**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 7. Quantum Mechanics – II
(A nanobiotechnológia fizikai alapjai )
(Kvantum mechanika)
Árpád I. CSURGAY,
Ádám FEKETE, Kristóf TAHY, Ildikó CSURGAYNÉ
TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 2
Table of Contents
6. Quantum Mechanics II
1. Wave Pocket Propagation
2. Electron Reflection, Transmission and Tunneling
3. Single Electron in a One-dimensional Periodic Potential 4. Quantum Well, Quantum Wire, Quantum Dot
5. Hydrogen and Hydrogen Like Atoms
1. Wave pocket propagation in one-dimension
A free particle (electron in a force-free space) moves in a constant (can be zero) potential energy space, thus the Schrödinger
equation reads as
This differential equation has a k -dependent solution for any E > 0
2011.05.31. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 4
,
d d
2
22 2
x E
x x
m
. 1 2
where 2 ,
d ψ
d
22 2
mE ψ k
ψ k mE
x
e ,
jkxk k
ψ x A .
2
2 2
m k E
k
There is a quadratic relation between the frequency (energy ) and the wave-number k (wavelength ).
The general solution is a superposition of the solutions belonging to different k -s.
Note that for fixed t, is the Fourier transform of , and
h
E 2 π / k
e
j kx-Etd .
ψ x,t A k k
x,t
ψ A k , t
1 e
jd .
2π
kx Et
A k,t ψ x, t x
At fixed t the spatial dependence of a wave function, and the k (or p) dependence of its „amplitude‟ are related by Fourier transformation.
At t = 0 we can localize the electron: let be the probability
amplitude in an interval (a/2 < x < a/2) constant, and zero outside the interval:
2011.05.31. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 6
), ( x
j 0
1 e , if
( , 0) 2 2 .
0 otherwise
p x
a a
x a x
The Schrödinger equation is satisfied, and
thus the electron can be found in the interval.
The Fourier transform of this function, i.e. A(k), using and as a function of p
, 1 d
2 /
2 /
a
a
x
k p k
p d
d
. ) 2 (
) 2 (
sin )
(
0 0
p a p
p a p
h p a
A
The wave pocket is put together from components with p mainly in the interval
(This is an illustration of Heisenberg’s relation)
Note (and check) that for wave-pockets, a Gauss-function wave- form can be put together from components forming a Gauss-
function like spectrum.
2011.05.31. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 8
, /
/
00
a p p a
p
/ 2, / / 2.
x a p a x p
2 2
0 0
( ) ( )
( ) x A e
x xA k ( ) B e
k k.
2. Reflection, transmission and tunneling (in one-dimensional configuration space)
If a wave pocket is approaching a potential barrier, we first decompose the wave pocket into components of specific wave-numbers (i.e. energy). We calculate the reflection and transmission of a given fixed energy wave, then we construct the response.
We start with an electron of energy E flying toward a potential barrier of potential E
0.
There are two cases E E
0, and E > E
0.
Example 1: Electron of energy E flying toward a potential barrier of potential E0.
2011.05.31. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 10
2 2
2
pot pot
2 2 2
d d 2
2 d d
x x m
E x x E x E E x x
m x x
2 2
2
2 0
2 , if <0,
d d 2
, if 0.
m E x x
x x m
E E x x
2 2
2
2 0
2 , if <0,
d d 2
, if 0.
m E x x
x x m
E E x x
0 x
Epot
E0
E
x
Epot
E0
E
0 x
x
,1 2 ,
1 2
0
1 mE m E E
k
k1 1 2mE, k2 1 2m
E E0
,
,e e
e j
ej 1 1
x
x D C
x B k
x A k
x
1 1
2 2
j j
e e
( ) ,
j j
e e
k x k x
A B
x k x k x
C D
0.
x C D 0. No electron from right.
1 10 ,
' 0 j j ,
A B D
k A k B D
j j j ,
,
2 1
1A k B k C
k
C B A
j . 2 1
j ,
2 1 1 1
k
B D k
A D
2 . ,
2 1
1 2
1
2 1
k k A k k C
k k Ak
B
2011.05.31. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 12
At the x > 0 region, which region was prohibited classically
probability is not zero !
All components of energy smaller than the barrier will be reflected.
At x > 0 we get a wave representing the electron flying into the +x
direction.
The particle either continues its flight (as classically) or it is reflected by the barrier.
(Classically the particle is never reflected).
1 1
1 1
j j
1 e
2 if 0
j j
1 e
2
e x if 0
D k x
k x
x D k x
k
D x
0 j if
2 e
, 0 j if
e ej
2 2
1 1
1 2
1
2 1
1
x x k k
k A k
x x k k
k k A k
x A k
x
0 ,
e 2
D D x x
.
1
A A
B R B
Example 2: Electron’s tunneling through a potential barrier
For the particles move into the +x direction, thus F = 0.
At and are continuous.
pot 0
0, if 0,
, if 0< < ,
0, if > ,
x
E E x
x
2 2 1
1
2 2
2
0 1
1 2 , if 0,
d , if 0<x< ,
1 d
2 , if ,
k x x
k mE
x x
m E E x k x x
x
,
x (x) and '(x)
E0
0 x
x E
x
0 x
2011.05.31. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 14
1
1
1
j
j , 1
, 0 continuous,
j , 0 continuous,
e e e , continuous,
e e j e continuous.
k
k
A B C D
k A B C D
C D E
C D k E
1
1
1 1
j j 1
1 1 1 0 1
j 0 j
0 e e e 0 .
0 e e j e 0
k k
B
k C k
D A
k E
.
>
j if j e
e
, 0
if e
e
0,
<
j if j e
e
1 1
1 1
x x
F k x E k
x x x D
C
x x B k
x A k
x
E0
0 x
x E
x
The T transfer function describes a typical quantum mechanical phenomenon:
tunneling classically would be impossible !
The transfer-coefficient of tunneling is approximately:
This formula approximates T accurately if , 4 sinh
1 1
1
2 2 2
2 1 1
2 k a
k k k
A k A
E T E
,
4 sinh 1 1
4 sinh 1
2 2 2
2 1 1
2
2 2 2
1 1
2
a k k
k k
k
a k k
k k
k R
.
1
T R
. e
1
16 2 0
2
0 0
m E E
E E E
T E
1.1 2
m E E
k
3. Single electron in a one-dimensional periodic potential
Example 3: Single electron in a one-dimensional “crystal”
Period of a periodic potential:
2011.05.31. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 16
0,2 d
d
2 p 2
2
x E m E
x
. 0
if ,
0
, 0 if
) ,
( p0
p x a
x b x E
E
x
0
Ep
Epot
E
a 0
b
j j
e e , if 0,
( )
e e , if 0 ,
x x
x x
A B b x
x
C D x a
2 .
0
p E
m E
2 , m E
Bloch function
Constants A, B, C and D can be determined from the following four conditions:
x uk
x e ,
j j
j j
e e , if 0,
( )
e e , if 0 .
k x k x
k k x k x
A B b x
u x
C D x a
1 0 2 0, i.e. + = + ,
k x k x
u u A B C D
1 2
0 0
d ( ) d ( )
j j j j ,
d d
k k
x x
u x u x
k A k B k C k D
x x
j j j j ,
1 2 e k b e k b e k a e k a ,
k x a k x b
u u A B C D
j j
1 2
d ( ) d ( )
j e j e
d d .
k b k b
k k
x a x b
u x u x
k A k B
x x
Trivial solution: No electron in the crystal.
We can find an electron in the crystal if and only if the determinant of this equation is zero!
2011.05.31. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 18
j j j j
j j j j
1 1 1 1
j j j j
e e e e 0.
j e j e j e j e
k a k a k b k b
k b k b k a k a
A
k k k k B
C
k k k k D
sh j sin ch j cos cos
.j 2
j 2 2
b a k a
b a
b
.
0
B C D A
Let Ep0 ; b 0, but in such a way that
a is the periodicity in „lattice-space”
If the dispersion relation does not hold, then there is no electron in the crystal !
, 2 0
2 2
2
b
b a ab
b ab
, cos cos
ch b a a , sin cos
cos a
a a a
k
p 0
2
0
lim ,
2
E b
ab
2m ,
E
. 0 ,
e π ,
2 j
kx k
a a a a
x
a a a a
k
sin cos
cos
In periodic potential the solution of the „single-electron” problem leads to admitted energy bands separated by forbidden bands. At energies in the forbidden band no electron can exist in the periodic potential field.
2011.05.31. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 20
2 4 6 8 10 12
-1 1 2
a a a
a
sin cos
5 10 15 20 25
-0.75 -0.5 -0.25 0.25 0.5 0.75
a k a
k cos
a a a a
k
sin cos
cos
2 4 6 8 10 12
-1 1 2 3
5 10 15 20 25
-0.75 -0.5 -0.25 0.25 0.5 0.75
„Forbidden bands”
„Allowed bands”
4. Quantum well, quantum wire, quantum dot
Example 4: Three-dimensional ideal potential box
The single electron in a three-dimensional ideal potential box is a structure a lot to be learned from
Let us look for the solution as
otherwise.
,
<
0 ,
<
0 ,
<
0 ha ,
, 0
pot ,
c z b
y a
z x y x E
r E
r ,
r t,
r e jEt,
H
.2
2
r E m r
r
H
x y z, ,
x y z .
z
a y c
b
This relation holds if and only if the three functions on the left side of the equation are all constants.
Let us introduce three new constants as :
2011.05.31. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 22
2 2 2
1 2 3 1
2 3
2 2 2 2 2
d ,
d
x y z
y z
x y z x x
1 2 3
3
1 2
2 3 2 1 3 2 1 2 2
d
d d
d d d ,
x y z
y z x z x y
x y z
2 , d
d 1 d
d 1 d
d 1
2 2
3 2
3 2
2 2
2 2
1 2
1
m E z
z z
y y y
x x
x
3.
2
1 E E
E
E
The stationary states of the electron inside the box:
1
1 1 1 1 1
2 2
2 2
2 2 2 2 2
2 2
2 3
3 3 3 3 3
2 2
, sin π , 1, 2, ,
d
d 2
, sin π , 1, 2, ,
d
d 2
, sin π , 1, 2, ,
d
E x x A n n
x a
y m y
E y y B n n
y b
z m z
E z z C n n
z a
, ,
1
2 3 sin 1π sin 2π sin 3π , c n z bn y a
n x A
z y
x z
y
x
2 2 2 2
1 2 3
0 0 0
sin π sin π sin π d d d =1 8 .
a b c
x y z
A n n n x y z A
a b c abc
x,y,z;t
8 sinn π x sinn π y sinn π z ejEn1n2n3t .
The admissible energy-levels of the electron inside the box :
Degenerate stationary state:
More than one eigenfunction belong to an eigenvalue. If c = b = a , then there are degenerate states
2011.05.31. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 24
8 . h
2 2 3 2
2 2 2
2 1 2
3 2
1 3
2
1
c
n b
n a
n E m
E n E
n En
2
2 2 2
1 2 3 1 2 3
1 2 3 2
h , 1, 2,...; 1, 2,...; 1, 2...
E 8 n n n n n n
n n n ma
π . π sin
π sin 8 sin
3 2
3 1 3
2
1 a
n z a
n y a
n x n a
n
n
States having the same eigenvalue but different quantum numbers, i.e. different state-functions are degenerate states .
If c = b = a , then the number of degenerate states increases because in the expression of commuting n1 and n2, n1 and n3, and n2 and n3 results in the same energy eigenvalue.
An eigenfunction defines the state, thus at an energy (eigenvalue) more than one stationary states can be present. At an energy level the electron can be in
2
2 2 2
1 2 3 1 2 3
1 2 3 2
h , 1, 2,...; 1, 2,...; 1, 2...
E 8 n n n n n n
n n n ma
3 2 1n n
En
,8
h 2
3 2
2 2
2 1 2 3
2 1
n n
ma n n
n
En
π . π sin
π sin 8 sin
3 2
3 1 3
2
1 a
n z a
n y a
n x n a
n
n
An eigenfunction defines the state, thus at an energy more than one stationary states can be present. At an energy level the electron can be in different
stationary states. In a cubic box
the ground-state (null-point) energy, E1, belongs to the quantum numbers (1, 1, 1);
the next energy level is E2 = 2E1 . There are three eigenfunctions belonging to E2: (2, 1, 1), (1, 2, 1) and (1, 1, 2);
at energy level E6=(14/3)E1 there are six different stationary states:
(1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1),(3, 1, 2), (3, 2, 1).
2011.05.31. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 26
To every (n1, n2, n3) integer triple belong two different functions (spin quantum numbers +1/2 and 1/2)
The state function depends not only on the spatial coordinates but on the spin coordinates as well.
The to the ground state (nullpoint) energy E1 , with quantum numbers (1, 1, 1), two eigensates (+1/2,1/2) belong.
To the energy level E6= (14/3)E1 with six different quantum number trios, there are 6 · 2 = 12 states.
1 2 3 1 2 3
1 1
2 2
1 1
, , , , , .
2 2
n n n x y z s n n n x y z s
2011.05.31. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 28
E
E1
1
2 2E
E
1
3 3E
E
1
4 3
11E E
1 5
4E E
1
6 3
14 E E
2 / 1 111 2
/ 1
111
2 / 1 211 2
/ 1 211 2
/ 1 121 2
/ 1 121 2
/ 1 112 2
/ 1
112
2 / 1 221 2
/ 1 221 2
/ 1 212 2
/ 1 212 2
/ 1 122 2
/ 1
122
2 / 1 331 2
/ 1 331 2
/ 1 313 2
/ 1 313 2
/ 1 133 2
/ 1
133
2 / 1 222 2
/ 1
222
2 / 1 321 2
/ 1 321 2
/ 1 3121 2
/ 1 312 2
/ 1 231 2
/ 1 231
2 / 1 213 2
/ 1 213 2
/ 1 132 2
/ 1 132 2
/ 1 123 2
/ 1 123
A potential box confines the electron in space. Confinement is characterized by the size of the box compared to the electron‟s de Broglie wavelength.
If the confinement is one-dimensional, i.e. in a very „flat‟,
otherwise very „big‟ box, the structure is called „quantum well‟.
In this case, the electron is „free „ in two dimensions, and bounded in one.
If we bound the electron in two dimensions in a „long‟ box with
„small‟ cross section, the structure is called „quantum wire‟.
If the electron is confined in three dimensions, the spectrum will
be discrete, and the structure is called quantum dot.
5. Hydrogen and hydrogen like atoms
The single electron problem can be exactly solved for the potential field of a positive nucleus. The nucleus is in the center of a
sphere, and the electron moves in the Coulomb field of the nucleus.
The potential energy of the electron is spherically symmetric:
The Schrödinger-equation:
2011.05.31. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 30
2
pot e
0
e 1
4π . E q U Z
r
2 e
2
0
2 e
4π 0.
m Z
E r
This eigenvalue problem for E and , in spherical coordinates.
The partial differential equation is reduced to three ordinary differential equations for R, and functions.
1 ,
2 , ,
,
r rr
2 2
2 , 2 2
1 1 1
, sin
sin sin
r
r
r r r
r , , R r Φ Θ .
,
,
r
Example 5: Hydrogen like atoms
2011.05.31. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 32
2
2 2
1 d ,
d m
( ) ejm ,
m A
m 0,1,2,
2 2
1 d d
sin sin sin
d d m
lm( ) Pm
cos
; m .2
2 2
2 2 2
0
1 d d 2 e
d d 4π 0.
R m Z
r r E R
r r r r r
2 4 2 2 2 0
e , 1, 2,
8 h
n
E mZ n
n
2 11
0 0 22 90
e 2 2 , , 0.0528 10 m.
π e
n n n
Z r h
R A L a
a m
. 1 ,
, 2 , 1 ,
0
n
2 1
j, , ( ) ,
e 2 2 cos e
m
n m n m n
m m
n
r A B R Y
A L P
. , , , , , , , , m
n
n1,2,3,; 0,1,2,, 1; 2 1 0 1 2
2 0
0 2
0
, ,
e
h Z r
a m n a
, , .
n m
r R
nr Φ
mΘ
m
, cos
jm m m m m
Y Φ Θ B P
m e
Constant A can be determined by the normalization of
2 2
0 0 0
sin
n m, ,
n m, , d d d 1.
r r r r
Example 6: How many different solution belong to a given value of n ? ℓ goes from 0 to (n 1). For a given ℓ, m can be 2 ℓ +1 from ℓ to ℓ.
Thus to a given eigenvalue, i.e. to a given energy it belongs
different eigenfunctions, i.e. (“degenerate” stationary eigenstates).
To every triple n, ℓ, m two spin quantum numbers
2011.05.31. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 34
21 0
1
2 n
n
12
, , 1 ,
n m r s 2
1
2
, , 1 .
n m r s 2
, ,
e
2
2
cos
en m r A Ln P
2 0 9
0 2
0
, 0.0528 10 m
π e Z r h
a a m
3/2 100
0
1, 0, 0, 1 e
π
n m Z
a
3/ 2
/ 2 200
0
2, 0, 0, 1 2 e
32π
n m Z
a
3/ 2
/ 2 210
0
2, 1, 0, 1 e cos
32π
n m Z
a
3/ 2
/ 2 j
21 1
2, 1, 1, 1 e sin e
64π
n m Z
a
2011.05.31. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 36
2
/3300
0
3, 0, 0, 1 27 18 2 e
81 3π
n m Z
a
3/ 2
2 /3
300
0
1 2
3, 1, 0, 6 e cos
81 π
n m Z
a
3/ 2
2 /3 j
31 1
0
1 1
3, 1, 1, 6 e sin e
81 π
n m Z
a
3/ 2
2 /3 2
320
0
1 1
3, 2, 0 e 3cos 1
81 6π
n m Z
a
3/ 2
2 /3 j
32 1
0
1 1
3, 2, 1, e sin cos e
81 π
n m Z
a
3/ 2
2 /3 2 j2
32 2
0
1 1
3, 2, 2 e sin e
162 π
n m Z
a
