Lecture notes including summary of the Quantum Concepts course
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Course
Quantum Concepten (5092QUCO3Y)
Institution
Universiteit Van Amsterdam (UvA)
Comprehensive lecture notes on the Quantum Concepts course given by Jorik van de Groep in the 2nd year of the bachelor's degree in Physics and Astronomy at the UvA/VU. At the end of the lecture notes, an overview/summary is available with all important concepts clearly listed and described for each...
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> nanomaterialen , , ,
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Herhaling QM 1
In Classical Mechanics , we solve Newton's and law :
F =
ma =
m =X(t)
↳ know is
exactly where particle .
In Quantum Mechanics particles have wave behaviour described function :
,
by the wave
↑ (X ,
t)
In Quantum Mechanics ,
we solve
Schrodingers equation to
get the wave function :
~ external potential
it
ONteA + VP
↳ time
derivative
↓ and derivative
in
space
What does it mean ?
INR a
·
lik probability density
=
"
↳
position of electron as a
function of
time is described by the wave function
b &
ab X
(IN( ,
t) ax :
gives probability of finding the
particle in this
range .
A
> Statistical interpretation
Notes : Particle is still a particle its location is described the wave function
by
·
,
· Inherent in determinacy
Particle has to be somewhere
~ Normalisation of the wave function :
A
& INIX tiax ,
= 1c use to find constants of wave function
-
↳
**4
9 dx =
Exercise :
Mix t
Ac Siweh
real constants
=
,
↳ find Ak
~
Normaliseer de integraal : * =
In is door-i. ( etiwt)liwt
de complex geconjugeerde vervangen we alle
J he
&
↳ tijds afhankelijkheid vervalt
!
.
1Tax
= naal A
N
- e
buiten de inte
>
graal ! 6 ausische integraal
-
want constante
& Lax
.
Gausische integraal met standaardoplossing : eaxdxY
In dit geval : a = +
E en dus wordt de
oplossing ,
toc
A2 ,
10
Upon measurement :
INEx
12
m S
=
I I S
X X
↳ stort in
golffunctie
het moment dat
>
Op je meet weet je precies waar het
deeltje is de onzekerheid is opeens verdwenen ->
golf functie stort in
-
.
,
golffunctie wordt deltafunctie
(c)
* No more
uncertainty ,
one specific location
Ware function
*measure "collapses" upon measurement
again : same result
, Before measurement : What do we expect location to be ?
"expectation Value" (verwachtingswaarde) > <X)
~ Waar verwacht je dat het deeltje zich bevindt op het moment dat je het meet >
verwachtingswaarde
S
(x) =
( x (i(x ,
t))2ax
-
↳insert in front of
prob density
What does this mean !
> <X)
- is the
average of many measurements of X, of particles that have the exact same ware function
↳dus p identiefunctieswaarvanweallemaal metemen gemiddeldeen
daar neem het een
je
Mathematical tools
Define position "Operator" "working" ,
on a ware
function
* =
x = )
(y)
=
I 4
*
2xYNdx
&
-
2
=
xx =
Jy xydx*
Can we also define <
p >?
(p) = m(v) = mo(x)
↳ Schrödinger eq .
relates de to :
Pit De
s
↳ <p) = -
ih(*a
[p] = -
ingx
Now WecanDefine Dynamicvariables
in terms of xa!,
(T) = ax
↳
-
Kinetische energie
Verwachtingswaarde
The uncertainty principle
Can we determine the location and momentum of a particle with
arbitrary accuracy ? (Like in Classical Mechanics
de Broglie formula :
P = hk =
22
↳ wave length
↑ X ? plaats kan hieruit niet worden afgeleid
.
I Golflengte
↑? > -
kan hieruit niet worden
afgeleid
3
&
X
& Heisenberg's uncertainty principle
Ex Op sh
↳ o =
(X -
(12] Standard deviation
Zowel impuls als plaats en
energie zijn operatoren in de QM , maar
tijd is dat niet
Tijd At levens duur elektron
S best
:
Energie = Zw : DE
Ok , so how do we find the actual form of the wave function ?
Let's revisit the Schrödinger equation :
~ external potential
it
O -t o = + VP
↳ time
derivative
↓ and derivative
in
space
↑ (x , t) function of X and t
↳ in reality ,
most potentials are independent of time .
, Separation of variables :
split X and t
M(x t) ,
=
((x)y(t)
in p d y +
vo
by =
7
+ V =
EY
↳ constant
in Et
in
-
=
Ey =
y(t) =
e
Time independent
-
Schrödinger equation :
+V =
Now ,
recall that =
-in and me
-hany + Vo =
Eq
2mdX
↳ kinetic ↳
potential ( total
energy
energy energy
↳ Hamiltonian operator J
82
2 =
- x
+ V(x) = jtp =
Ep
Eigenvalue problem :
↓ Eigenvalue
50 =
Eq
1-
Energy of eigen state
& &
S & ↑
Operator Eigenvector
E shape of
eigenstate
Examples of potentials :
Infinite square well
V(X) /
co
n= 3
~
20
oexa
vix =
M =2
~ otherwise
n=1
X
↳ En =
Methhe
The Harmonic Oscillator
xVIXI
~ V(x) = kx2
-
En =
(n 2) kw
+
-
& &
X
=>
Energy Spectra become discreet
The solution is
general a linear combination
of eigenstates :
M(x
o
Gifn'tS energy of staten
o
(nOn(Xigenstaten
,
ti =
n=
14
amplitude
coefficient
, Hoorcollege 2
-
Thema 7 : Quantum opsluiting in nanomaterialen
Leerdoelen :
·
Beschrijven van
quantumopsluiting in <D , 2D en 3D ; wat dit betekent voor de
golffunctie van elektronen
·
Voorbeelden noemen van nanomaterialen voor deze 3
categorieën
·
Quantumopsluiting <>
Deeltje-in-doosje
·
Schatten bij welke grootte van nanomaterialen
opsluiting een rol speelt
·
Praktische toepassingen
Experimenten beschrijven
·
waarin Lichtbaar is
quantumopsluiting .
Energieniveau's uitrekenen
·
van een elektron in een
quantumdot :
bijbehorende verwachtingswaarde voor de optische transitie
Contents of
today
:
·
Size does matter
Confinement and the ware function
·
· Dimensions of confinement
·
Free particle
Quantumwell >
-
1 D
Quantum wire -
> 2D
·
Quantum dot >
-
3D
Degeneracy and total
energy
·
Excitons , binding energy and Bohr radius
Optical properties
·
Size does matter
Properties of materials are
independent of their size...
Silicon "ingot" "wafer" >macroscopischobjectteomrooster
-
as the same properties as silicium as
↳ like : electrical conductivity
density
refractive index
heat transfer
-
down to a certain size
. Size does matter in the nanoworld :
Squantum dots
↳ in nanowereld quantumopsluiting vindt plaats
S
relevant >
maat is opeens
-
hiermee schuiven elektronenroosters en
↳ in kleine volumes
proppen van elektron golffuncties daarmee de
eigenschappen
Ok , then what is nano ?
In nanostructures , the electronic wave function can be confined
"Quantum Confinement "
↳ interfering wave functions cause standing wave patterns (staande golf)
&
" staande gof interfereert met zichzel
e
↳
Scanning tunneling microscope
golf Functie 1112
↳
Je meet hier direct de
How does confinement enter the
Schrödinger equation ?
t +
V)x = 2x =
24
↳
↳ kinetic
energy
potentiaals
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