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2. Quantum theory: techniques and applications
2.1.Translational motion
2.1.1 Particle in a box
translation
2.1.2 Tunnelling
2.2. Vibrational motion
vibration
2.2.1 The energy levels
2.2.2 The wavefunctions
2.3. Rotational motion
2.3.1 Rotation in 2 dimensions
2.3.2 Rotation in 3 dimensions
2.3.3 Spin
rotation
The energy in a molecule is stored as molecular
vibration, rotation and translation.
2.1 The translational motion
For a free particle (V=0) travelling in one
dimension, the Schrödinger equation has a
k= Aeikx + Be-ikx
2
2
general solution k, where k is a value      E
2 2
2

k
Ek 
characteristic of the energy (eigenvalue) of  2m  x 
2m
the particle Ek.
For a free particle, all the values of k, i.e. all the energies are possible: there is no quantization
2.1.1 Particle in a box
Particle of mass m is confined in an infinite square
well. Between the walls: V=0 and the solution of the
SE is the same as for a free particle.
k= C sinkx + D coskx
NB: with D= (A+B) C= i(A-B)
A. boundary condition (BC): The difference with
the free particle is that the wavefunction of a confined
particle must satisfy certain constraints, called
boundary conditions, at certain locations.
 BC1: k(0)=0 → k (0) = C 0 + D 1=0 → D=0
→ after BC1: k= C sinkx
 BC2: k(L)=0 → k (L) = C sinkL =0
→ absurd solution: C=0, it gives k(x)=0 and |k(x)|2=0… the particle is not in the box!
→ physical solution: kL= n with n=1,2,… (n0 is also absurd)
The wavefunction n(x) of a particle in an infinite square well is now labeled with “n”
instead of k. Because of the boundary conditions, the particle can only have particular
energies En:
nx
 n ( x)  C sin
; n  1,2,...
L
2
 2 n / L 
n2h2
En 

; n  1,2,...
2
2m
8mL
B. Normalization: Let’s find the value of the constant C such that the wavefunction is
normalized.
L

0
L
L
 nx 
 nx 
2 1
2 L
 n ( x) dx  C 2  sin 2 
x
sin  2
1
 dx  C
 C
2
4n
2
 L 
 L 0
0
L
2
2
C  
 L
1
2
C. Properties of the solutions
 2
 L
1
2
 n ( x)    sin
 The solutions are labeled with n, called
“quantum number”. This is an integer that
specifies the energetic state of the system. In order
to fit into the cavity, n(x) must have specific
wavelength characterized by the quantum number.
With an increase of n, n(x) has a shorter wavelength
(more nodes) and a higher average curvature → the
kinetic energy of the particle increases.
nx
;
L
for 0  x  L
n2h2
En 
; n  1,2,...
8mL2
 The probability density to find the particle at a position x in the box is
 n2 ( x) 
2
 nx 
sin 2 

L
L


The larger n , the more uniform 2n(x): the situation is close
to the example of a ball bouncing between two walls, for
which there is no preferred position between the two walls.
 The classical mechanics emerges from quantum
mechanics as high quantum numbers are reached.
 The zero-point energy: because n>0, the lowest energy is
not zero but E1=h2/(8mL2).
That follows the Uncertainty Principle: if the location of the
particle is not completely indefinite (in the well), then the
momentum p cannot be precisely zero and E >0.
h2
 The energy level separation E= En 1  En  (2n  1)
increases with n.
2
8mL
E decreases with the size L of the cavity  for a molecule in gas phase free to move in a
laboratory-sized vessel, L is huge and E is negligible: the translational energy of a
molecule in gas phase is not quantized and can be described in classical physics.
2.1.2 Tunnelling
If the energy E of the particle is below a finite barrier
of potential V, the wavefunction of the particle is nonzero inside the barrier and outside the barrier.
 there is certain probability to find the particle
outside the barrier, even though according to classical
mechanics the particle has insufficient energy to
escape: this effect is called “tunnelling”.
X=0
X=L
 Transmission probability of the particle through the barrier.
 For x<0: the wavefunction is that of a free particle: (x<0)= Aeikx + Be-ikx with kħ=(2mE)1/2.
Aeikx represents the incident wave, Be-ikx corresponds to the reflected wave bouncing on the
wall.
For x>L: V=0, it’s like for a free particle: (x>L)= A’eikx + B’e-ikx with kħ=(2mE)1/2. But, the
direction of the transmitted wave is (Left Right), hence B’=0 since B’e-ikx is a wave
travelling in the (Right  Left) direction. A’eikx represents the transmitted wave.
x
X=0
X=L
 For 0<x<L: the wavefunction must be solution of the SE for a particle in a constant
potential V.
 2 2



V

  E
2
 2m  x

The general solutions are (0<x<L)= Ceqx + De-qx with qħ=[2m(V-E)]1/2. NB: here, the
two exponentials are real!
 The probability to find the particle in the barrier decreases exponentionally with the
distance x.
 The probability to find a particle in the region x<0, which travels LR, is proportional to |A|2
 The probability to find a particle in the region x<0, which travels RL, is proportional to |B|2
 The probability to find a particle in the region x>L, which travels LR, is proportional to |A’|2
 The probability that the particle crosses the potential barrier from x<0 to x>L is given by
the transmission probability: T=|A’/A|2
 The probability to be reflected on the barrier is characterized by the reflection
probability: R= |B/A|2
Since if the particle is not reflected, it is transmitted: T+R=1
Considering that the wave function must be continuous at the edges of the barrier (for x=0
and L), as well as the derivative of the wave function; it is possible to extract the
transmission probability:


qL
 qL 2 

e

e

T  1 

16 (1   ) 

with =E/V and
q=(1/ħ) [2m(V-E)]1/2
For a thick barrier qL>>1:
T≅ 16(1- )e-2qL
For a thick barrier qL>>1: T≃
16(1- )e-2qL
The transmission probability decreases exponentially with the thickness of the barrier and
with m1/2.
 T is increased also when the energy of the particle E is higher.
 Tunnelling is important for electrons, moderately important for protons (quick acid-base
equilibrium reaction), and negligible for heavier particles.
J  q*L 
L
2mV  E 

J=2
J=4
J=10
A large value of J corresponds to a
heavy particle or a wide barrier L
Example 5: Resonant tunneling diodes
Moore’s Law
In 1965, after he assisted in the design of Intel’s 8088
processor, Gordon Moore proposed that transistor
density per die would double every year after that.
“Moore’s Law”, as it was coined, led computer
manufacturers to reduce the size of transistors at a
rapid rate. The benefits from smaller transistors are
threefold:
1. Smaller transistors switch faster which leads to
faster processing speeds.
2. Smaller transistors allow more complex processors
to be built in the same space.
3. Smaller transistors allow for a greater number of
processors to be built within the same space. As a
result of these economic and technical factors Intel’s
first PC chip, the 8088, had 29,000 transistors with a
critical dimension of 3 microns (micrometers). The
Intel Pentium II processors has 7.5 million transistors
with a critical dimension of .25 microns. For thirty
years Intel and other chip makers have spent billions
in research and development to continue product
maturation at the rate explained by Moore.
Resonant Tunneling Diode
The use of a barrier to control the flow of electrons from one lead to the other is the basis of transistors. The
miniaturization of solid-state devices can’t continue forever. That is, eventually the barriers that are the key to transistor
function will be too small to control quantum effects and the electrons will tunnel when the transistor should be off. This
is a consequence of the particle-wave duality of electrons, and the single electron characterization of Schrodinger’s
equation. At the quantum level the wave nature of the electron will allow the electrons to tunnel through the barriers and
create a current. Quantum effects are seen at dimensions less then a micron, but the tunneling effect is expected to be
dominant when the critical dimensions approach the wavelength of an electron (approx. 10nm).
Ingenious devices exploit the quantum effects of miniature structures to control electrical current. These devices operate
by single electron control, and they require that electron movement be confined to two (quantum well), one (quantum
wire), or zero (quantum dot) dimensions. In these devices small voltages heat electrons rapidly, inducing complex
nonlinear behavior; the study of “hot” electrons, as they are termed, is central to the further development of these
devices. Two such devices are the Resonant Tunneling Diode and the Resonant Tunneling Transistor. These devices create
a new “switching” mechanism that requires controlled quantum tunneling to function.
The Resonant Tunneling Diode (RTD) consists of an emitter and a collector separated by two barriers with a quantum
well in between these barriers. The quantum well is extremely narrow (5-10nm) and is usually p doped. Resonant
tunneling across the double barrier occurs when the energy of the incident electrons in the emitter match that of the
unoccupied energy state in the quantum well. An illustration of the double barrier Resonant Tunneling Diode is shown in
Figure 4 . When the quantum well energy level is below E0, no current may flow by the tunneling mechanism. When the
bias is such that the energy level in the quantum well is aligned with a population of electrons above E0 in the emitter, the
electrons may tunnel from the emitter, to the quantum well, and through to the collector. As the voltage is increased, the
flow of electrons drops as the electrons are unable to tunnel above the resonant level. As the voltage bias continues to
increase, the current begins to increase again, this time as a result of the electrons flowing over the top of the barriers.
What results is an S shaped IV curve for the Resonant Tunneling Diode shown in Figure 5 .
There are several proposed applications of the resonant tunneling diode. The interesting S shaped IV characteristic makes
multistate memory and Logic circuits a possibility. Several resonant tunneling diodes can be combined to form multiple
peaks. The implication is that there can be multiple operating points for a circuit. Rather then determining if the memory
cell or logic state is a one or a zero, we can determine if it is any number of states.
The tunneling diode has not yet been fabricated using Silicon based technology, and the operating temperature of the
GaAs devices fabricated is below room temperature. Repeatable control of the size of the quantum well and other
structures is not yet realizable with current technologies. These and other manufacturing issues must be resolved before
the resonant tunneling diode is a widely used component.
http://www.mitre.org/research/nanotech/quantum_dot_cell1.html
Forms of carbon:
diamond
graphite
fullerenes
nanotubes
Carbon nanotube single-electron transistors
“ Single-electron transistors (SETs) have been proposed as a future alternative to conventional Si electronic components. However,
most SETs operate at cryogenic temperatures, which strongly limits their practical application. Some examples of SETs with roomtemperature operation (RTSETs) have been realized with ultrasmall grains, but their properties are extremely hard to control. The use of
conducting molecules with well-defined dimensions and properties would be a natural solution for RTSETs. We report RTSETs made
within an individual metallic carbon nanotube molecule. SETs consist of a conducting island connected by tunnel barriers to two
metallic leads. For temperatures and bias voltages that are low relative to a characteristic energy required to add an electron to the
island, electrical transport through the device is blocked. Conduction can be restored, however, by tuning a voltage on a close-by gate,
rendering this three-terminal device a transistor. Recently, we found that strong bends ("buckles") within metallic carbon nanotubes act
as nanometer-sized tunnel barriers for electron transport. This prompted us to fabricate single-electron transistors by inducing two
buckles in series within an individual metallic single-wall carbon nanotube, achieved by manipulation with an atomic force microscope
(AFM)(Fig. C and D). The two buckles define a 25-nm island within the nanotube.”
in “Carbon nanotube single-electron transistors at room temperature” by Postma-HWC; Teepen-T; Zhen-Yao; Grifoni-M; Dekker-G in
Science. vol.293, no.5527; 6 July 2001; p.76-9.
2.2 The vibrational motion
Classical mechanics
V
F 
  kx
x
V 
1 2
kx
2
A particle undergoes harmonic motion
if it experiences a restoring force
proportional to its displacement
Quantum mechanics
 2 2
1 2


kx   E

2
2
 2m  x

Eigenvalues:
1

E      ;   0,1,2,...
2

1/ 2
 Energy separation : constant = ħω
k
  
m
 Zero-point energy : E(=0)=½ ħω
 classical limit : for a huge mass m, ω is small and the energy levels form a continuum
A. The form of the wavefunctions
  ( x)  N H ( y )e  y
2
/2
;
1/ 4
 2 

y
and   

 mk 
x
 1 ( x)  N1 2 y e y
  12 ( x) 
4 N12
2
2
/2

x 2 e x
2 N1
x e x
2
/ 2 2
 N0 e x
2
/ 2 2

2
/ 2
N is the normalization constant
 0 ( x)  N 0 e  y
2
/2
  02 ( x)  N 02 e  x
NB: < x >= 0  the oscillator is equally likely to be
found on either side of x=0, like a classical oscillator.
1

 x 2     ( x) x 2   ( x) dx  (  )
2 mk 1/ 2
2
/ 2
B. The virial theorem
In a 1-dimensional problem with a potential V(x)= xn, the expectation values of the
kinetic energy <T> and the potential energy <V> verify the following equality:
2 <T> = n <V> ; with the total energy: <E>= <T> + <V>
 The harmonic oscillator, V=½kx2, is a special case of the virial theorem since n=2
and we have seen that
1/ 2
1
1
k
   (  ) 
2
2
m
1

E    
2

1
1
1
 V  k  x 2  (  ) 
2
2
2
we also know that
 V 
1
E
2
<E>= <T> + <V>
<T> = <V>
C. Quantum behavior of the oscillator
 The probability to find an oscillator (in its ground state:
=0) beyond the turning point xtp (the classical limit), is:

P    20 ( x) dx  0.08
xtp
V  Vmax
1
 E  kxtp2
2
1/ 2

 2E 
xtp   

 k 
xtp
Quantum
behavior
Quantum
behavior
Classical
behavior
xtp 0 -xtp
0
Quantum
behavior
xtp
Classical
behavior
 In the harmonic approximation, a diatomic molecule
in the vibration state = 0 has a probability of 8% to be
stretched (and 8% to be compressed) beyond its
classical limit. These tunnelling probabilities are
independent of the force constant and the mass of the
oscillator.
 Classical limit: for huge  (the case of macroscopic
object), P  0
2.3 The rotational motion
2.3.1 Rotation in 2 dimensions
Lz
2
2
2
p
L
L
z
 Classical mechanics: V  0; T  E 

 z
2
2m 2mr
2I

The angular momentum |Lz|= ∓pr
The moment of inertia I= mr2
 In quantum mechanics: not all the values of Lz are permitted,
and therefore the rotational energy is quantized. Where does
this quantization come from?
h
hr
Lz   pr and p 
 Lz  

No physical meaning

 The wavelength  of the wavefunction () cannot have
any value. When  increases beyond 2, we must have ()=
(+2), such that the wavefunction is single-valued: |()|2 is
then meaningful.
 The wavelength  should fit to the circumference 2r of
the circle. The allowed wavelengths are = 2r/ml ; where ml
is an integer that is the quantum number for rotation.
m hr
mh
hr
Lz     l   l   ml 

2r
2
A. Schrödinger equation for rotation in 2D
Go to cylindrical coordinates:
2  2
2 
 2 

H 
2 
2m   x
y 
x= r cos; y= r sin
2  2 1 
1 2 
2 2
 2 

H 
 2
2 
2m   r
r  r r  
2mr 2   2
because r  0
 2 2 
 ( )  E ( )
 Schrödinger equation: 
2
2
I




 The normalized general solutions:  ( ) 
e iml
2 1 / 2
have to fulfill the cyclic boundary condition ()= (+2):
 (  2 ) 
e iml (  2 )
2 1 / 2
ei 2ml  ei 
2 ml

e iml e i 2ml
2 1 / 2
 e i 2ml  ( )
 cos   i sin  
2 ml
  1
2 ml
1
2ml = an even integer  ml = 0, ∓1, ∓ 2, ∓3, ...
L2z ml2 2
 The eigenvalues are given by E 

2I
2I
NB: With ml2, the energy does not depend on the sense of rotation
 ( ) 
e iml
2 1 / 2
 For an increasing ml, the real part of the
wavefunction has more nodes
 the wavelength decreases and consequently, the
momentum of the particle that travels round the
ring increases (de Broglie relation): p=h/
 The probability density to find the particle in 
is a constant: |()|2=1/2

knowing the angular momentum precisely
eliminates the possibility of specifying the
particle’s location: the operator position and
angular momentum do not commute: uncertainty
principle.
Plots of the real part of
the wavefunction ()
B. The angular momentum operator Lz
Correspondence
cylindrical coordinates:
Classical mechanics
x= r cos; y= r sin
principles (chap 1)
ux u y uz
 
 
  

Lz   x
y
Lz  r  p  x
y
z  xpy  yp x
Lz   
i  y
x
i   
px p y pz
Quantum mechanics
ux, uy, uz are
because p z  0, z  0
unitary vectors
What are the eigenfunctions and eigenvalues of Lz?
Let’s apply Lz to the wavefunctions that are solutions
of the Schrödinger equation:
  
 1
Lz ml    ml  iml
eiml  ml  ml
1/ 2
i   
i 2 
Vector representation of angular momentum: the magnitude
of the angular momentum is represented by the length of the
vector, and the orientation of the motion in space by the
orientation of the vector
 The solutions of the Schrödinger equation, eigenfunctions of the Hamiltonian operator,
are also eigenfunctions of the angular momentum operator Lz: H and Lz are commutable:
the energy and the angular momentum can be known simultaneously.
 ml() is an eigenfunction of the angular momentum operator Lz and corresponds to an
angular momentum of mlħ.
z
2.3.2 Rotation in 3 dimensions
A particle of mass m free to travel (V=0) over a sphere of radius r.
2  2
2
2 
 2 
H 
 2 
2
2m   x
y
z 
spherical coordinates:
r
x= r sincos; y= r sin sin; z= r cos

2  2
2 
1
 2 
H 
 2 2 
2m   r
r r r

∂r = 0 (the particle stays on the sphere)
x
1
2
1  
 

 is the Legendrian
 

sin

  
sin 2    2 sin    
2
 The Schrödinger equation is :
Since I = mr2, we can write:
1 2
2mE


(

,

)


 ( , )
2
2
r

2   
with

2 IE
2
y
 We consider that (,) can be separated in 2 independent functions:  ( , )  ( )( )
→ the Hamiltonian can be separated in 2 parts → the SE is divided into 2 equations
1 d 2  sin  d 
d 
2



sin




sin
 + ml2 -ml2
2


 d
 d 
d 
At the moment, ml2 is just
introduced as an arbitrary constant
sin  d 
d 
 sin 
   sin 2   ml2
 d 
d 
1 d 2
 ml2
2
 d
d 2
2


m

l
2
d
Same as for the rotation in 2-D
with
 ml ( ) 
e iml
2 1 / 2
 l ,m
The solutions  should also fulfill the cyclic
boundary condition: ()=(+2); because of
that another quantum number “l” appears and
is linked to ml. Plm(cos ) is a polynomial called
the associated Legendre functions. Nlm is the
normalization constant.
( )  N lm Pl m (cos  )
N lm
iml
m
 ( )( ) 
e
P
(cos  )  Yl ,m
l
1/ 2
(2 )
l = 0, 1, 2, 3,…
|m|≤l
 The normalized functions lm(,)=Ylm(,) are called spherical Harmonics
The figure represents the amplitude of the
spherical harmonics at different points on the
spherical surface.
Note that the number of nodal lines (where
lm(,)=0) increases as the value of l
increases: a higher angular momentum
implies higher kinetic energy.
2
 From the solution of the SE, the energy is restricted to: E  l (l  1) ; l  0,1,2,...
2I
→ The energy is quantized and is independent of ml. Because there are (2l+1) different
wavefunctions (one for each value of ml) that correspond to the same energy, the energetic
level characterized by “l” is called “(2l+1)-fold degenerate”.
Spherical harmonics
ml = 0: a path around the vertical z-axis of the
sphere does not cut through any nodes. For
those functions, the kinetic energy arises from
the motion parallel to the equator because the
curvature is the greatest in that direction.
http://www.sci.gu.edu.au/research/laserP/livejava/spher_harm.html
http://mathworld.wolfram.com/SphericalHarmonic.html
Vector representation of the angular momentum
 The comparison between the classical energy E=L2/2I and
the previous expression for E, shows that the angular
momentum L is quantized and has the values (→ length of the
vector):
L={l(l+1)}1/2 ħ ;
l= 0, 1, 2,...
 As for the rotation in 2-D, the z-component Lz is also
quantized, but with the quantum number ml (→ orientation of
the vector L):
Lz= ml ħ ;
ml= l, l-1, …, -l
 For a particle having a certain energy (e.g. characterized by
l=2), the plane of rotation can only take a discrete range of
orientations (characterized by one of the 2l+1 values ml)
 The orientation of a rotating body is quantized
Cone representation of the angular momentum
While L2 and Lz commute, Lz and Lx (or Ly) do not commute
 Lz and Lx (or Ly) cannot be measured accurately and simultaneously
 If Lz is known precisely, Lx and Ly are completely unknown: representation with
a cone is more realistic than a simple vector. It means that once the orientation of
the rotation plane is known, Lx and Ly can take any value.
Notation: L is also often written J in textbooks
2.3.2 Spin of a particle
The spin “s” of a particle is an angular momentum characterizing the
rotation (the spinning) of the particle around its own axis.
 The wavefunction of the particle has to satisfy specific boundary conditions for this
motion (not the same as for the 3D-rotation). It follows that this spin angular momentum is
characterized by two quantum numbers:
 s (in place of l) > 0 and ∈ R → the magnitude of the spin angular momentum: {s(s+1)}1/2ħ
 ms ≤ |s| → the projection of the spin angular momentum on the z-axis: msħ
NB: In this course the spin is introduced as such. But in the Relativistic Quantum Field Theory, the
spin appears naturally from the mathematics.
 Electrons: s = ½ → the magnitude of the spin angular
momentum is 0.8666 ħ. The spins may lie in 2s+1= 2 different
orientations (see figure). The orientation for ms= +½, called  and
noted ; the orientation for ms= -½ is called  and noted .
 Photons: s = 1 → the angular momentum is 21/2 ħ
 The properties of fermions are described in
the statistic of Fermi-Dirac.
 The properties of bosons are described in the
statistic of Bose-Einstein.