PH0008 Quantum Mechanics and Special Relativity Lecture ?? (Quantum Mechanics) 020516 TEST FILE

1. PH0008Quantum Mechanics and Special RelativityLecture ?? (Quantum Mechanics)020516TEST FILE Prof Rick Gaitskell Department of PhysicsBrown University Main source at Brown Course Publisher background material may also be available at http://gaitskell.brown.edu

2. Recommended Reading

3. Reading - Complete Summary • Please note that Ch 13 is NOT on the list now • I have also indicated areas of background interest only Background reading only - not examined

4. Wave Function - Starting Point… • Proposition: A propagating particle has an associated wave function • This appears as a “reasonable” guess, given our previous studies of waves • Experimental evidence indicated matter has wave like properties • Why is the complex amplitude necessary? • In order to extract the kinetic energy (p2/2m) and total energy (E) in the non-relativistic Schrödinger equation from the wave function we require a second order derivative w.r.t. space, and a first order derivative w.r.t. time • A expression formed from a linear combination of sin() & cos() does not have the desired behaviour • We cannot form an eigen-equation for the Total Energy, which has to be first order derivative w.r.t. time in order that E (or w) drops out Empirically determined He took relationship from photons, and generalised to massiveparticles

5. FAQ - Schrödinger Equation • Why does the Sch. Eq. have the form it does? • As horrible as it sounds - because it works so well (for non-relativistic particles) when used to predict their behaviour in experiments • If we assume that a free particle has the formthen the differential operators naturally provide expressions for the Kinetic, Potential and Total Energy • The Sch. Eq. also has the desirable property of being linear, meaning that if Y1 and Y2 are separately solutions of the Sch. Eq. then aY1 + bY2 is also a solution • If we consider the wave function Y to be a probability “amplitude”. |Y|2 is then interpreted directly as the probability of the particle being at (x,t). “Copenhagen Interpretation” • This interpretation seems very natural and (again) works well in our formalism of quantum mechanics - therefore we use it ! • Remember we never know certain outcome, just the probability distribution of outcomes

6. Heisenberg Uncertainty Principle • Heisenberg proposed the Uncertainty Principle • “It is impossible to design an apparatus to determine which hole the electron passes through, that will not at the same time disturb the electrons enough to destroy the interference pattern”. • The Uncertainty Principle is a necessary for Quantum Mechanics to stay intact • Contradictions arise if we are able to measure both the position and the momentum of a particle with arbitrary accuracy • e.g. See Double Slits discussions

7. A few constants you should be comfortable using… • You will be given constants, but make sure you know how to use them…

8. Solving Sch. Eq. in a Infinite Square Potential (2) • Solutions:- x=0 x=L

9. Reflection at Step Up or Down - Review • Wave Incident on step up You need to know why this naive guess is wrong (see L13 - currents)

10. Superposition Demonstration - Review

11. Resolving Crisis: The beginning… • Planck 1900 • Suggest that “if” it is assumed that energy of normal mode is quantised such that E=hn (h is an arbitrary constant, Planck’s arbitrary constant, experimentaly determined so that theory fits data) then higher frequency (shorter wavelength) modes will be suppressed/eliminated. • Planck suggests ad hoc that the radiation emitted from the walls must happen in discrete bundles (called quanta) such that E=hn . Mathematically this additional effect generates an expression for spectrum that fits data well. • The Planck constant is determined empirically from then existing data • The short wavelength modes are eliminated • In a classical theory, the wave amplitude is related to the energy, but there is no necessary link between the frequency and energy • Classically one can have low freq. waves of high energy and vise versa without constraint • Planck is unable to explain how such an effect could come about in classical physics • Einstein 1905 • Based on Photoelectric effect, Einstein proposed quantisation of light (photons) • Photons are both emitted and absorbed in quanta

12. Watching the Electrons (6) • Let’s repeat the previous 2 slit experiment, but we will include a strong light source so that we can see which slit the electrons go through… ElectronGun • Electrons are charged and so scatter light • Every time we detect a “click” on the far right wall • We will also see a flash of light from near the slits • If we tabulate the results we see P1 and P2 distns as for the case of single slit • What about the combined probability distn?