2. Quantum theory: techniques and applications

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2. Quantum theory techniques and applications
2.1.Translational motion 2.1.1 Particle in a
box 2.1.2 Tunnelling 2.2. Vibrational
motion 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
The energy in a molecule is stored as molecular
vibration, rotation and translation.
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2.1 The translational motion
For a free particle (V0) travelling in one
dimension, the Schrödinger equation has a general
solution ?k, where k is a value characteristic of
the energy (eigenvalue) of the particle Ek.
?k Aeikx Be-ikx
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 V0 and the
solution of the SE is the same as for a free
particle. ?k C sinkx D coskx NB with D
(AB) 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 10 ? D0 ? after BC1 ?k C sinkx
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? BC2 ?k(L)0 ? ?k (L) C sinkL 0 ? absurd
solution C0, it gives ?k(x)0 and ?k(x)20
the particle is not in the box! ? physical
solution kL n? with n1,2, (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
B. Normalization Lets find the value of the
constant C such that the wavefunction is
normalized.
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C. Properties of the solutions
? 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.
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? The probability density to find the particle at
a position x in the box is
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 ngt0, the lowest
energy is not zero but E1h2/(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 gt0.
? The energy level separation ?E increases
with n. ?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.
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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 non-zero 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.

• ? Transmission probability of the particle
  through the barrier.
• ? For xlt0 the wavefunction is that of a free
  particle ?(xlt0) Aeikx Be-ikx with
  kh(2mE)1/2.
• Aeikx represents the incident wave, Be-ikx
  corresponds to the reflected wave bouncing on the
  wall.
• For xgtL V0, its like for a free particle
  ?(xgtL) Aeikx Be-ikx with kh(2mE)1/2. But,
  the direction of the transmitted wave is (Left
  ?Right), hence B0 since Be-ikx is a wave
  travelling in the (Right ? Left) direction.
  Aeikx represents the transmitted wave.

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? For 0ltxltL the wavefunction must be solution of
the SE for a particle in a constant potential
V. The general solutions are ?(0ltxltL) Ceqx
De-qx with qh2m(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 xlt0, which travels L?R, is proportional to
A2 ? The probability to find a particle in the
region xlt0, which travels R?L, is proportional to
B2 ? The probability to find a particle in the
region xgtL, which travels L?R, is proportional to
A2
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? The probability that the particle crosses the
potential barrier from xlt0 to xgtL is given by the
transmission probability TA/A2 ? The
probability to be reflected on the barrier is
characterized by the reflection probability R
B/A2 Since if the particle is not reflected, it
is transmitted TR1
?Considering that the wave function must be
continuous at the edges of the barrier (for x0
and L), as well as the derivative of the wave
function it is possible to extract the
transmission probability
with ?E/V and q(1/h) 2m(V-E)1/2
For a thick barrier qLgtgt1 T? 16?(1- ?)e-2qL
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• For a thick barrier qLgtgt1 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.

A large value of J corresponds to a heavy
particle or a wide barrier L
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Example 5 Resonant tunneling diodes
Moores Law
In 1965, after he assisted in the design of
Intels 8088 processor, Gordon Moore proposed
that transistor density per die would double
every year after that. Moores 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 Intels 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.
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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 cant 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 Schrodingers 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.
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http//www.mitre.org/research/nanotech/quantum_dot
_cell1.html
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Forms of carbon diamond graphite fullerenes nano
tubes
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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 room-temperature 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.
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2.2 The vibrational motion
Classical mechanics
Quantum mechanics
A particle undergoes harmonic motion if it
experiences a restoring force proportional to its
displacement
Eigenvalues
? Energy separation constant h? ? Zero-point
energy E(?0)½ h? ? classical limit for a
huge mass m, ? is small and the energy levels
form a continuum
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A. The form of the wavefunctions
N? is the normalization constant
NB lt x gt 0 ? the oscillator is equally likely
to be found on either side of x0, like a
classical oscillator.
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B. The virial theorem
In a 1-dimensional problem with a potential V(x)
?xn, the expectation values of the kinetic energy
ltTgt and the potential energy ltVgt verify the
following equality 2 ltTgt n ltVgt with the
total energy ltEgt ltTgt ltVgt
? The harmonic oscillator, V½kx2, is a special
case of the virial theorem since n2
and we have seen that
ltTgt ltVgt
we also know that ltEgt ltTgt ltVgt
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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
0
xtp
xtp
Quantum behavior
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
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2.3 The rotational motion
2.3.1 Rotation in 2 dimensions
Lz
? Classical mechanics
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?
? 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.
No physical meaning
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A. Schrödinger equation for rotation in 2D
Go to cylindrical coordinates x r cos? y r
sin?
? Schrödinger equation
? The normalized general solutions have to
fulfill the cyclic boundary condition ?(?)
?(?2?)
2ml an even integer ? ml 0, 1, 2, 3, ...
? The eigenvalues are given by
NB With ml2, the energy does not depend on the
sense of rotation
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? 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) ph/?
? The probability density to find the particle in
? is a constant ?(?)21/2? ? knowing the
angular momentum precisely eliminates the
possibility of specifying the particles
location the operator position and angular
momentum do not commute uncertainty principle.
Plots of the real part of the wavefunction ?(?)
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B. The angular momentum operator Lz
Classical mechanics
Correspondence principles (chap 1)
cylindrical coordinates x r cos? y r sin?
Quantum mechanics
ux, uy, uz are unitary vectors
What are the eigenfunctions and eigenvalues of
Lz? Lets apply Lz to the wavefunctions that are
solutions of the Schrödinger equation
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 mlh.
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2.3.2 Rotation in 3 dimensions
A particle of mass m free to travel (V0) over a
sphere of radius r.
spherical coordinates x r sin?cos? y r sin?
sin? z r cos?
?r 0 (the particle stays on the sphere)
is the Legendrian
? The Schrödinger equation is
Since I mr2, we can write
with
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? 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
ml2 -ml2
At the moment, ml2 is just introduced as an
arbitrary constant
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.
Same as for the rotation in 2-D with
l 0, 1, 2, 3, ml
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? 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.
? From the solution of the SE, the energy is
restricted to ? The energy is quantized and is
independent of ml. Because there are (2l1)
different wavefunctions (one for each value of
ml) that correspond to the same energy, the
energetic level characterized by l is called
(2l1)-fold degenerate.
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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.htm
l
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Vector representation of the angular momentum
? The comparison between the classical energy
EL2/2I and the previous expression for E, shows
that the angular momentum L is quantized and has
the values (? length of the vector)
Ll(l1)1/2 h 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 h ml
l, l-1, , -l
? For a particle having a certain energy (e.g.
characterized by l2), the plane of rotation can
only take a discrete range of orientations
(characterized by one of the 2l1 values ml) ?
The orientation of a rotating body is quantized
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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
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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) gt 0 and
? R ? the magnitude of the spin angular momentum
s(s1)1/2h ? ms s ? the projection of the
spin angular momentum on the z-axis msh
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 h. The
spins may lie in 2s1 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 h
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? The properties of fermions are described in the
statistic of Fermi-Dirac.
? The properties of bosons are described in the
statistic of Bose-Einstein.