- Demonstrate that

where represents a classical Poisson bracket. Here, the and are the coordinates and corresponding canonical momenta of a classical, many degree of freedom, dynamical system. - Verify that

where represents either a classical or a quantum mechanical Poisson bracket. Here, , , , etc., represent dynamical variables (i.e., functions of the coordinates and canonical momenta of a dynamical system), and represents a number. - Consider a Gaussian wavepacket whose corresponding wavefunction is
- Suppose that we displace a one-dimensional quantum mechanical system a distance
along the
-axis. The
corresponding displacement operator is
Let , and let denote an eigenket of the operator belonging to the eigenvalue . Demonstrate that