PH 401 Dr. Cecilia Vogel Lecture 1

Review Outline  light waves  matter waves  duality, complementarity  wave function  probability  Review 301 quantum section

Light and Quantum  Quantum Mechanics began historically with light  Light was known to be an EM wave  obey wave equation  2 E=  ∂ 2 E/∂t 2.  solutions sin(kx-  t) and cos(kx-  t) and any combo thereof  Light also found to have particle nature  individual, indivisible photons  Each photon has E=hf=    and p = h/ =  k  Where EM wave has large amplitude  bright, many photons  any one photon likely to be ther

Wave-particle duality  Light (and matter( have both wave and particle properties  gives people issues due to perceptions:  IF you perceive a wave as having a wavelength  IF you perceive a particle as having a position  that’s not the whole story  Reality is a gray area between these extremes  the wave/particle has a spread of positions and a spread of wavelengths  All well-behaved functions are like this:   x  k>1/2

Matter  Matter particles, like electrons, have particle properties (of course)  individual, indivisible particles  energy & momentum  Matter particles, also have wave properties!  They diffract!  They interfere!  Obey a wave equation = Time-dependent Schroedinger eqn

Duality equations  wave and particle properties are related  wavefunction is a function of  (kx-  t)

Complex Wavefunction  If you have issues with a physical quantity like wavefunction being complex  all measureable quantities will be real  like probability density = |  | 2.  OR you can think of wavefucntion as having two components  like light has E-field and B-field  each component will be real  but you will have two components to calculate with two coupled differential eqns  complex functions make the math easier!

Wave Function  Wave nature described by wavefunction  NOT like water or sound wave  where matter actually moves  More like light, wavefunction is a field  electric field (and B( x,t ) = magnetic field).  has a value for every point in space  For matter the wave function is  ( x,t )  like nothing we’ve encountered before.  Not an EM wave.  does not have a direction in space.

Wavefunction Interpreted  For light beam, where the wave function (E-field) is large,  the light is bright  there are lots of photons  For beam of matter particles, where the wave function is large  there are lots of particles.  The “bright” spots in interference pattern  are where lots of photons or matter particles strike.

Probability Interpretation  If you have one particle, rather than a beam,  the wavefunction only gives probability density  P(x,t) = |  (x,t)| 2.  there is no way to predict precisely where it will be.  Where the wave function is large  the particle is likely to be.  The “bright” spots in interference pattern  are where a photon or matter particle is likely to strike.

Probability and Probability Density  P(x,t) = |  (x,t)| 2 is the probability density at position x at time t  like mass density.  To get probability, must have finite region of space  the probability of the particle being in a volume of space