1. Quantum Mechanics

2. Electrons behave as waves (interference etc) and also particles (fixed mass, charge, number)
Lessons on or

3. Science at the end of ~1900: Classical Mechanics

4. Leading to Mech. Engg., Civil Engg., Chem. Engg.

5. Science at the end of ~1900: Electromagnetics
Rainbows
Polaroids
Lightning
Northern
Lights
Telescope
Laser
Optics

6. Science at the end of ~1900: Electromagnetics
Electronic Gadgets

7. Science at the end of ~1900: Electromagnetics
Chemical Reactions
Neural Impulses
Ion Channels
Biological Processes
Chemistry and Biology

8. But there were puzzles !!! What does an atom look like ???
Dalton (1808)
What does an atom look like ???

9. Solar system model of atom
Continuous radiation from orbiting electron
mv2/r = Zq2/4pe0r2
Centripetal
force
Electrostatic
force
Pb1: Atom would
be unstable!
(expect nanoseconds
observe billion years!)
Pb2: Spectra of
atoms are discrete!
Spectrum of Helium
Transitions E0(1/n2 – 1/m2) (n,m: integers)

10. Bohr’s suggestion From 2 equations, rn = (n2/Z) a0
Only certain modes allowed (like a plucked string)
nl = 2pr (fit waves on circle)
Momentum ~ 1/wavelength
(DeBroglie)
p = mv = h/l
(massive classical particles  vanishing l)
This means angular momentum is quantized
mvr = nh/2p = nħ
From 2 equations, rn = (n2/Z) a0
a0 = h2e0/pq2m = Å (Bohr radius)

11. Bohr’s suggestion
E = mv2/2 – Zq2/4pe0r
Using previous two equations
En = (Z2/n2)E0
E0 = -mq4/8ħ2e0 = eV
= 1 Rydberg
Transitions E0(1/n2 – 1/m2) (n,m: integers)
Explains discrete atomic spectra
So need a suitable Wave equation so that imposing boundary
conditions will yield the correct quantized solutions

12. What should our wave equation look like?
∂2y/∂t2 = v2(∂2y/∂x2)
String y x w k
Solution: y(x,t) = y0ei(kx-wt)
w2 = v2k2
What is the dispersion (w-k) for a particle?

13. What should our wave equation look like?
Quantum theory:
E=hf = ħw (Planck’s Law)
p = h/l = ħk (de Broglie Law)
and E = p2/2m + U (energy of a particle) w k
Thus, dispersion we are looking for is
w  k2 + U
∂2y/∂t2 = v2(∂2y/∂x2)
So we need one time-derivative
and two spatial derivatives X

14. Wave equation (Schrodinger)
iħ∂Y/∂t = (-ħ22/2m + U)Y
Kinetic Potential
Energy Energy
Makes sense in context of waves
Eg. free particle U=0
Solution Y = Aei(kx-wt) = Aei(px-Et)/ħ
We then get E = p2/2m = ħ2k2/2m w k

15. For all time-independent problems
iħ∂Y/∂t = (-ħ22/2m + U)Y = ĤY
Separation of variables for static potentials
Y(x,t) = f(x)e-iEt/ħ
Ĥf = Ef, Ĥ = -ħ22/2m + U
Oscillating solution in time
BCs : Ĥfn = Enfn (n = 1,2,3...)
En : eigenvalues (usually fixed by BCs)
fn(x): eigenvectors/stationary states