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
where , , and are real numbers. Demonstrate that
What are the physical significances of these results?
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 .