1. Chapter 5: Quantum Mechanics
Limitations of the Bohr atom necessitate a more general approach
de Broglie waves –> a “new” wave equation
“probability” waves
classical mechanics as an approximation
Wave Function Y
probability amplitude

2. Mathematical properties of the wave function

3. More mathematical properties of the wave function

4. The classical wave equation as an example of a wave equation:

6. Time dependent Schrödinger Equation
linear (in Y) partial differential equation

7. Expectation values (average values)

8. If the potential energy U is time independent,
Schrödinger equation can be simplified by “factoring”
separation of variables
Total energy can have a constant (and well defined) value
Consider plane wave:
An eigenvector, eigenvalue problem!

9. The time independent Schrödinger equation
Allowed values for (some) physical quantities such as energy are related to the eigenvalues/eigenvectors of differential operators
eigenvalues will depend on the details of the wave equation (especially in U) and on the boundary conditions

10. Particle in a box: (infinite) potential wellU
Particle in a box: (infinite) potential well L V0 x

11. Wavefunction normalization

12. Example Find the probability that a particle trapped in a box L wide can be found between .45L and .55L for the ground state and for the first excited state.
Example 5.4 Find <x> for a particle trapped in a box of length L

13. Particle in a box: finite potential wellU
Particle in a box: finite potential well L V0 E x I II
III

14. U
Boundary Conditions L V0 E x I II
III

16. c = 4
c = 100
c = 1600

17. Tunneling U L V0 E x I II
III

18. U
Boundary Conditions L V0 E x I II
III

20. Example 5. 5: Electrons with 1. 0 eV and 2
Example 5.5: Electrons with 1.0 eV and 2.0 eV are incident on a barrier 10.0 eV high and 0.50 nm wide.
(a) Find their respective transmission probabilities.
(b) How are these affected if the barrier is doubled in width?

21. Harmonic Oscillator: classical treatment

22. Quantum Oscillator

24. Quantum Harmonic Oscillator Solutions

25. Operators

26. Example 5. 6: An eigenfunction of the operator d 2 /dx 2 is y = e2x
Example 5.6: An eigenfunction of the operator d 2 /dx 2 is y = e2x. Find the corresponding eigenvalue.

27. Chapter 5 exercises: 4, 5, 6, 11, 23