1. Chapters Q6 and Q7 The Wavefunction and Bound Systems.

2. The difference between classical and quantum physics. Physical Reality Experimental phenomena SG Conceptual model (our understanding) F=ma Mathematical model Mathematical model ψ(x) Classical PhysicsQuantum physics At the quantum physics level we have no “feeling” for what will happen There is no easy link between the math and the experiment.

3. Wavefunction rules 1)All we can know about a quanton (called its “state”) is contained in a wavefunction,. 2)For every possible numerical value of any observable there is a normalized wavevector that corresponds to that state which we call an “eigenvector” (characteristic vector). The possible values of the state are called the eigenvalues.

4. Rule 3, the collapse rule This rule is the furthest from our macroscopic experience and the most difficult to accept. 3) When we perform any experiment to determine the value of one of the quanton’s observables, the experiment determines the observable’s value by “collapsing “ the quanton’s state to a randomly selected one of that observable’s eigenfunctions and yields the observable value corresponding to that state. Ψ(x) Before collapse After collapse, only one position is possible (The quanton can only be in one place.) x

5. 4) The outcome probability rule. In any experiment, the probability of any given result (finding the position of the particle) is the absolute square of the inner product of the quanton’s original state vector and the result’s eigenvector. This means the probability of finding the particle between x 0 and x 1 is given by: In the limit as dx 0.

6. Bound States A bound state in classical physics exists when E<E 0. This means the particle cannot escape. E0E0 E x x The particle must stay in the range marked “x” below. If E increases, the range of x will also increase. V(x)

7. Quanton of mass m in a box of width L with infinitely high sides. (It can never get out.) ∞ ∞ The three lowest eigenvalues for ψ(x) are: n=1 n=2 n=3 The general solution is: The general solution for the allowed energies is: where n=1,2,3,… L

8. The simple harmonic oscillator The potential of a simple harmonic oscillator is V(x) =½mω 2 x 2 The allowed energies are: n=0,1,2,3,… V(x) E0E0 ψ E0 (x) E1E1 ψ E1 (x) Note that there is a finite probability the particle will be outside the well. This result is impossible in classical physics!

9. The Bohr Model of the Hydrogen Atom. Suppose the electron must make an even (integral) number of de Broglie wavelengths as it orbits the proton in the hydrogen atom. This will happen when the radius satisfies the relationship 2πr=nλ, where n=1, 2, 3, … From classical mechanics we know the centripetal force must be equal the electrical force so: From de Broglie’s equation Combining these two equations gives the final result for the possible radii of the atom as:

10. Results of the Bohr Model If an electron circles the nucleus it is constantly being accelerated and gives off radiation –This cannot be the case or the electron would run out of energy. The Bohr model requiring closed orbits solves this problem. The energy levels of the hydrogen atom (spectra) are very well explained by this very simple model.

11. Problems due Wednesday Q6B.3, Q6B.4, Q7B.1 and Q7B.5