Читать книгу Quantum Mechanical Foundations of Molecular Spectroscopy - Max Diem - Страница 34

The following trigonometric integral relationships are needed for these problems:

1 Show that the function f(x) = cos(bx) is an eigenfunction of the operator d2/dx2. What is the eigenvalue?

2 Show that the function e−x2/2 is an eigenfunction of the operator (d2/dx2) − x2. What is the eigenvalue?

3 Show that the function e−4ix is an eigenfunction of the operator d2/dx2. What is the eigenvalue?

4 What is the probability P of finding a ground‐state PiB in the center third of the box? What is P for the same range for a classical particle?

5 For the PiB in the ground state, determine the expectation values of x and px.

6 What is the expectation value of the kinetic energy operator T for the ground‐state PiB?

7 What is the probability P of finding a particle in the first excited state in the left half of the box with length L within the PiB approximation?

8 Consider an electron in a one‐dimensional box with a length of 0.1 nm.Calculate the energy of the 1st, 2nd, and 3rd energy levels for this electron.Calculate the wavelength of a photon required to promote the electron from the 2nd to the 3rd energy level.

9 Describe in your own words why the particle‐in‐a‐box model results in quantized energy levels.

10 What is quantum mechanical tunneling?

11 Calculate the commutator [Tx, px] where Tx is the kinetic energy operator in the x‐direction and px is the momentum operator in the x‐direction. Can the kinetic energy and the momentum be determined simultaneously in a quantum mechanical system?

12 Show by analytical integration that ψ1(x) and ψ2(x) are orthogonal for the one‐dimensional PiB.

13 Using a graphics program such as Excel, plot the first and second wavefunctions for a particle in a box. Show by graphical integration that these functions are orthogonal.