**Molekuláris bionika és Infobionika Szakok tananyagának komplex fejlesztése konzorciumi keretben
***A projekt az Európai Unió támogatásával, az Európai Szociális Alap társfinanszírozásával valósul meg.
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 ***
PHYSICS FOR NANOBIO-TECHNOLOGY
Principles of Physics for Bionic Engineering
Chapter 6. Quantum Mechanics – I
(A nanobio-technológia fizikai alapjai )
(Kvantum mechanika)
Árpád I. CSURGAY,
Ádám FEKETE, Kristóf TAHY, Ildikó CSURGAYNÉ
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
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 Cathode Rays (J. J. Thomson) 1895 Röntgen‟s X-ray
1897 Electron (J. J. Thomson)
1900 Planck‟s Law – Black-body Radiation 1902 Photoelectric Effect
1905 Einstein‟s – Photon, Special Relativity, Brown Motion 1911 Rutherford: Nucleus of the Atoms
1915 Bohr‟s Model of the Hydrogen Atom
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.
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.
Incident photon
Scintillator
Light
Photocathode Electrons
Focusing
Dynode Photomultiplier Anode
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
Richard P. Feynman, QED – The Strange Theory of Light and Matter, Princeton University Press, 1985
Photocathode
one photon Amplifier
Speaker
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.
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 %. B
A
100 4
glass 96
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.
B A
100 0 to 16
100 to 84 Thickness of glass
Percentage of reflection 16%
8%
0%
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.
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).
Probability is predicted from the sum over paths.
The absolute square-value of the sum gives the probability.
A
stopwatch
front reflection arrow
0,2
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.
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.
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
f S
2π
S
A ~ ej
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 „wave- like”, however, as particles, they move in a probabilistic way
(only probabilities can be known).
A „wave-icle” state is represented by a probability complex amplitude
Probability that a wavicle at time t is at position x
A single electron moves in one direction:
1. Let us draw every path
2. Calculate the action for each path
constant of normalization
r,t .
, d 1.) ,
( t t 2
x P x x
).
, ( )
,
( x t )]
( [x t S
), (t x
Θ ; ) 1
, , , (
path
)]
( j [
)]
( j [
t S x t
x S
e t
x U
e A
Θ:
The propagator:
The complex amplitude of the wavicle at can be calculated from the complex amplitude at as
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).
( )
,Θ ) 1
, , , (
)]
( j [
t x D e
t x U
t x S
) , (x t )
, (
. d ) , ( )
, , , ( )
,
(
x t U x t
In the propagator every path, including the classical one, gets the same weight. Modulus of every path is 1.
If 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 for an electron Consider a free particle that leaves the origin at and arrives at
, 1 /
S
, 10
~
/ 25
class
S Sel / 1.
) 0 (t ).
s 1 ,
cm 1
(x t
Example 1: Free particles
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 Consider another path Phase change
For a macroscopic particle For an electron ,
t x
0.01 2 0.01
1
2 1
2 40 0
1 1
/ 0.02 d 0.01 d / 0.16 10 / .
2 2
x t x t
S S m t t m t m
26
1 g
1.6 10 rad m
9.1 10 31 kg 0.14 rad
m
2. t x
1,1
cmx
st
1
1
Mechanics
Quantum Mechanics
The “weight” of each path is equal.
The phase is determined by S/ . Classical Mechanics
The probability of the classical path is 1.
The probability of all other paths is 0.
0 1 /
S
2
34 Ws
10 05
.
1
1
~ / S
(t) 0S r
Prof. A. Zeilinger
http://www.quantum.univie.ac.at/zeilinger/
C60
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).
( , )
( , , , ) ( , ) d , x t
U x t
2 2
j 2 .
2 V
t m x
In three dimensions
Time-dependent Schrödinger equation for a single particle ),
, ) (
,
j ( x t
t t
x
H
.
2 2
2 2
V
x m
H
).
, ( )
, , , (
) , , , ( )
, , , ( )
, , , ( )
, (
2 2
2 2
2 2
2 2
t t
z y x
z
t z y x y
t z y x x
t z y x x
t x
r
), , ) (
,
j ( t
t
t H r
r
( ) .
2 pot
2
r
H V
m
r, t.
Time-independent Schrödinger equation for a single particle Let us look for the solution as ( , )x t Ψ( ) ( ).r t
) Ψ( )
) ( ) (
Ψ(
j r t H r
t
t
t E t
t
Ψ( ) constant
) Ψ(
1 )
( )
(
j 1 H r
r
) Ψ( )
Ψ(r r
H E ( ) j ( )
E t t
t
( ) j
j ( ) ( )
Et
t E
t t e
t
Eigenvalue problem
General solution:
Discrete (bounded electron )
Continuous (free electron)
Erwin Schrödinger, Quantisierung als Eigenwertproblem, Annalen der Physik, 361 – 376, 1926
Eigenvalues Eigenfunctions Stationary solutions
) Ψ( )
Ψ(r r
H E
i
E t j i
i i
i
e c
E , Ψ (r)
1 2
1 2
, , ... ,
Ψ ( ), Ψ ( ), ... Ψ ( )
i i
E E E
r r r
1 2
1 2
1 2
( , ) ( , ) ,..., ( , )
Ψ ( ) , Ψ ( ) ,..., Ψ ( ) ,
i
i
E
E E
j t j t j t
i
t t t
e e e
r r r
r r r
Example 2: Single electron in a one-dimensional configuration space
2 2
2
d ( )
( ) 0 ,
2 d
( ) 0 0, .
x E x if x a
m x
x if x a x
), d (
) (
d 2
2 2
x x k
x
k 2mE (x) Asinkx Bcoskx,
, 0
) 0
( B
(a) Asinka 0ka nπ.
8 , 2
2 2 2 2
2
ma n k h
En m n π
( ) sin ,
n
x A n x
a 2 2 π 2
sin n d 1 .
A x x A
a a
) (x Vpot
x 0 a
x
Inside the box:
Outside the box:
Ground-state energy
E t n
n
e a x
n t a
x π j
2 sin )
,
(
0 ) , (x t
n
2 2
1 8ma
E h En n2E1
6 6
1 9
1 10
1
15 6
1
10 m 1mikron, 0.376 10 eV,
10 m 1 nm, 0.376 eV,
10 m 0.1 nm, 37.6 eV,
10 m 10 nm, 376 GeV,
a E
a E
a E
a E
~ atom
~ nucleus E1
E2
Epot
x 0 a
x
5. Measurements and Operators
The expectation value of the measurement of the position of a wavicle is
because the probability that it is at position (x, x + dx) is
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
, d )
(x 2 x x
x
. d )
(x 2 x
. d )
( p 2 p p
p
Let us assume that we are measuring observable „a‟.
Let us also assume that we can get the complex probability amplitude for „a‟ from as
The expectation value of the measurement of „a‟
x a(a)
Ua(x)
x d x.
aU x U x a x x
xx
a x
x x
U x
x x
U a
a a
a a
a a
a a
a a
a a
a a
a
d ' d ) ' ( d
) ' ( )
( )
(
d d
) ( d
) (
d ) ( ) ( d
) ( 2
) ' , ˆ(x x A
' d ') ' , ˆ(
ˆ
A x x x x AIf 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 d x
a
A Aˆ
Aˆ(x, x')
x' d x'.Aˆ
x Xˆ x(x) Xˆ x
p x x x
j ) ˆ
j (
ˆ
P P
, d
d
2 x x
x x
x
. d d
j d
x x
p x
Observables and Operators
x Xˆ x (x) Xˆ x
px x x x
j ) ˆ
j (
ˆ
P P
)
pot(x
V ˆ ( ) ( ) ˆ ( )
pot pot
pot x x V x
V V
V
) 2 pot(
2
x m V
E p
) 2 (
ˆ
) ( ) ) (
( 2
2 pot 2 2
2 pot 2 2
x x V
m
x x
x V x m
H
H
6. Dirac Formalism
The state of a mechanical system is represented by the complex probability (wave) function
Dirac introduced a shorthand notation for the state, and he called it
“ket”
The complex conjugate of the state function is called “bra”.
The integral form of the expectation value is called “bra-ket”
The eigen-functions are frequently represented by their index )
, ,...,
(q1 qf t
x t
q
q ,..., f , ) ,
( 1
. ,
) , ,...,
(q1 qf t x
.
dx a
A An u
u ,
The total energy of a stationary state Frequency, period in time
Momentum
Wavenumber, wavelength, period in space The time-dependent
Schrödinger equation
Time-independent Schrödinger equation
H
t
j H E
h E
Et
t e
ej j
,
p k
k h
p
k 2π
The expectation value change in time as
The operators can change in time Commutator of operators H and L
L L
VL
V
d
t t
t L
L d L
d
H L L
H
H
j j
d j d
t L t
HL LH HL LH
j j
j d
d t L
H,L
L
j d
d
t
H,L
HL LHExample 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;
b) the expectation values of the squares of position and momentum;
c) check the validity of the Heisenberg‟s uncertainty principle.
x t x x t x x
, , d
x t xt x x
p
, j , d
x t x
x t x x2
, 2 , d
x t xt x x
p , 2 , d
2 2
2
2 2 2 2
p p
x x
p
x
2
0 0
1 cos
2 π 2
sin d d
2 2
a a
x a a
x x x x x
a a a
3 3 3
2 2 2 2 2
2
0 0
π 2 1 2π 2
sin d cos d 0, 2833
6 2 6 4π
a a
x a x a a
x x x x x a
a a a a
0
2 π π π
sin j cos d 0
a x x
p x
a a a a
p2
x,t 2 x22
x,t d x
a a
x x
x
2 2 0,2833 0,25 0,18 2 2 π
p p p
a
57 2
,
2 0
2 2
2
x p x x p p
