A particle is confined to move on the surface of a circular cone with its axis on the z axis, vertex at the origin (pointing down), and half-angle α. The particle's position can be specified by two generalized coordinates, which you can choose to be the coordinates (ρ, φ) of cylindrical polar coordinates. Write down the equations that give the three Cartesian coordinates of the particle in terms of the generalized coordinates (ρ, φ) and vice versa.

A particle of mass m has the wave function $\Psi(x, t)=A e^{-a\left[\left(m x^{2} / \hbar\right)+i t\right]}$, where A and a are positive real constants. (a) Find A. (b) For what potential energy function, V(x), is this a solution to the Schrödinger equation? (c) Calculate the expectation values of x, $x^{2}$, p, and $p^{2}$. (d) Find $\sigma_{x}$ and $\sigma_{p}$. Is their product consistent with the uncertainty principle?

Consider a mass m constrained to move in a vertical line under the influence of gravity. Using the coordinate x measured vertically down from a convenient origin O, write down the Lagrangian L and find the generalized momentum p = ∂xL/∂ẋ. Find the Hamiltonian H as a function of x and p, and write down Hamilton's equations of motion. It is too much to hope with a system this simple that you would learn anything new by using the Hamiltonian approach, but do check that the equations of motion make sense.)

Consider a bead that is threaded on a rigid circular hoop of radius R lying in the xy plane with its center at O, and use the angle φ of two-dimensional polar coordinates as the one generalized coordinate to describe the bead's position. Write down the equations that give the Cartesian coordinates (x, y) in terms of φ and the equation that gives the generalized coordinate φ in terms of (x, y).

Consider a mass m moving in two dimensions with potential energy U (x, y) = 1/2 kr², where r² = x² + y². Write down the Lagrangian, using coordinates x and y, and find the two Lagrange equations of motion. Describe their solutions.

Let the position of a point P in three dimensions be given by the vector r = (x, y, z) in rectangular (or Cartesian) coordinates. The same position can be specified by cylindrical polar coordinates, ρ,φ,z, which are defined as follows: Let P' denote the projection of P onto the xy plane; that is, P' has Cartesian coordinates (x, y, 0). Then ρ and φ are defined as the two-dimensional polar coordinates of P' in the xy plane, while z is the third Cartesian coordinate, unchanged. (a) Make a sketch to illustrate the three cylindrical coordinates. Give expressions for ρ,φ,z in terms of the Cartesian coordinates x, y, z. Explain in words what ρ is ("ρ is the distance of P from ______ "). There are many variants in notation. For instance, some people use r instead of ρ. Explain why this use of r is unfortunate. (b) Describe the three unit vectors ρ̂,φ̂,ẑ and write the expansion of the position vector r in terms of these unit vectors. (c) Differentiate your last answer twice to find the cylindrical components of the acceleration a = r̈ of the particle.