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Homework Statement
In ##1+1##-dimensional spacetime, two objects, each with charge ##Q##, are fixed and separated by a distance ##d##.
(a) A light object of mass ##m## and charge ##-q## is attached to one of the massive objects via a spring of spring constant ##k##. Quantise the motion of the light charge and obtain the ground state energies and wavefunctions. What approximations did you use, if any?
(b) The same light object of mass ##m## and charge ##-q## is now attached to both the massive objects via springs of spring constants ##k_1## and and ##k_2##. Repeat the analysis in (a) for this new system.
Homework Equations
The Attempt at a Solution
(a) To quantise the motion of the light charge, we need to use the potential energy of the system in the Schrodinger equation to obtain the wavefunctions and the energy spectrum of the theory. Therefore, the first objective is to find the potential energy of the theory.
There are two interactions in the theory. The first is due to the spring which connects the object with charge ##Q## and the object with charge ##-q##. The potential energy due to the spring is ##\frac{1}{2}kx^{2}##.
The second interaction is due to the charges of the objects. Now, Gauss's law in one dimension gives the electric field as
##\int \vec{E}\ \cdot{d\vec{A}}=\frac{-q}{\epsilon_{0}} \implies 2E = -\frac{q}{\epsilon_{0}} \implies E = -\frac{q}{2\epsilon_{0}}##,
since the electric field in integrated over the two endpoints of the Gaussian "surface" on the one spatial dimension. Letting the distance from the equilibrium position of the charge ##-q## to the fixed position of one of the charges ##Q## be ##x_{eq}## and the displacement (from the equilibrium position) of the charge ##-q## in the direction of the other charge ##Q## be ##x##, the potential energy due to the object of charge ##-q## is
##-\frac{qQ}{2\epsilon_{0}}[(w_{eq}+x)+(d-x-x_{eq})]=-\frac{qQd}{2\epsilon_{0}}##.
Therefore, the electric potential energy is a constant and can be ignored.
Am I correct in my analysis so far?