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PETER PAZMANY
CATHOLIC UNIVERSITY
SEMMELWEIS
UNIVERSITY
Development of Complex Curricula for Molecular Bionics and Infobionics Programs within a consortial* framework**
Consortium leader
PETER PAZMANY CATHOLIC UNIVERSITY
Consortium members
SEMMELWEIS UNIVERSITY, DIALOG CAMPUS PUBLISHER
The Project has been realised with the support of the European Union and has been co-financed by the European Social Fund ***
**Molekuláris bionika és Infobionika Szakok tananyagának komplex fejlesztése konzorciumi keretben
***A projekt az Európai Unió támogatásával, az Európai Szociális Alap társfinanszírozásával valósul meg.
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Peter Pazmany Catholic University
Faculty of Information Technology
www.itk.ppke.hu
PHYSICS FOR NANOBIO-TECHNOLOGY
Principles of Physics for Bionic Engineering
(A nanobio-technológia fizikai alapjai )
Chapter 6. Quantum Mechanics – I
(Kvantum mechanika)
Árpád I. CSURGAY,
Ádám FEKETE, Kristóf TAHY, Ildikó CSURGAYNÉ
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Chapter 6. Quantum Mechanics – I
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Table of Contents
6. Quantum Mechanics I
1. A Glimpse of the Quantum Story
2. Experimental foundation
3. Feynman‟s Path Integral
4. Schrödinger Equation
5. Measurements and Operators
6. Dirac Formalism
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1.A Glimpse of the Quantum Story
In the late 1890‟s and early 1900‟s new technologies (vacuum
technology, optical spectroscopy) and new “layers” of matter were
discovered:
1890
1895
1897
1900
1902
1905
1911
1915
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Cathode Rays (J. J. Thomson)
Röntgen‟s X-ray
Electron (J. J. Thomson)
Planck‟s Law – Black-body Radiation
Photoelectric Effect
Einstein‟s – Photon, Special Relativity, Brown Motion
Rutherford: Nucleus of the Atoms
Bohr‟s Model of the Hydrogen Atom
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Attempts to explain the new experiments with classical laws
failed, e.g.
“Ultraviolet Catastrophe” ( Lord J. W. S. Rayleigh, James Jeans )
“Thermodynamic Paradox“ ( Ludwig Boltzmann )
The laws of classical mechanics and classical electrodynamics
could not explain the outcome at the new experimental frames
(deeper layers)
Why does the electromagnetic energy not change
in a cavity continuously?
Why does the electron not fall into the nucleus?
Why the atomic spectrum is discrete?
Why the chemical “forces” are so mysterious?, etc.
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2. Experimental foundation
For a human eye it takes only ~(5 – 6) photons to activate a nerve
cell and to send a signal to the brain.
Single photons can be detected by photomultipliers.
Photocathode
Incident
photon
Electrons
Anode
Scintillator
Light
photon
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Focusing
electrode
Dynode
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Photomultiplier
tube (PMT)
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When a photon strikes the photocathode of a
photo-multiplier, an electron is knocked loose
and attracted to positively a charged plate,
knocking more electrons loose. This is
repeated many times, until significant number
of electrons is loosed, the anode is reached,
and through an amplifier a click can be heard.
Clicks of uniform loudness are heard each
time a photon hits the photocathode
Amplifier
Speaker
one photon
Photocathode
Richard P. Feynman, QED – The Strange Theory of Light and Matter,
Princeton University Press, 1985
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If we put a lot of photomultipliers around and let the intensity of
light to be very dim, then the light goes into one multiplier or
another and makes a click of full intensity.
There is no splitting of light into “half particles” that go to
different sensors: we experience particles of light.
If we do reflection and transmission experiments with very dim
light, we learn that only probabilities can be predicted.
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Partial reflection of light by a single glass surface
 We send 100 photons from a
light source toward a glass surface.
 Two detectors, A and B,
are set to count photons.
 We hear a click either
in A or in B.
 Only probabilities can be predicted,
in this example we find 4 %.
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A
100
4
glass
96
B
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Partial reflection of light by two surfaces of glass
 The probability of reflection depends
on the thickness of glass!
 Instead of 4 %, the experiment
gives 0 to 16% probs.
A
100
0 to 16
Percentage of
reflection
16%
8%
100 to 84
0%
B
Thickness of glass
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Chapter 6. Quantum Mechanics – I
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Fundamental question of quantum mechanics:
 Given that a particle is located at x1 at a time t1, what is the
probability that it will be at x2 at time t2?
 The experimental results of partial reflection can be predicted by
assuming that the photon explores all paths between emitter and
detector, paths that include single and multiple reflections from
each glass surface.
 The hand of an imaginary „quantum stopwatch‟ rotates (phase)
as the photon explores each path.
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Rotation rate for the hand of the photon quantum stopwatch:
 the frequency of the
corresponding classical wave.
 (Classical wave optics
can predict the result)
(Interference).
A
stopwatch
0,2
front reflection
arrow
 Probability is predicted from the sum over paths.
 The absolute square-value of the sum gives the probability.
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Calculation of the probabilities
 The probability of an event is always the absolute square of the
complex probability amplitude.
 Each path is equally possible, nature explores all possibilities,
thus we have to calculate the complex probability amplitude for
each path.
 If an event could happen in two alternative ways, we add the
complex amplitudes.
 If an event could happen in two consecutive ways, we multiply
the complex amplitudes.
 The virtual stopwatch determine the phase changes of the
individual complex amplitudes.
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Single electrons can be detected by electron-multipliers.
Slit experiments show that also in case of electrons only
probabilities can be predicted.
Given that an electron is located at x1 at a time t1, what is the
probability that it will be at x2 at time t2?
The similarity between electron interference and photon
interference suggests that the behavior of the electron may also be
correctly predicted by assuming that it explores all paths between
emission and detection.
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Exploration along each path is accompanied by the rotating hand
of an imaginary stopwatch.
From the interference experiments we can learn that the number of
rotations that the quantum stopwatch makes as the particle
explores a given path is equal to the action S along that path
divided by Planck‟s constant h (quantum of the action).
Complex probability amplitude
A~e
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j
S

S
  2π  f 

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3. Feynman’s Path Integral
Totality of experiments suggests that small particles (e. g.
electrons, protons, atoms) behave as particle-like and as wave-like
objects. R. P. Feynman calls them
wave + particle = “wavicle”.
Interactions with themselves and with their environment is „wavelike”, however, as particles, they move in a probabilistic way
(only probabilities can be known).
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A „wave-icle” state is represented by a probability complex
amplitude  r,t .
Probability that a wavicle at time t is at position x
P(x, t )   x, t 
2



  dx  1.
A single electron moves in one direction: ( , )  ( x, t ).
1. Let us draw every path x(t ),
2. Calculate the action for each path S[ x(t )]
A  e
j
S [ x ( t )]

j
1
 U ( , , x, t )   e
Θ path
S [ x ( t )]

;
Θ : constant of normalization
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The propagator:
S [ x ( t )]
1 j 
U ( , , x, t )   e
Dx (t ),
Θ
The complex amplitude of the wavicle at ( x, t ) can be calculated
from the complex amplitude at ( , ) as
 ( x, t )   U ( x, t ,  , )  ( , ) d  .
If we increase the size of a small particle
(nano → micro → macro),
then the movement of a wavicle gradually approaches the
trajectory of a classical particle (correspondence principle).
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In the propagator every path, including the classical one, gets the
same weight. Modulus of every path is 1.
If S /   1, the classical path prevails.
In the neighborhood of the classical path the action is close to
stationary. In the vicinity of the classical path the paths contribute
„coherently‟.
In macro case Sclass /  ~ 1025 , for an electron Sel /   1.
Consider a free particle that leaves the origin at (t  0)
and arrives at ( x  1 cm, t  1 s).
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x  cm
Example 1: Free particles
1
Consider a free particle that leaves the origin
at t = 0 and arrives to x = 1 cm, at t = 1 s.
The classical path is x  t ,
Consider another path x  t 2 .
Phase change
S
x  0.01t 2
 S x 0.01t

1
t s 
1
1 1

2
2 1
/    m  0.02t  d t    0.01 m d t  /  0.16 104 m / .
2
0
0 2

For a macroscopic particle
For an electron
m 1g
m  9.11031 kg
 0.14 rad
 1.6 1026 rad
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1,1
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Mechanics
S /   1
0
Classical Mechanics
 S r(t )  0
The probability of the
classical path is 1.
The probability of all
other paths is 0.
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S / ~1
  1.05 1034 Ws 2
Quantum Mechanics
The “weight” of each
path is equal.
The phase is
determined by S/ .
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C60
Prof. A. Zeilinger
http://www.quantum.univie.ac.at/zeilinger/
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4. Schrödinger Equation
In classical mechanics, from the principle of least action we get
the local differential equations of motion (Newton).
In quantum mechanics, if a single particle of mass m moves in a
potential field V(x), then  it can be shown  from the Feynman
path-integral we can get a local partial differential equation of the
probability complex amplitude (Schrödinger equation).
 ( x, t ) 
  U ( x, t ,  , )  ( , ) d  ,
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
  2  2
j
 
 V  .
2
 t  2m  x

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 ( x, t )
j
 H ( x, t ),
t
 2 2

H  
 V .
2
 2m  x

In three dimensions
 2 ( x, t )
 2 ( x, y, z, t )  2 ( x, y, z, t )  2 ( x, y, z, t )



2
2
2
x
x
y
z 2
  ( x, y, z, t )   (r, t ).
Time-dependent Schrödinger equation for a single particle
 (r, t )
j
 H (r, t ),
t
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 2

H  
  Vpot (r ).
 2m

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r, t.
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Time-independent Schrödinger equation for a single particle
Let us look for the solution as  ( x, t )  Ψ(r)   (t ).
 (t )
jΨ(r )
  (t )HΨ(r )
t
1  (t )
1
j

HΨ(r )  constant  E
 (t )  t
Ψ(r)
HΨ(r)  EΨ(r)
 (t )
E
  j  (t )
t

E
j t
 (t )
E
  j  (t )   (t )  e
t
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Eigenvalue problem
Eigenvalues
E1 ,
Eigenfunctions
Stationary
solutions
HΨ(r)  E  Ψ(r)
E2 ,
...
Ei ,
Ψ 2 (r),
...
Ψi (r)
 2 (r, t ) 
,...,
Ψ1 (r),
 1 (r, t ) 
 Ψ1 (r )  e
j
E1
General solution:
 Discrete (bounded electron )
 Continuous (free electron)
t
,  Ψ 2 (r)  e
Ei ,
j
E2
t
,...,
 i (r, t ) 
 Ψ i (r)  e
 c Ψ (r)  e
i
j
j
Ei
t
,
Ei
t

i
i
Erwin Schrödinger, Quantisierung als Eigenwertproblem,
Annalen der Physik, 361 – 376, 1926
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Example 2: Single electron in a one-dimensional configuration space
V pot (x)
d 2 ( x)

 E ( x) if
2
2m dx
 ( x)  0 if
2
0  x  a,
   x  0, a  x  .
x0
d 2 ( x)
 k 2 ( x),
2
dx
 (0)  0  B,
k 
2mE

x
 ( x)  A sin kx  B cos kx,
 (a)  A sin ka  0  ka  nπ.
nπ
2 2
h2 2

(
x
)

A
sin
x,
En 
kn 
n
,
n
2
a
2m
8ma
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a
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


A2 sin 2
nπ
2
xd x 1 A 
.
a
a
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Epot
E
Inside the box:
j n t
2
nπ
 n ( x, t ) 
sin
xe 
a
a
E2
Outside the box:  n ( x, t )  0
h2
E1 
8ma 2
Ground-state energy
E1
En  n E1
2
a  106 m  1mikron,
E1  0.376 106 eV,
a  109 m  1 nm,
E1  0.376 eV,
a  1010 m  0.1 nm,
E1  37.6 eV,
~ atom
a  1015 m  106 nm,
E1  376 GeV,
~ nucleus
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x0
a
x
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5. Measurements and Operators
The expectation value of the measurement of the position of a
wavicle is
2
x   x ( x) d x ,
because the probability that it is at position (x, x + dx) is
 ( x) d x.
2
We would like to know the expectation values of the
measurements of the momentum, angular momentum, energy, etc.
of the wavicle. If we knew the complex probability amplitude for
the momentum p, we could calculate it as
p   p  ( p) d p.
2
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Let us assume that we are measuring observable „a‟.
Let us also assume that we can get the complex probability
amplitude for „a‟ from  x  as a (a)   U a ( x) x d x.
The expectation value of the measurement of „a‟
a   a  a (a) da   a a (a) a (a) d a
2
  U ( x) xd x  U ( x) xd xd a
  ( x)   aU ( x)U ( x' ) d a  ( x' ) d x' d x

 a
a

a

a
a
Aˆ ( x, x' )
ˆ   Aˆ ( x, x' ) x' d x'
A

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ˆ  d x,
a    A
ˆ   Aˆ ( x, x' ) x' d x'.
A

If we know the operators  we can calculate the expectation
value of „a‟ from the prob. amplitude of the position.
Without proofs we present the results:
Observable Operator
ˆ   x  ( x)  X
ˆ  x
x    x  dx 
X
x
  x dx,
2
p
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

Pˆ     x ( x)  Pˆ   x
j
j
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


 x dx 

j

d
  
dx.
j
dx
p 
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Observables and Operators
x
ˆ   x  ( x)  X
ˆ  x
X
px
ˆP      ( x)  Pˆ   
x
x
j
j
Vpot ( x)
ˆ   V ( x) ( x)  V
ˆ  V ( x) 
V
pot
pot
pot
2
p
E
 Vpot ( x)
2m
2

 ( x)

H  
 Vpot ( x) ( x) 
2
2m  x
2
2
ˆ  
H
 Vpot ( x)
2
2m  x
2
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Physics for Nanobio-technology:
Chapter 6. Quantum Mechanics – I
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6. Dirac Formalism
The state of a mechanical system is represented by the complex
probability (wave) function  (q1 ,..., q f , t )
Dirac introduced a shorthand notation for the state, and he called it
“ket”  (q1 ,..., q f , t )   , x
The complex conjugate of the state function is called “bra”.
  (q1 ,..., q f , t )   , x .
The integral form of the expectation value is called “bra-ket”


 A dx 
 A  a .
The eigen-functions are frequently represented by their index
un
un , n
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Physics for Nanobio-technology:
Chapter 6. Quantum Mechanics – I
www.itk.ppke.hu
The time-dependent
Schrödinger equation
j

t
Time-independent
Schrödinger equation
H  E
 H
E  h     
The total energy of a stationary state
Frequency, period in time
e  j t  e
Momentum p   k , p    k 
E
j t

h

Wavenumber, wavelength, period in space
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k
2π

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Physics for Nanobio-technology:
Chapter 6. Quantum Mechanics – I
www.itk.ppke.hu
The expectation value change in time as
d L
L   L d V   L 



L   L
dt
t
t

V
d L

j
j
j
  H 
 H L    L H
t

dt


d L
j
j
j
  HL    LH    HL  LH 
dt



The operators can change in time
Commutator of operators H and L
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dL j
 H, L
dt 
H, L  HL  LH
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Physics for Nanobio-technology:
Chapter 6. Quantum Mechanics – I
www.itk.ppke.hu
Example 2: Electron in a one-dimensional box
For an electron in a one-dimensional ideal “box” of size a calculate
a) the expectation values of the electron‟s position and momentum;




p    x,t    j   x,t dx
x 


x   x,t  x  x,t  d x


b) the expectation values of the squares of position and momentum;


x 2    x,t  x 2  x,t  d x
p

2
 2 2 
 x,t  d x
  x,t    
2 
x 



c) check the validity of the Heisenberg‟s uncertainty principle.
x p 
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x2  x
2
p2  p
2
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Physics for Nanobio-technology:
Chapter 6. Quantum Mechanics – I
www.itk.ppke.hu
a
x 
a
2
2
2 πx
x
sin
d
x

x


a0
a
a0
a
x 2  
0
a
3
3
3




π
x
2
a
1
2π
x
2
a
a
2
2
2
x sin
d x     x cos
dx     2   0, 2833  a 2
a
a 6 20
a
 a  6 4π 
a
2
πx
p   sin   j
a0
a
x
x2  x
2
π
πx
 cos dx  0
a
a

p2
2




2
 x,t  d x
  x,t    
2 
x 


 0,2833  0,25  a  0,18  a
x p 
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2π x
a dx  a
2
2
1  cos
x2  x
2
p2  p
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p 
2
p2  p
 0,57   
2
 π
 
 a 

2
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