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### Exercises

1. Monochromatic light with a wavelength of passes through a fast shutter that opens for sec. What is the subsequent spread in wavelengths of the no longer monochromatic light?

2. Calculate , , and , as well as , , and , for the normalized wavefunction Use these to find . Note that .

3. Classically, if a particle is not observed then the probability of finding it in a one-dimensional box of length , which extends from to , is a constant per unit length. Show that the classical expectation value of is , the expectation value of is , and the standard deviation of is .

4. Demonstrate that if a particle in a one-dimensional stationary state is bound then the expectation value of its momentum must be zero.

5. Suppose that is complex. Obtain an expression for and from Schrödinger's equation. What does this tell us about a complex ?

6. and are normalized eigenfunctions corresponding to the same eigenvalue. If where is real, find normalized linear combinations of and which are orthogonal to (a) , (b) .

7. Demonstrate that is an Hermitian operator. Find the Hermitian conjugate of .

8. An operator , corresponding to a physical quantity , has two normalized eigenfunctions and , with eigenvalues and . An operator , corresponding to another physical quantity , has normalized eigenfunctions and , with eigenvalues and . The eigenfunctions are related via       is measured and the value is obtained. If is then measured and then again, show that the probability of obtaining a second time is .

9. Demonstrate that an operator which commutes with the Hamiltonian, and contains no explicit time dependence, has an expectation value which is constant in time.

10. For a certain system, the operator corresponding to the physical quantity does not commute with the Hamiltonian. It has eigenvalues and , corresponding to properly normalized eigenfunctions      where and are properly normalized eigenfunctions of the Hamiltonian with eigenvalues and . If the system is in the state at time , show that the expectation value of at time is    Next: One-Dimensional Potentials Up: Stationary States Previous: Stationary States
Richard Fitzpatrick 2010-07-20