PHYSICS 102N Spring Week 12 Quantum Mechanics and Atoms

Transcription

1 PHYSICS 102N Spring 2009 Week 12 Quantum Mechanics and Atoms

2 Quantum Mechanics 1. All objects can be represented by waves describing their propagation through space 2. The wave length is λ=h/p and frequency is f = E/h 3. Simultaneous measurement of position and momentum is limited by Heisenberg s uncertainty principle 4. Stable orbits = standing waves (see later) 5. If you build an apparatus that registers a localized interaction (atom excitation, electron emission, pixel activitation ), you can only predict probability for particle to be at a given point: P(r) = Wave Intensity(r) (Amplitude squared) 6. Interaction is all or nothing (quantized)

3 How can we see atoms? Need particles with wave length < size of atom (10-10 m) light ruled out; gamma rays too hard to image electrons (microscope -> accelerator) possible, but require lots and lots (small cross section for interaction); difficult to disentangle from atomic electrons Rutherford: Used high-momentum 4 He nuclei ( alpha particles ) emitted from radioactive substances (see later) Findings: Most are deflected only a tiny bit ( refraction by thin medium = electron cloud ), some are deflected a LOT (even thrown back) => Point source (Huygen s principle!) = atomic nucleus; must contain most of mass Wave length: 4 MeV = J, mass = kg => E = 1 2 mv 2 " v = 2E m =14 #106 m/s ; p = mv = 9#10 $20 kgm/s " % = h / p = 6.63#10 $34 Js / 9#10 $20 kgm/s = 7 #10-15 m = 7 fm

4 How do we know the properties of electrons? Cathode ray tube experiments (J.J. Thompson): get ratio charge/mass (q/m) Deflection by magnetic field B is q/p = q/(mv) Deflection by electric field E is 1 / 2 at 2 q/(mv 2 ) Oil drop experiment (Millikan): get charge directly (q) measure electric field needed to balance gravity for electrically charged oil drops (known mass m) qe = mg => q Combine both: get q and m!

5 Put electrons and nuclei together => atoms Already studied Bohr s model: Electrons can only circle nucleus on a fixed set of orbits (not many are possible) as in the solar system no radiation while they are IN these orbits Each orbit has a fixed total energy (kinetic plus electrostatic binding In hydrogen, the possible values are E n = ev / n 2 (n = 1,2,3, ; n = is ionized H) Possible photon frequencies are given by energy differences between any 2 levels: f = ΔE/h In hydrogen = 13.6 ev x (1/n final 2-1/n initial2 ) Balmer series: n final = 1 (all UV); n final = 2 are visible Heavier nuclei: Larger energies => X-rays, UV, light, IR

6 How to build atoms Rules: Nuclear charge (# of protons) determines # of electrons Electrons only in stable orbits Each orbit can take only up to a maximum number of electrons (Pauli Exclusion Principle): lowest orbit: 2, next: 8, then 8+10=18, =32 Most stable: filled orbits (He, Ne, Ar, Kr, Xe) explains periodic table of elements explains (at least in part) chemistry As orbits are filled, Z increases and atoms become smaller until next orbit starts getting filled

7 What are these orbits really? Must have connection to wave nature of particle motion Remember slide Quantum Mechanics : Stable orbits = standing waves (see later) As electron goes around nucleus it is like a wave on a closed string - to get constructive interference, it must have same phase after going around once => length must be equal to integer multiple of wave length λ = h/p straightforward math shows and E = " mq4 1 8# 2 0 h 2 n 2 = n 2 ev for H r = h2 n 2 " 0 # mq 2 = n2 $ 0.53$10 %10 m for H not all orbits are circles; also ellipses allowed (hence more than 2 electrons for higher orbits)

8 but really really? Remember: waves in space spread out into all 3 dimensions it makes no sense to talk about classical orbits at all - electrons can be anywhere picture cloud = 3-dimensional amplitude-distribution (around a classical orbit) each orbit has different size and geometry of cloud amplitude distribution fulfills Schrödinger s wave equation in dispersive medium of electrostatic field around the nucleus Surprisingly, numerical values remain correct!

9 What does this cloud mean? Again, remember : probability for particle to be at a given point: P(r) = Wave Intensity(r) (Amplitude squared) => cloud represents wave intensity all over space = probability to find the electron exactly THERE can only predict average outcome from many experiments So what is exactly waving here? The square root of a probability distribution??? Something even more abstract than the electromagnetic field - a mathematical construct (a complex-number function called ψ> = psi ) living in an abstract space called Hilbert Space