1. The meaning of it all

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

3. Two slit experiment Dq ~ Dpy/py ~ l/2d

4. Can’t measure position and momentum together !
Flash detects particle
Deflects it, scrambles momentum
Washes off interference
Longer wavelength preserves particle
Momentum unaffected
Cannot ‘see’ it anymore !

5. Wave particle duality p = h/l = ħk k Monochromatic plane wave
x, k are Fourier transform pairs
k, l well defined, what about x ???

6. Make it look more like a particle
p = h/l = ħk k
Wavepacket
x more narrowed down, but
p, k undefined

7. What equation does this wave satisfy?

8. 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?

9. 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

10. Wave equation (Schrodinger, 1925)
iħ∂Y/∂t = (-ħ22/2m + U)Y
Kinetic Potential
Energy Energy
Eg. free particle U=0
Solution Y = Aei(kx-wt) = Aei(px-Et)/ħ
We then get E = p2/2m = ħ2k2/2m w k
Note that Y must be complex !!

11. What does Y(x,t) represent?
|Y(x,t)|2dx: Probability amplitude of finding
particle between x and x+dx at time t
(y like phasor electric field E
P like intensity)
Must have ∫P(x,t)dx = 1
at all times
Max Born, 1926

12. High Y*Y here Each measurement yields a definitive result
Probability shows up through ensemble histogram

13. Action of a measurement operation

14. Copenhagen Interpretation, 1927

15. http://graphics8. nytimes
krugman1/053011krugman1-blog480.jpg

16. xy |y> py p = -iħ What do we measure? iħ∂Y/∂t = (-ħ22/2m + U)Y ⌃
Kinetic Potential
Energy Energy
|y> ⌃
p = -iħ ⌃ py

17. Readings .. Eigenstates (like modes)
Oyn = Onyn ⌃ O1 O2 O ⌃ O3

18. Example … position operator⌃
Eigenstate of x is a delta function in x
(since that gives an unambiguous reading!)
x d(x-x0) = x0 d(x-x0) ⌃ x x0
Here we have a continuous set of eigenvalues
since x0 can have any value

19. Example … position operator
x d(x-x0) = x0 d(x-x0) ⌃
Since we can construct any function out of spikes,
x Y(x) = dy x d(y-x) Y(y)
= x dyd(y-x)Y(y)
= xY(x) ⌃  
ie, a simple multiplication
not an
eigenstate
of x

20. A generic solution is a superposition⌃
Oyn = Onyn
y = c1y1 + c2y2 + c3y3 + …
Much like a Fourier series, except
what we actually see in a single shot
measurement is never such a
superposition, but only one of the
eigenvalues. The superposition only
shows up when aggregated over many measurements
(ie, on average)

21. A generic solution is a superposition⌃
Oyn = Onyn
y = c1y1 + c2y2 + c3y3 + …
Measure O1 with probability |c1|2
Measure O2 with probability |c2|2

22. Complementarity of measurements
Y cannot be simultaneous eigenstates of x and p,
since we know that measuring both disrupts the state
and is behind the x,p uncertainty ⌃ ⌃ ⌃
xpy ≠
pxy ⌃ ⌃ ⌃ ⌃ ⌃ ⌃
x,p xp - px ⌃ [ ] = ≠

23. Complementarity of measurements
If we choose Y as the eigenstate of x (ie, delta
function), then p must alter it immediately ⌃ = x x ⌃ = p p

24. Complementarity of measurements
[x, px]y = x(-iħy/x) + iħ(xy)/x = iħy
(xp ⌃
px) -
y = iħy
x,p = [ ] iħ
This equation only makes sense with the y in there

25. What does Y(x,t) represent?
Y itself hard to measure so overall phase irrelevant
Probabilities allow us to compute
expectations !!
<O> = ΣnOnPn
Charge density r(x,t) = qP(x,t)
Current density J(x,t) = qvP(x,t) (“v” to be defined later)

26. Averages O = ∫|y(x,t)|2O(x)dx O = ∫y*(x,t)Ô(x)y(x,t)dx
Symmetrized!
Ehrenfest
One can show that averages
follow ‘classical rules’
sx.sp ≥ ħ/2
(Uncertainty Principle)
d<x>/dt = <v>
md<v>/dt = <-dU/dx > l

27. Some meaningful expectations
n = Y*Y (Electron Density)
v = p/m = -iħ/m
J = q<v> = q∫y*(x) [-iħ/m] y(x)dx (Symmetrize !!!)
J = iħ/2m(YY*/x – Y*Y/x)
Symmetrized version of <v>
(v + v*)/2, which gives a negative sign in the
middle because of the ‘i'

28. Verifying Current Density
For plane waves Y = Y0eikx
J = n(ħk/m) = nv, as expected
Also easy to check:
n/t + J/x = 0 (Continuity Equation)

29. For all time-independent problems
iħ∂Y/∂t = (-ħ22/2m + U)Y = ĤY
Separation of variables for static potentials
Y(x,t) = y(x)e-iEt/ħ
Ĥy = Ey, Ĥ = -ħ22/2m + U
Oscillating solution in time
BCs : Ĥyn = Enyn (n = 1,2,3...)
En : eigenvalues (usually fixed by BCs)
yn(x): eigenvectors/stationary states

30. Relation between Y and yn ?
yn s are allowed solutions (like mode shapes of a fixed string)
Their energies (‘frequencies’) are the eigenvalues En
Aside: They are orthogonal (independent), like modes of a string
∫y*n(x)ym(x)dx = 0 if n ≠ m
and normalized
∫y*n(x)yn(x)dx = 1
Y shows the actual solution (superposition of allowed ones)
In general, Y(x,t) = Sn an yn(x)e-iEnt/ħ

31. Summary
Electron dynamics is inherently uncertain. Averages of observables can be computed by associating the electron with a probability wave whose amplitude satisfies the Schrodinger equation.
Boundary conditions imposed on the waves create quantized modes at specific energies.