550, Chem. Spring 2005 Instructor: Nancy Makri
PROBLEM SET 2 Due: March 4th 1. Consider the classical one-dimensional harmonic oscillator, for which the Lagrangian is & L ( x, x ) = 121 & mx m 2 x 2 . 2 2
a) Construct the Hamiltonian and solve Hamiltons equations of motion, i.e., find x (t ) as a function
of the initial conditions x0 , p0 . b) By integrating the Lagrangian along trajectories, find Hamiltons principal function S ( x0 , t0 ; x1 , t1 ) . Also show that S = p1 , x1 S = p0 . x0
c) For the same system find ( x0 , t0 ) , ( x1 , t1 ) pairs such that there is exactly one solution to Lag-
ranges boundary value problem; no solution; and an infinite number of solutions. These results are unique to the harmonic oscillator; anharmonic potentials do not possess such perfect focusing properties. 2. The classical Hamiltonian for a diatomic molecule AB is H=
2 pA + p2 B + V ( | R A R B |) 2mA 2mB
where R A,B are the positions of the two nuclei and PA,B the conjugate momenta. We have expressed the potential in the above equation in terms of the distance | R A R B| r between the two atoms, assuming that there are no external fields acting on the system. Perform a canonical transformation to new coordinates R and r, where R is the center-of-mass coordinate, R= mA R A + mB R B mA + mB and r = RA RB
and show that the Hamiltonian can be expressed in terms of these new coordinates in the form H=
2 2 PR pr + + V (r ) 2M 2
where M = mA + mB , = mA mB / M , and PR , pr are the momenta conjugate to R and r, respectively. The first term in the new Hamiltonian corresponds to the kinetic energy of the center of mass of the molecule and represents rigid translation in space, while the part depending on r describes rotation and vibration.