Download Brief introduction to quantum mechanics

A very elementary approach to Quantum mechanics
„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, I think it is safe to say that
no one understands quantum mechanics“
R.P. Feynman
The Character of
Physical Law (1967)
 let´s approach some aspects of qm anyway
Experimental facts:
Light has wave (interference) and particle properties
Ekin
max
  
Plot from
Radiation modes in a
hot cavity provide
a test of quantum theory
Existence of photons
E 
Frequency
Planck’s const.
Energy of the quantum
Energy of a free particle
E  mc  c p  m0 c
2
2
2 4 where m 
2
m0
1 v / c
2
; p  mv
Consider photons
m0  0
  cp
and
vc
with
E 

h h
p k
or
p


c
c 
  c    ck
Dispersion relation for light
Electrons (particles) have wave properties
Figures from
p k
applicable for “particles”
de Broglie
Today: LowEenergyElectronDiffraction standard method in surface science
LEED
Fe0.5Zn0.5F2(110)
232 eV
top view
(110)-surface
Implications of the experimental facts
Electrons described by waves:
 ( x, t )  A ei ( kxt )
 ( x, t )dx   ( x, t ) ( x, t )dx   ( x, t ) dx
2
*
Wave function
(complex for charged particles like electrons)
Probability to find electron at (x,t)
Which equation describes the temporal evolution of
 ( x, t )
Schroedinger equation
Can’t be derived, but can be made plausible
Let’s start from the wave nature of, e.g., an electron:
Erwin Schroedinger
 ( x, t )  A ei ( kxt )
and take advantage of
p  k ;E  
 ( x, t )  A e

x
i ( px  Et ) /

ip
ip
 ( x, t )  A ei ( px  Et ) /   ( x, t )
x

i
 ( x, t )  p  ( x, t )
x
p  i
i

p
x
In complete analogy we find the representation of E
 ( x, t )  A ei ( px  Et ) /

t

iE
iE
i ( px  Et ) /
 ( x, t )   A e
   ( x, t ) i
t


i
 ( x, t )  E  ( x, t )
E i
H
t
t
Schroedinger equation for 1 free particle
p2
Hamilton function of classical mechanics H 
2m

p
x

E i
H
t
p  i
In 3 dimensions

2
2m
; H=E total energy
of the particle
2



(
x
,
t
)

i
 ( x, t )
2
2m x
t
2
1-dimensional
    
p   i  , ,   i   p
 x y z 
 ( r , t )  i

 (r , t)
t
2
where   
Schroedinger equation for a particle in a potential
Classical Hamilton function: H 
p2
2m
V (r )

H ( r , t )  i
 (r , t)
t
2






V
(
r
)

(
r
,
t
)

i
 (r , t)


2
m

t


H
Time dependent
Schroedinger equation
Hamilton operator
If H independent of time like H  
2
2m
 V (r )
only stationary Schroedinger equation has to be solved
Proof:
i
 Et


(
r
,
t
)


(
r
)
e
H ( r , t )  i
 (r , t)
Ansatz:
(Trial function)
H ( r ) e
i
 Et
t
 i Et
 i (r ) e
t
H ( r )  E ( r ) Stationary Schroedinger equation
Solving the Schroedinger equation (Eigenvalue problem)

Solution requires: -Normalization of the wave function according

 (r ) d 3r  1
2

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:
2
 nx
sin
L
L
 n ( x) 
En 
 2 n2
2
2
2mL
Eigenfunctions
n  1, 2,3,... Eigenenergies
Quantum number
x
Details see homework
Heisenberg‘s uncertainty principle
It all comes down to the wave nature of particles
 ( x, t )  A ei ( px  Et ) /
Wave function given by a single wavelength p 
h

Momentum p precisely known, but where is the particle position
-P precisely given
-x completely unknown
Wave package
p1 
h
1
p2 
h
2
Fourier-analysis
Particle somewhere
in the region
x
Particle position known with uncertainty
x
x p 
Particle momentum known with uncertainty  p
In analogy
2
Fourier-theorem
E t 
2