Consider a system made up of two atoms of masses m1 and mg. The vibrations of this system can be described by a one-dimensional Schrodinger equation for a system of reduced mass [1, evolving in a potential V(r) that depends on the relative distance r of the two atoms. The potential War) is repulsive at short distances, has a minimum Vo at some value re and goes to zero when r > oo. Sketch the potential, describe the physical meaning of re for a classical system and write the time-independent Schrodinger equation for this system. In order to study the small deviations around re, we introduce the variable z = r re and expand the potential: V(r) : V0 + %V\"(re)z2 + 0(z3). Explain the origin of the expansion above. What can you say about V\"(re )? Compute the dimensions of the parameter V\"(re) a) : . it Write the time-independent Schrodinger equation for small oscillations, using a) and neglecting terms that are cubic or higher order in z. State the energy spectrum of the system in this approximation. What can be learned about the potential V by measuring the energy eigenvalues? Let us denote by In) the nth energy eigenstate. Compute the typical size of the uctuations cal,1 = l (nl2 ln) of the position. You can use the fact that for the harmonic oscillator: (nllolm = %<>