„There was a time when newspapers said that only
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Transcript of „There was a time when newspapers said that only
„There was a time when newspapers said that only
twelve men understood the theory of relativity.
I do not believe that there ever was such a time...
On the other hand,
A very elementary approach to Quantum mechanics
R.P. Feynman The Character of Physical Law (1967)
I think it is safe to say that
no one understands quantum mechanics“
let´s approach some aspects of qm anyway
Experimental facts:
Light has wave (interference) and particle properties
Plot from
maxkinE
Existence of photons
E
Energy of the quantumPlanck’s const.
Frequency
Radiation modes in ahot cavity providea test of quantum theory
Energy of a free particle
2E mc 22 2 40c p m c where
0
21 /
mm
v c
Consider photons
0 0m v cand E with
cp h h
pc c
c ck
or p k
; p mv
Dispersion relation for light
Electrons (particles) have wave properties
Today: LowEenergyElectronDiffraction standard method in surface science
Figures from
p k applicable for “particles” de Broglie
LEED
Fe0.5Zn0.5F2(110)
232 eV
top view
(110)-surface
Implications of the experimental facts
Electrons described by waves:( )( , ) i kx tx t A e
Wave function (complex for charged particles like electrons)
2*( , ) ( , ) ( , ) ( , )x t dx x t x t dx x t dx Probability to find electron at (x,t)
Which equation describes the temporal evolution of ( , )x t
Schroedinger equation
Erwin Schroedinger
Can’t be derived, but can be made plausible
Let’s start from the wave nature of, e.g., an electron:
( )( , ) i kx tx t A e and take advantage of p k ;E
( ) /( , ) i px Etx t A e x
( ) /( , ) i px Etipx t A e
x
( , )
ipx t
i
( , ) ( , )i x t p x tx
p i px
In complete analogy we find the representation of E
t
( ) /( , ) i px Etx t A e
( ) /( , ) i px EtiEx t A e
t
( , )
iEx t
i
( , ) ( , )i x t E x tt
E i H
t
Schroedinger equation for 1 free particle
Hamilton function of classical mechanics2
2
pH
m ; H=E total energy
of the particle
p i px
E i Ht
2 2
2( , ) ( , )
2x t i x t
m x t
1-dimensional
In 3 dimensions , ,p i i px y z
2
( , ) ( , )2
r t i r tm t
where 2
Schroedinger equation for a particle in a potential
Classical Hamilton function:2
( )2
pH V r
m
2
( ) ( , ) ( , )2
V r r t i r tm t
H
Hamilton operator
( , ) ( , )H r t i r tt
Time dependentSchroedinger equation
If H
independent of time like 2
( )2
H V rm
only stationary Schroedinger equation has to be solved Proof:
Ansatz: ( , ) ( )iE t
r t r e
( , ) ( , )H r t i r tt
(Trial function)
( ) ( )i iE t E t
H r e i r et
( ) ( )H r E r
Stationary Schroedinger equation
Solving the Schroedinger equation (Eigenvalue problem)
Solution requires: -Normalization of the wave function according2 3( ) 1r d r
Physical meaning: probability to find the particle somewhere in the universe is 1
-Boundary conditions of the solution:
and have to be continuous when merging piecewise solutions
Note: boundary conditions give rise to the quantization
Particle in a box:
x
2( ) sinn
nxx
L L
Eigenfunctions
2 2 2
21,2,3,...
2n
nE n
mL
Eigenenergies
Quantum numberDetails see homework
Heisenberg‘s uncertainty principle
It all comes down to the wave nature of particles
( ) /( , ) i px Etx t A e
Wave function given by a single wavelength h
p
Momentum p precisely known, but where is the particle position
-P precisely given
-x completely unknown
Wave package
Particle somewhere
in the region x
Fourier-analysis
11
hp
22
hp
Particle position known with uncertainty x
Particle momentum known with uncertainty p2
x p
Fourier-theorem
In analogy2
E t