Download Quantum Concepts and Quantum Concepts and Mechanics

Quantum Concepts and
Mechanics
What everyone should know
Patrik Callis
Department of Chemistry and Biochemistry
Montana State University
Presented to MSU Philosophy Group April 27, 2009
and Chmy373 January 13, 2010
Outline
Snippets of wisdom and insights
insights, gathered during 48 years
of studying quantum mechanics and quantum chemsitry.
1. The attraction of nuclei and electrons is ENORMOUS
2. Why doesn’t the electron fall into the nucleus?
3. A simple way to look at the Schrodinger Equation, and
how it predicts ZERO POINT energy (thus predicting the
electron won’t
won t fall) and TUNNELING.
TUNNELING
4. “Understanding” quantum mechanics
5 An amazing
g thing
g about quantum interference in the
double slit experiment.
6. The secret life of wavefunctions (they oscillate in time
with a real and imaginary part
part.))
A Working Theory of Chemistry, Biology, and Geology
Matter: Nuclei (+) and electrons (-)
Potential Energy:
Energ
Electromagnetic -->
> Coulomb’s
Co lomb’s Law
La
= constant * charge1
g * charge2
g /distance
Mechanics: Quantum
Atoms: electrons seeking a more
positive environment
Chemical Energy: electrons
seeking a more positive environment
by pulling nucleli together
_
+
+
_
+
The Coulomb Force is Enormously Strong
an electron 0.1 Angstrom
g
from a proton
p
would “weigh” 2.3 milligrams
i e a MACROSCOPIC FORCE!!
i.e.,
The forces (and therefore the energies) are the SAME
in quantum mechanics and classical mechanics
Birth of Quantum Theory
1905 Planck
Quantization of energy ΔE = hν in matter to
emit light of freq. ν
1905 Einstein: Quantization of energy
ΔE = hν for LIGHT to absorb or
emit light of freq. ν
and Particle nature of light: momentum = h/wavelength
‘
1912 Rutherford’s scattering of alpha particles revealed the nature of atoms:
dense nucleus surrounded by empty space with electrons.
This revealed a new concept: Zero Point Energy, i.e. the electron for some
reason will not fall into the nucleus.
Other radical new quantum concepts:
~1920 DeBroglie: Wave nature of particles
wavelength = h/momentum
~1925 Heisenberg: Uncertainty Principle Δmomentum x Δposition ~ h
The lowest energy
gy
state of the Hydrogen
atom.
+
proton
electron
1 Angstrom
The electron will not fall into
the nucleus!
Figure 3. Density slice through the 1s orbital
The blue line is the square of the
wavefunction.
wavefunction
1925
Schrodinger’s Equation:
A simple equation that was discovered (not derived)
Classical Mechanics
Kinetic Energy
+
Potential Energy
=
Total Energy
Quantum Mechanics (Schrodinger’s Equation without time) translated into
English:
-h2/8pi2mass x Curvature of Wavefunction + Potential Energy x Wavefunction =
Energy x Wavefunction
curvature operation
(2nd derivative
h/2pi
mass
Kinetic energy
wavefunction
potential energy
Total energy
Time independent Schrodinger Equation :
⎛ ∂2 ∂2
∂ 2 ⎟⎞
⎜
- 2×
+ 2 Ψ + potential E × Ψ = total E × Ψ
2
2
⎜
8π
∂z j ⎟⎠
all particles j ⎝ ∂x j ∂y j
or : kinetic energy operator × Ψ + classical potential energy × Ψ
h2
∑
= total energy × Ψ
HΨ = EΨ , where H = Hamiltonian = total energy operator
Ψ * Ψ = probablility density for finding particle locations
Ψ * is
i the
h complex
l conjugate
j
. i.e.,
i change
h
all
ll i - - > - i
i = -1
Potential energy EXACTLY same
as in Classical mechanics
Three things are different from Classical mechanics:
1) The wavefunction (Schrödinger did not know what its physical meaning was at
the time he published). Later the consensus was reached that the absolute square
of the wavefunction gives the probability density for finding the particle.)
2) Kinetic energy is represented by the CURVATURE of the Wavefunction.
In calculus, that is the 2nd derivative (i.e., the slope of the slope of the function)
3) h, Planck
Planck's
s constant, which was empirically adjusted so that the Schrödinger
Equation gives agreement with experiment.
This simple equation embodies the 5 seemingly distinct new "quantum concepts"
focus on zero-point energy:
which
hi h is
i so powerful
f l that
th t it kkeeps th
the electrons
l t
((-)) ffrom falling
f lli iinto
t th
the
nucleus(+) of an atom despite the enormous attraction. (Classically, the
electron would sit in the nucleus at ordinary temperatures).
When a force confines a particle, this means that its wavefunction is
localized, which in turn means its CURVATURE is large.
B
0.5
Wavefunction
Wavefunction
W
10
1.0
0.0
1
. 0
0
. 5
0
. 0
-0.5
-
0
. 5
-1.0
-
1
. 0
0
0
10
20
30
40
50
1
0
2
0
3
X
X
Large curvature =
HIGH kinetic E
Small curvature =
LOW kinetic E
Note that when wavefunction positive, the curvature is negative and vice
versa, so the kinetic E is always positive
0
4
0
This, coupled with small mass, means kinetic energy is very high for
confined electrons.
Chemical energy is virtually all the change of zeropoint energy of
electrons during chemical reactions.
The dark energy of the universe that is responsible for the accelerating
expansion of the universe is now thought by many to be quantum zero
point energy.
Understanding Quantum Mechanics?
Richard
Ri
h d Feynman
F
lecturing
l t i to
t a lay
l audience
di
att
Cornell, circa. 1965:
“There
There was a time when the newspapers said that only twelve
men understood the theory of relativity. I do not believe
there ever was such a time... After they read the paper, quite
a lot
l t off people
l understood
d t d the
th th
theory off relativity...
l ti it On
O the
th
other hand, I think it is safe to say that no one
p saying
y g to
“understands” qquantum mechanics... Do not keep
your self “But how can it be like that?”, because you will get
“down the drain” into a blind alley from whihc nobody has
yet escaped
escaped. NOBODY KNOWS HOW IT CAN BE LIKE
THAT. “
--Richard P. Feynman
Chapter 6, The Character of Physical Law
23rd Printing, 1998
Feynman, from Lectures on Physics III :
Quantum Mechanics exactly describes the
g
behavior electrons and light.
“Electrons and light do not behave like anything
we have ever seen
seen.”
“There is one lucky break, however—electrons
behave just like light”
TIME DEPENDENT Schrodinger Equation :
HΨ = EΨ , where H = Hamiltonian = total energy operator
∂Ψ
∂t
= −i
h
HΨ
2π
Ψ is proportional to e
−i
2πEt
h
All wavefunctions oscillate with frequency such that E = hυ
Time dependent Schrodinger Eq.
Wavefunction = Space part x e-i2π(E/h)t
Square of wavefunction has NO time
dependence for energy eigen states
Actual experimental results of a
double-slit-experiment performed by
Dr. A. Tonomura showing the buildup of an interference pattern of
single electrons. Numbers of
electrons are 10 (a), 200 (b), 6000
(c) 40000 (d)
(c),
(d), 140000 (e)
(e).
(Provided with kind permission of
Dr. AkiraTonomura.)
http://www.youtube.com/watch?v=oxknfn97vFE
Click on the link below to see calculations and movies of
an electron exhibiting interference when passing through
two slits.
http://rugth30.phys.rug.nl/quantummechanics/diffint.htm