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# 2.P: Exercises

1. 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.
2. 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.

3. Consider a Gaussian wavepacket whose corresponding wavefunction is

where , , and are constants. Demonstrate that Here, and are a position operator and its conjugate momentum operator, respectively.

1.

2.

3.

4.

4. Suppose that we displace a one-dimensional quantum mechanical system a distance along the -axis. The corresponding displacement operator is

where is the momentum conjugate to the position operator . Demonstrate that

[Hint: Use the momentum representation, .] Similarly, demonstrate that

Hence, deduce that

where is a general function of .

Let , and let denote an eigenket of the operator belonging to the eigenvalue . Demonstrate that

where the are arbitrary complex coefficients, and , is an eigenket of the operator belonging to the eigenvalue . Show that the corresponding wavefunction can be written

where for all .