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# Exercises

1. Demonstrate that
Here, represents a classical Poisson bracket. Moreover, the and are the coordinates and corresponding canonical momenta of a classical, many degree of freedom, dynamical system.

2. Verify that
Here, represents either a classical or a quantum mechanical Poisson bracket. Moreover, , , , et cetera, represent dynamical variables (i.e., in the classical case, functions of the coordinates and canonical momenta), and represents a number.

3. Let be an operator whose eigenvalues can take a continuous range of values. Let the be the corresponding eigenstates. Let be a function of that can be expanded as a power series. Demonstrate that

and

where is the same function of the eigenvalue that is of the operator . Let be a function of the operator that can be expanded as a power series, and let and commute. Demonstrate that

4. Consider a Gaussian wavepacket whose corresponding wavefunction is

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

5. Let and be operators that displace a quantum mechanical system the finite distances and along the - and -directions, respectively. Demonstrate that

and

What are the physical significances of these results?

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

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

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

where is a non-negative integer. Hence, deduce that

where is a function of that can be expanded as a power series.

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 .

Next: Quantum Dynamics Up: Position and Momentum Previous: Displacement Operators
Richard Fitzpatrick 2016-01-22