Chapter 6, Problem 6.1P

Write the equations that describe the simple harmonic motion of a particle moving uniformly around a circle of radius8​units, with linear speed 3units per second.

For a one dimensional system, x is the position operator and p the momentum operator in the x direction.Show that the commutator [x, p] = ih

Consider the van der Waals potential U(r)=U0[( R 0 r)122( R 0 r)6] , used to model the potential energy function of two molecules, where the minimum potential is at r=R0 . Find the force as a function of r. Consider a small displacement R=R0+r and use the binomial theorem: (1+x)n=1+nx+n( n1)2!x2+n( n1)( n2)3!x3+ , to show that the force does approximate a Hooke’s law force.

If Force B on the x-z plane is equal to 300N and h = 4m and v = 10m, then what is the i and k components of Force B?

Find the equations of motion for the LagrangianL = eγt(mq˙2/2 − kq2/2)How would you describe the system? Are there any constants of motion?Suppose a point transformation is made of the forms = eγt/2q.What is the effective Lagrangian in terms of s? Find the equation of motion for s. What do these results say about the conserved quantities for the system?

A frictionless table has a mass m on it. The table has a hole in it, where mass m is connected to a string that passes through the hole, attached to a hanging mass M.(a) What is the Lagrangian for this system? Using this Lagrangian, find the equations of motion (a = function of position) for each of the generalized coordinates used.(b) Under what conditions does m make a circular motion?(c) What is the frequency of small oscillations (in r) about this circular motion? Further, assume the oscillations are small: use a Taylor Expansion at the point we choose to use the small angle approximation.)

Let's consider a simple pendulum, consisting of a point mass m, fixed to the end of a massless rod of length l, whose other end is fixed so that the mass can swing freely in a vertical plane. The pendulum's position can be specified by its angle Φ from the equilbrium position. Prove that the pendulum's potential energy is U(ϕ)=mgl(1−cosϕ). Write down the total energy E as a function of Φ and ϕ˙. Show that by differentiating E with respect to t you can get the equation of motion for Φ. Solve for Φ(t). If you solve properly, you should find periodic motion. What is the period of the motion?

A particle of mass m described by one generalized coordinate q movesunder the influence of a potential V(q) and a damping force −2mγq˙  proportional to its velocity with the Lagrangian  L = e2γt(1/2 * mq˙2 − V (q)) which gives the desired equation of motion. (a) Consider the following generating function: F = eγtqP - QP.Obtain the canonical transformation from (q,p) to (Q,P) and the transformed Hamiltonian K(Q,P,t). (b) Let V (q) = (1/2)mω2q2 be a harmonic potential with a natural frequency  ω and note that the transformed Hamiltonian yields a constant of motion.  Obtain the solution Q(t) for the damped oscillator in the under damped case γ < ω by solving Hamilton's equations in the transformed coordinates. Then, write down the solution q(t) using the canonical coordinates obtained in part (a).

If we have two operators A and B possess the same common Eigen function, then prove that the two operators commute with each other