k Three equal masses are so connected by springs that they move along a circle (see figure below). Point A is fixed. Find the normal coordinates and eigenfrequencies of the system. www

Calculate the commutator 2m between the kinetic energy and the position operators for a particle moving in one dimension. Please notice that in the answers (hbar) is h/2T. Oa. (hbar)? m2 O b (hbar) m OC Px Od. (hbar)? d m dx 2m

**Problem 3.23 Consider the following Hermitian matrix: 2 i 1 T = -i i 1 -i 2 (a) Calculate det(T) and Tr(T). (b) Find the eigenvalues of T. Check that their sum and product are consistent with (a), in the sense of Equation 3.82. Write down the diagonalized version of T. (c) Find the eigenvectors of T. Within the degenerate sector, construct two linearly independent eigenvectors (it is this step that is always possible for a Hermitian matrix, but not for an arbitrary matrix-contrast Problem 3.18). Orthogonalize them, and check that both are orthogonal to the third. Normalize all three eigenvectors. (d) Construct the unitary matrix S that diagonalizes T, and show explicitly that the similarity transformation using S reduces T to the appropriate diagonal form.

The relationship,v = vo + at , between the velocity, v, initial velocity, vo, acceleration, a, and time, t, is valid: O Only if vo is zero. O Only if the acceleration is constant. O For all values of acceleration.