Transcript Quantum Mechanics

Good Evening to all of
YOU!
The strange (and beautiful) world of
Quantum Mechanics
Very
frightening
Just
beautiful!
Outline
1. Wave or Particle?
..Two slit experiments
2. Uncertainty Principle
3. Standing Waves
4. Stationary States - Atomic Orbitals
5. Do we understand Quantum Mechanics?
6. The Path Integral – Random Walk
7. The Copenhagen Interpretation
8. The Schroedinger’s Cat
9. The EPR Paradox
Two Slit experiments
With particles (bullets)
I1
I2
I12
Behavior of bullets
is easy to understand
- LUMPINESS!
I1
I12
I2
Behavior of bullets
is easy to understand
- LUMPINESS!
With waves (water waves)
I1
I2
I12
With waves (water waves)
I1
I2
I12
Waves interfere!
NO LUMPINESS!
1 + 2  12
With electrons
I1
I2
I
I12
12
Electrons are like
bullets - lumps
LUMPINESS!
INTERFERENCE!
1/1000 mm
http://www.hqrd.hitachi.co.jp/em/doubleslit.cfm
God does
not play
dice!
This type of behavior
was observed first in
the case of light!
History
Light is a beam
of particles
Exhibits
interference,
hence waves!
Huygens
Newton
EM Theory,
hence waves
Einstein
In photoelectric effect,
light behaves like particles
In my scattering expts,
light - beam of particles
Compton
Maxwell
On Mondays, Wednesdays and
Fridays, it is a PARTICLE!
On, Tuesdays,Thursdays and
Saturdays it is a WAVE!
Oh LORD, please, is it a
PARTICLE or a
WAVE!!???
On Sundays
It is both! It has a DUAL
NATURE
Not only electrons and photons,but
EVERYTHING has this dual nature!
Exptly shown for protons, neutrons, He
atoms, even C60!
C60
Prof. A. Zeilinger
http://www.quantum.univie.ac.at/zeilinger/
Duality
Whom do you see
in this picture?
A young girl?
Old woman?
De Broglie - Heisenberg
de Broglie
Heisenberg
Reflection
Modes of a String - Standing waves
Standing waves of a bridge
On a rectangular membrane
Stretched
membrane
Time dependent wave
Stretched
membrane
One node
No nodes
Two nodes
Infinite modes
One node
Four nodes
Box
Particle in a box
Standing wave
patterns in 2D can
be formed!
From work of Eigler (IBM)
From work of Eigler (IBM)
From work of Eigler (IBM)
Box
+
Hydrogen Atom: ebound to proton
Standing wave
patterns in 3D can
be formed!
Atomic Orbitals
pz
dz2
dxy
Standing waves
in 3 dimensions
fz 3
gz4
leads to bonding
Electron density goes
into internuclear region!
Constructive interference!
Electron density goes away
from the internuclear region!
Destructive interference!
Do we understand Quantum Mechanics?
How does it
know that both
the slits are
open!!??
I don’t understand
it!!
At least this time,
he is sensible!
Nobody does!!!
Does it go
through both
the slits?
How does an electron move
(propagate)?
Through all
possible paths!
He is crazy!
More slits
Source of
particles
Even more
Complex setup
In the limit of  walls
each with  slits…….
All paths
contribute!
Action
Quantum mechanics and the
drunken walker
Haddock in the Evening
We can only talk of
Probabilities!
BAR
Some paths are more
probable than others
BAR
I have always
known that the
electron is weird!
REFERENCES
1. G. Gamov, Tompkins in Paperback, Canto Books. CUP, 1993
2. R.Gilmore, Alice in Quantum Land, Affiliated East West Press Ltd.
1994
3. Gribbin, Schrodinger’s Cat, Black Swan, London, 1984
4. Cropper, The Quantum Physicists, OUP, 1971
5. J. Gribbin, Schrodinger’s Kittens, Little, Brown & Co. London, 1995
6. G. Gamov and R. Stannard, The New World of Tompkins, Cambridge
Univ. Press, 1999
7. R.P. Feynamn, QED, Princeton University Press, 1988
8. R.P. Feynman and A.R. Hibbs, Quantum Mechanics and Path Integrals,
McGraw Hill, New York, 1965.
See: www.colorado.edu/physics2000
I think it is safe to say that no
one understands quantum mechanics.
Richard Feynman
The reason universities have students is so
they can teach the professors,
and Feynman was one of the best (students).
John Wheeler
What do you think of Quantum
Mechanics?
OR
Random Walk
Haddock in the Evening
We can only talk of
Probabilities!
BAR
Examples
Colloidal particle – Brownian motion
Long Chain molecules in solution
Some paths are more
probable than others
BAR
@ ¡!
P ( x ; t) = D r 2 P ( ¡!x ; t)
@t
Random Walk in the City
BAR
RW with
absorption
Path (Functional) integrals
form the language of Modern Physics
If you have wave
phenomena, then there
must be a wave equation!
He is crazy!
Matter waves
obey my
equation
@ ¡!
!
¡
2
P ( x ; t) = D r P ( x ; t)
@t
@ !¡
~2
i ~ ª ( x ; t) = ¡
r 2 ª ( !¡x ; t)
@t
2m
Time dependent Schrodinger Equation
If you know
then you can calculate
My dear Quantum,
Why don’t stop this
blabbering!??
Postulate I
The state of a system is
specified, as fully as is possible
by the state function
ª (x; y::; t):
Prof. S.N. Datta
(IITB)
Probability of finding the
system in a volume element
d¿ = dx dy
ª
There he goes!
is given by
¤
ª d¿
Postulate II
Corresponding to every
observable, there is a linear,
Hermitian operator
!!!!


To find the operator, write
down the classical mechanical
expression for the observable,
and make the following
replacments:
x! x
@
px ! ¡ i ~
@x
Postulate III
Measurement of an observable
would give one of the eigenvalues
of the corresponding operator.
Average of a large number of
measurements is given by:
R
b
¤ Aª
d¿ª
< Ab > = R
d¿ª ¤ ª
http://www.ipod.org.uk/reality/index.asp
Decoherence
Postulate IV
The state function obeys the
time dependent Schroedinger
Equation:
@
ª
i~
@t
= Hb ª
You are crazy, my dear
Calculus!
That is enough!
Consequence
If the V(x) is time independent,
then there exists special states stationary states
i ni ti al
af ter a ti me t
Time Evolution
In general, the state
function changes
Special situation
i ni ti al
af ter a ti me t
The state function remains
unchanged!
Stationary State
How to½ prove this?¾
@ª (x; t)
i~
=
@t
Let
~2 @2
¡
+ V (x) ª (x; t)
2m @x 2
ª (x; t) = Ã(x)T (t)
½
¾
@T (t)
~2 @2
i ~Ã(x)
= T (t) ¡
+ V (x) Ã(x)
@t
2m @x 2
½
¾
1 @T (t)
1
~2 @2
i~
=
¡
+ V (x) Ã(x)
T (t) @t
Ã(x)
2m @x 2
½
¾
1
~2 @2
¡
+ V (x) Ã(x) = E
Ã(x)
2m @x 2
1 @T (t)
i~
= E
T (t) @t
1 @T (t)
i~
= E
T (t) @t
T(t) = e¡
E t =~
½
¾
1
~2 @2
¡
+ V (x) Ã(x) = E
Ã(x)
2m @x 2
½
¾
~2 @2
¡
+ V (x) Ã(x) = E Ã(x)
2m @x 2
Hb Ã(x) = E Ã(x)
½
¾
~2 @2
¡
+ V (x) Ã(x) = E Ã(x)
2m @x 2
Time independent
Schrödinger Equation