Path Integral Formalism To The Evaluation Of The Density Of States Of A Four Dimensional Electron System In A Classically Smooth Potential

Path Integral Formalism To The Evaluation Of The Density Of States Of A Four Dimensional Electron System In A Classically Smooth Potential

S.K.Suman1 , V.S.Giri2

1*Assistant Professor, Department of Pure and Applied Physics, Guru Ghasidas University ( A Central University), Bilaspur, Chhatisgarh, India

2 Ex- Reader, University Department of Physics, Ranchi University,Ranchi, India

ABSTRACT

We have defined the Greens function for the four-dimensional electron System in a classically smooth potential. Then the method of path integral formalism have been used for the evaluation of the density of states (DOS) of a four-dimensional electron system in a classically smooth potential. Finally we have calculated the time coordinate representation of the DOS, and as a conclusion of this study we have found the manner in which the DOS of the fourdimensional electron system varies with respect to the energy E of the system.

Key words : path integral, density of states, four-dimensaional electron system, smooth potential.

.

1. Introduction

   Density of states[1] refers to the number of energy states in each unit interval of energy and in each unit volume of the crystal. For the electrons[2] confined into a cube of side length L, The density of states in the absence of any external field may be written as ,

   =

   2

   2

   for =

   2

   and =L3

   Classically smooth potential[3] means the potential whose variation with the space coordinate is small. In this paper we have defined the Greens function for the four dimensional electron system under classically smooth potential, then Path integral approach[4] is applied for the evaluation of the DOS of the four-dimensional electron system. Finally we have deduced the expression for the DOS of the four-dimensional electron system in the time- coordinate representation.

2. Method

   Hamiltonian for the system under study is time independent as it is clear from its expression given below :

   H = 2 2 + U( q ) ……………..(1)

   2m

   Here U( q ) represents the classically smooth potential depending on the spatial coordinate q in the q-

   representation. Therefore for the system, Greens function[5] will satisfy the equation

   ……………..(2)

   Where and are the spatial co- ordinates in the q- representation. Solution of equation (2) may be written as and as follows:

   …………….(3A)

   with the initial condition that ; and

   ……………(3B)

   with the initial condition that .

   therefore,

   ………………(4)

   Let the intermediate times between 0 and t be represented as with the corresponding coordinates respectively,in such a way that,

   , for k = 2,3,4, …….n . …………….(5)

   With the boundary condition that as .

   See the figure (1) . Now our task is to evaluate , For this the method of path integral formalism is used

   here.Hence from the equation (4) we may write,

   ………

   .

   ……………..(6)

   In writing the equation (6) we have used the completeness condition of quantum mechanics[6]. Now for the

   four-dimensional system, the matrix element of the operator in the q-coordinate representation may be simplified as,

   …………….(7)

   Therefore from the equations (6) and equation (7) ,

   .

   ………………(8)

   Where,

   .

   If we take

   and .

   For the classically smooth potential we can explicitly write from equation (8) that,

   ……………..(9)

   Where c is a constant, which is interpreted as the correlation function for the case of classically smooth potential . Therefore density of states of the four-dimensional electrons can be written as ,

   …………….(10)

   Using time-coordinate representation we simplify the equation (10) as,

   ……………..(11)

3. Results and Discussion

   After simplifying the equation (11) we get,

   Where and are confluent

   hypergeometric function[7] . We have plotted the graph of versus by assuming that

   . For simplicity we have drawn the graph for two different values of c namely , c=1/2 , and c=1 , See the figure (2) and figure(3) . From the graph for

   c=1/2 , and c=1 , it is evident that the density of states (DOS) of four-

   dimensional electron system increases by increasing E, in the manner which is shown in the graph.

4. Conclusion

   we have studied the four-dimensional electron system in which we have evaluated the density of states of the four-dimensional electron system, from which it is clear that the density of states (DOS) increases by increasing the energy E of the system and the rate of increase of the density of states also increases by increasing E for small positive values of E.

   References

   1. N. Davidson statistical mechanics ( McGraw-Hill Book Company, Inc.)(1962).

   2. D. J. Thouless J. Phys. C: Solid State Phys. 9 L603(1976) .

   3. V. J. Donnay Jounal of statistical physics Vol. 96, Nos. 5/6 (1999).

   4. L. H. Ryder Quantum field theory ( Cambridge University Press) (1996).

   5. Doniach Greens functions in quantum physics (Springer) (2006) .

   6. R. McWeeny Quantum mechanics: principles and formalism (Dover Publication, Inc. New York) (1972) .

   7. K. Aomoto and M. Kita Theory of hypergeometric functions (Springer) ( 2011).

Caption:

Figure 1 : Representation of the coordinates at different times for

.

.

Figure 1 : Representation of the coordinates at different times for

Figure 2 : Graph of versus for c=1/2 .

Figure 3 : Graph of versus for c=1