Onnes when a so called “blue boy” (local high school student recruited for the tedious job of monitoring experiments) noticed that the resistivity of Hg metal vanished abruptly at about 4 K. Although phenomenological models with predictive power were developed in the 30’s and 40’s of the last century [1], F and H London developed the successful phenomenological approach in 1935 describing the behavior of superconductors in the external magnetic field. Ogg Jr. proposed a root to high-temperature superconductivity (HTSC) introducing electron pairs in 1946 and Ginzburg and Landau proposed the phenomenological theory of the superconducting phase transition in 1950 providing a comprehensive understanding of the electromagnetic properties below T_c [2]. The microscopic mechanism underlying superconductivity was not discovered until 1957 by Bardeen Cooper and Schrieffer (BCS) [1]. Superconductors have been studied intensively for their fundamental interest and for the promise of technological applications which would be possible if a material which superconducts at room temperature was discovered. Until 1986, critical temperatures (T_c’s) at which resistance disappears were always less than about 23 K.

In 1986, Bednorz and Mueller published a paper, subsequently recognized with the 1987 Nobel Prize, for the discovery of a new class of materials called (HTSC) which currently include members with T_c’s of about 135 Kor more. Enormous numbers of studies have been carried out to clarify the mechanism of the high temperature superconductivity (HTSC) beyond the conventional BCS theory Figure 1.

One of the important HTSC is the cuprate compounds. The cuprate systems show not only high temperature superconductivity but also show various unusual behaviors when doped with holes where it is converted from an insulator to a superconductor [1]. The transition-metal oxides have been extensively investigated in recent years as materials that can be converted to superconductors. Understanding the nature of superconductors has been

the most challenging issue in condensed matter physics due to the difficulties inherent in the many-body interactions.

Although BCS theory explains several superconductors phenomena specially at low critical temperature T_c, but there are many setbacks associated with Bardeen, Cooper and Schrieffer BCS theory for high critical temperature, which was observed in some compounds specially CuO and Fe compounds.

There are many problems need to be solved. First of all, one observes that, till now, there is no well established theoretical expression in most celebrated SC models which shows how the resistance drops abruptly to zero below the critical temperature. The existence of an energy gap well above T_c with pressure and the substitution of O¹⁶ by its isotope O¹⁸ affecting T_c also need to be explained by a simple model also.

The aim of this work is to construct quantum mechanical model based on plasma equation to construct a quantum model which explains why the resistance vanishes below critical temperature. It also aimed to find a useful expression for the energy gap. These contribution are exhibited in Sections (5) and (6). Section (2) is devoted for the theoretical plasma equation.

According to plasma equation, a fluid of particles of mass m, number density n, velocity, force F and pressure P is given by

If F is a field force then

Where V is the potential of one particle. In one dimension

Thus according to Equation (1), in one dimension

Schrodinger equation can be derived by using new expression of energy obtained from the plasma equation to do this one can use (2) to get

Multiplying both sides by dx and integrating yields

Considering the pressure to be in general, thus

Hence

This constant conserved quantity looks like the ordinary energy beside the ordinary thermal energy term.

To find Schrödinger equation for it, consider the ordinary wave function

Differentiating both sides by t and x yields

Multiplying both sides of Equation (3) by yields

Substituting Equation (4), one gets

This equation represents Schrödinger equation when thermal motion is considered. The solution for time free potential can be

The time independent Schrödinger equation thus takes the form

For constant potential, the solution can be

,

Inserting this solution in Equation (5) yields

If one set the kinetic term to be, one can thus write the energy in the form

This quantum energy expression involves a thermal term beside kinetic and potential term.

The resistance, z, per unit length (L = 1) per unit area (A = 1) can be found from the ordinary definition of, z. The resistance z is defined to be the ratio of the potential, u, to the current per unit area, J, i.e.

With n and e standing for the free hole or electron density and charge respectively, while p represents the momentum of electron of mass m, where.

This resistance (it actually stands for resistivity) can be found by using the laws of quantum mechanics for a free charge which are responsible for generating the electric current, where the wave function takes the form

This selection of comes from the fact that the resistance property comes from the motion of the free charges. The potential u is related to the Hamiltonian H through the relation

Thus for freely moving charge one gets:

In view of Equation (8) and according to the correspondence principle V takes the form

While P becomes

Thus inserting Equations (9), (10) in (7) one obtains

where the expression for velocity is found by assuming charges to be waves, then following the electromagnetic theory (EMT), the speed of the waves is affected by electric permittivity and magnetic permeability through the relation

where the effect of medium changes the wave length, , while the frequency, f, is unchanged. Thus assuming the charge density, n, to be constant, the only change of, Z, can be caused by and.

It is also important to note that, in superconductors, the current can flow without the aid of deriving potential u. the role of u is confined only in enabling electrons to gain kinetic energy through the relations

where this potential can be applied between any two arbitrary points in the superconductors then remove it. The role of resistive force is neglected here as done in deriving London equations.

The expression for Z can also be found by inserting Equation (13) in to get

It is important to note that this quantum resistance expression resembles the ones found by Tsui [3] where one uses De Broglie hypothesis [4], i.e..

Consider holes in a conductor having resistive force F_r, magnetic force F_m and pressure force F_p, beside the electric force F_e, the equation motion then becomes [3]:

where

P, x, m, , , B, e and E stands for the pressure, displacement, mass, velocity, relaxation time, magnetic flux density, electron charge and electric field intensity respectively. Thus the equation of motion takes the form

The solution of this equation can be suggested to be:

Inserting (16) in (15) yields

This expression of x can be utilized in the formula which relates the electric polarization vector P to the susceptibility on one hand and to the number of atoms N via the following relation

Motivated by the important role of holes in HTSC, displacement can be assumed to result from the motion of holes or positive nuclear charges, thus inserting Equation (17) in (18) yields

The electric flux density assumes the following relation

The electric permittivity is given by

The electric permittivity is thus given according to Equation (20) to be

The resistance Z can be found by inserting (21) in (14) to get:

Thus the critical temperature is given by

If the internal field B results from N_o atoms each having a verge flux density then: [5].

Therefore T_c can take the form

In tight binding model [5] the energy of electrons in

the crystal is given by

where is the energy in the absence of crystal fieldwhile the other terms describe the effect of the crystal field. The energy can split into two terms the kinetic part which can describe the thermal motion in the form

beside the potential term for attractive force or bounded particle.

Thus one can write

represents the degrees of freedom.

The terms describing the effect of the crystal force are

In view of Equations (26) and (27)

Here stands for the crystal force Hamiltonian part, while and are the states of particles located at the site m and j respectively.

The superconductor is characterized by the existence of energy gap. This gap can be under stood here in two ways. If the electrons or holes are not free. This requires E to negative. Thus Equations (27) and (26) needs

Or the max value of where is less than zero, i.e.

For constant attractive crystal force

Thus

Thus the critical temperature is given by

Substituted Equation (33) beside Equation (32) in Equation (30) one gets

The energy gap s equal to the difference between zero energy in conduction band and the negative energy in the valence band. Thus

Since this relation holds for one can neglect T since it is small to get

Equation (30) can also be utilized to get the forbidden energy states which characterizes superconductors, where

The energy is forbidden when

Thus the critical temperature

The forbidden energy is thus related to the critical temperature through the relation

If the particle has a 4—degree of freedom, 3—translational and one vibration.

In view of Equations (32) and (28), since Plank constant is very small and for very small crystal field and for bound force, since the energy gap ∆ is the difference between bound valence energy E, and minimum free conduction electron energy zero. Thus

Which shows linear relation between ∆ and T_c, thus it resembles the empirical relations. Where the energy gap is found to be [6].

This model predicts that Schrödinger equation can be derived by using a new expression of energy obtained from the plasma equation. This expression includes thermal energy beside kinetic and potential energy according to Equation (6) It is very striking to note that this expression resembles the expression of the thermodynamic internal energy. A useful quantum expression for resistivity is also obtained in Equation (11) this expression resembles those obtained by Aharonove, Bohm and Berry as pointed out at the end of section 4. The model predicts that the resistivity of low and high T_c superconductors it drop abruptly when Z₁ = real = zero according to Equation (8). It also finds the critical temperature T_c beyond which the resistivity vanish according to Equations (23) and (24). A useful expression for the energy gap which is dependent on T_c is also obtained. This expression is in agreement with the empirical relation. 

The plasma equation is utilized to derive new energy expression in which thermal energy is added to the ordinary kinetic and potential energy. This quantum equation which is temperature dependent within the framework of this equation uses tight binding approximation besides the quantum impedance expression, and it is very easy to explain why resistance vanishes below a certain critical temperature and why the empirical relation between the energy gap and the critical temperature is linear.

This raises a hope that Schrodinger quantum temperature dependent model can be utilized, if a frictional effect can be incorporated in it, to construct a general theoretical frame work capable of describing Schrodinger phenomena. The result obtained indicates that the quantum plasma model can improve the theoretical model to explain some of the phenomena associated with the HTSC. Strictly speaking it can explain why the resistance drops abruptly below T_c, besides explaining some important effects like the relation between energy gap and critical temperature.