Two hydrogen atoms are located on the z-axis a distance l from each other. The c

Question

Two hydrogen atoms are located on the z-axis a distance l from each other. The charge distribution of their electron shells is given by rho_1 = - er4_1/3^8 pi alpha^7 e^-2r_1/(3alpha) sin^2theta_1cos^2 theta_1, rho_2 = - er4_2/4 middot 3^8 pi alpha^7 e^-2r_2/(3alpha) sin^4theta_2, where r_1 and r_2 are the distances from the center each atom to the observer, and theta_1 and theta_2 are the angles formed by the z-axis and the vectors from each atom to the observer (in this problem the "observer" happens to be the other atom). In the above expressions e is the elementary charge and a is the Bohr radius. At the center of each atom is a proton with charge +e. Find the lowest order term for the potential for each atom, and write down but do not fully calculate an expression for the electrostatic energy between them, in terms of the potentials only.

Explanation / Answer

We know that potential energy at a distance l is given by U=Kq1q2/l where K is the constant and q1,q2 are the point charges and l in the distance between them.

In this case charge distribution is given. For H2 atom the potential is given by expression V(r)= e^2/r where e is the electron here. Again from the colubic point of view we introduce atomic number and permittivity but atomic number for H2 is 1 so the expression becomes

V= 1.e^2/4pie*epsilon*r

This is the interatomic potential for the charge e

again from schrodinger solution of bound state energy the energy E=Z^2/n^2(me^4/8epsilon^2h^2)

but to get lowest order term we put n=1 then E= (me^4/8epsilon^2h^2)

now charge distribution is defined as charge density means mass per unit volume

so for potential energy for first atom at l = Krho2/l and potential energy for second atom krho1/l

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