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Physics 222
Schrodinger’s
Equation
(c)
D.S. Durfee
![](3D"slide0003_image003.jpg")
![](3D"slide0003_image004.gif")
5th Solvay
Conference, Brussels, 23-29 October 1927
=
Back
row: A Piccard, E Henriot, P Ehrenfest, D Durfee, Ed Herzen, Th De Donder,=
E
Schroedinger
=
E Verschaffelt, W Pauli, Wa=
ldo, W
Heisenberg, R H Fowler, L Brillouin.Middle Row: P Debye, M Knudsen, W L Bragg, H= A Kramers, P A M Dirac, A H Compton, = = L de Broglie, M Born, N Bohr.
Front Row: I Langmuir, M Planck, Mme Curie, = H A Lorentz, A Einstein, P Langevin, Ch E Guye,
=
C T R Wilson, O W Richardson.
![](3D"slide0003_image005.gif")
![](3D"slide0003_image006.gif")
![](3D"slide0003_image007.gif")
![](3D"slide0003_image008.gif")
![](3D"slide0003_image009.gif")
![](3D"slide0003_image010.gif")
![](3D"slide0003_image011.gif")
![](3D"master17_background.jpg")
Thought
Question
nWhat would happen to =
the
hydrogen spectrum if I replace the electron
with a muon (which is like an electron but heavier)?
A: No change
B: Almost no change
C: All of the lines will shift to
significantly longer wavelengths
D: All of the lines will shift to
significantly shorter wavelengths
E: I have no idea --- hey, its an ho=
nest
answer!
Thought
Question
nWhat would happen to =
the
hydrogen spectrum if I replace the nucleus
(a proton) with an alpha particle (two protons plus two =
neutrons… i.e. more charge, more mass)?
=
A: No change
B: Almost no change
C: All of the lines will shift to
significantly longer wavelengths
D: All of the lines will shift to
significantly shorter wavelengths
E: I have no idea
Thought
Question
nWhat would happen to =
the
hydrogen spectrum if I replace the nucleus
(a proton) with a deuteron (a proton plus a neutron… <=
span
style=3D'position:absolute;top:17.5%;left:4.68%;width:94.38%;height:5.5%'>=
i.e. same charge, more mass)?
A: No change
B: Almost no change
C: All of the lines will shift to
significantly longer wavelengths
D: All of the lines will shift to
significantly shorter wavelengths
E: I have no idea
Bohr T=
heory
Successes/Failures
=
<=
span
style=3D'font-size:75%'>☺Predicts x-ray emissions (Moseley’s law)
=
;
<=
span
style=3D'font-size:75%'>☺Gives an intuitive picture of what goes on in an
atom
=
<=
span
style=3D'font-size:75%'>☺The correspondence principle is obeyed... sort of&=
#13;
<=
span
style=3D'font-size:75%'>XIt can’t easily be extended to more complica=
ted
atoms
=
<=
span
style=3D'font-size:75%'>XNo prediction of rates, linewidths, or line
strengths
<=
span
style=3D'font-size:75%'>XFine structure (and hyperfine structure) not accou=
nted
for
<=
span
style=3D'font-size:75%'>XHow do atoms form molecules/solids?
<=
span
style=3D'font-size:75%'>XWhere did it come from? There must be a more general
underlying theory!
underlying theory!
X
<=
span
style=3D'font-size:75%;color:#66FF33'>☺It gave hints of a new, underlying theory
X
![](3D"slide0010_image012.gif")
What Will the Wave Equation for =
Particles Look Like =
(1D)?
![](3D"slide0010_image014.gif")
![](3D"slide0010_image016.gif")
For light waves (photons):
For ALL classical waves:
![](3D"slide0010_image018.gif")
For massive particles?
![3D"?"](3D"slide0010_image019.gif")
What Do We Require of our Wave =
Equation?
![](3D"slide0011_image021.gif")
![](3D"slide0011_image022.gif")
![](3D"slide0011_image024.gif")
Waves oscillate in space and tim=
e, any
ω and k possible...
=
... our wave equation must have =
derivatives in x and t
Waves interfere just like light =
waves,
with no interaction...
... our wave equation must be li=
near
![](3D"slide0011_image025.gif")
![](3D"slide0011_image027.gif")
![](3D"slide0011_image028.gif")
![](3D"slide0011_image029.gif")
![](3D"slide0011_image030.gif")
![](3D"slide0011_image031.gif")
![](3D"slide0011_image032.gif")
![](3D"slide0012_image033.gif")
![](3D"slide0012_image034.gif")
Solution, let Ψ be complex
That’s
not to say that Schrödinge=
r’s
equation is relativistic...
Problem 1: Dispersion Relation
Problem 2: Principle of (Galilean) <=
/span>Relativity?
If an electron is at rest, λ=3D∞, P(x)=3Dconstant
=
In a frame moving at speed v,=
3;
λ=3Dh/mv,
P(x)=3D[sin(2π m v x / h)]2
has
nodes
![](3D"slide0012_image036.gif")
Real vs. Complex Waves
![](3D"slide0013_image038.jpg")
![](3D"slide0013_image040.jpg")
![](3D"slide0013_image042.jpg")
Real
P(x)
Complex
=
It’s
still wavy, we can still see
things
like two slit interference...
But
a plane wave doesn’t have
to
have probability “nodes”
Consequences of Imaginary Ψ=
nWe can’t directly measu=
re Ψ
nWe canR=
17;t
even assign physical meaning!
nNot like t=
he
wave of a photon, which is its electric field, w=
hich
has physical “meaning” and can be
measured (at least for an ensemble of
photons...).
nThings that
are physical and can be measured can stil=
l be
extracted from Ψ
=
;
nP(x,t)dx =3D |Ψ|2dx =3D Ψ*Ψdx=
With th=
at in
mind…
![](3D"slide0015_image044.gif")
Schrödinger’s
Equation
![](3D"slide0016_image046.gif")
![](3D"slide0016_image047.gif")
Potential
Energy
Looks Like
p2/2m,
Kinetic Energy
Looks Like
hω,
Total Energy
![](3D"slide0016_image048.gif")
![](3D"slide0016_image050.gif")
Free Particle
![](3D"slide0016_image051.gif")
In General
![](3D"slide0016_image052.gif")
![](3D"slide0016_image053.gif")
=
The
kinetic energy comes from the curvature of a wavefunction...
the more wiggly in space, t=
he
faster it’s moving!