1. Coordinate TransformationS:

2. H wavefunctions and probabilities:

3. Expectation Values:

4. Expectation values

5. Types of expectation values:

6. Expectation Values – Examples: • We will consider several examples where expectation values can be calculated for energy, linear momentum, position and distance of the electron from the nucleus. Additional calculations are possible – for example, expectation values for various types of angular momentum. Initially, the form of the wave function is known.

10. PIAB – one dimension – momentum: • The result obtained for the linear momentum expectation value on the previous slide, zero, may seem strange. In the one dimensional PIAB the particle always has non-zero momentum and kinetic energy. However, the average or expectation value for momentum is zero since the particle moves (equal probability) in both directions.

11. Expectation value – position:

12. Expectation values – hydrogen atom:

13. Variational Calculations: • In all examples of expectation value calculations considered the exact form of the wave function was known. In some cases the precise form of the wave function is not known. We can still estimate values for energies and so on using the methods outlined and an approximate, guessed or trial wave function.

14. Variational Calculations: • Estimates for energies obtained using approximate wave functions are extremely important in chemistry. If the energy calculated using the true wave function is ψExact then with and approximate wave function, ψAPP, we would obtain an approximate estimate for energy, EAPPROX.

15. Variational Calculations:

16. Variational Theorem: • The variational theorem tells us that when any approximate wave function is used the estimate for energy obtained from an expectation value calculation will be greater than the energy value calculated with the true wave function. We will explore this using class and assignment examples.