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The nanoscale and quantum mechanics
At the macroscopic scale of meters, classical mechanics can be used to describe motion.
At the microscopic scale of atoms, classical mechanics fails to describe motion properly
and quantum mechanics must be used. Quantum mechanics is valid at all length scales. It
is possible to describe the motion of macroscopic objects with quantum mechanics; it is
just mathematically difficult to do so. Classical mechanics and quantum mechanics give
the same predictions for macroscopic objects so we usually use the simpler classical
mechanics to describe large objects. Nanometer scale objects lie near the boundary
between classical mechanics and quantum mechanics and sometimes it is necessary to use
quantum mechanics to describe phenomena on the scale of nanometers.
Richard Feynmann had this to say about quantum mechanics, "Quantum mechanics is the
description of the behavior of matter and light in all its details and, in particular, of the
happenings on an atomic scale. Things on a very small scale behave like nothing that you
have any direct experience about. They do not behave like waves, they do not behave like
particles, they do not behave like clouds, or billiard balls, or weights on springs, or like
anything you have ever seen."
Feynmann also said, "I think I can safely say that nobody understands quantum
mechanics."
Quantum mechanics is spectacularly successful in describing phenomena on a small scale.
It is also mathematically difficult. In quantum mechanics, everything moves as a wave but
exchanges energy and momentum as a particle. When an electron moves, it must be
treated as a wave that can interfere. The wavefunction that describes an electron has peaks
and valleys that move around and reflect off walls much like water waves. You can see the
electron waves in the left image of Fig. 1.1. When a peak meets a valley there is
destructive interference and when a peak meets a peak there is constructive interference.
While an electron is moving, don't think of it as a particle that follows a particular path
through space. A wave follows many paths simultaneously. Although an electron was used
as an example here, the same could be said about other particles like protons, neutrons, or
photons. It is even possible to observe the wave nature of larger objects such as atoms and
molecules.
All of the information that can be known about a particle is contained in its wavefunction.
For instance, the square of the amplitude of the wavefuction describes the probability that
an particle will be observed at a particular place. Other quantities such as the velocity of
the particle can also be determined from the wavefunction. The time evolution of the
wavefunction is described by the Schrödinger equation,
Here i2 = -1, m is the mass of the particle, and V(x,y,z,t) is a space and time dependent
potential that confines the particle. Ψ is the wavefunction. It is a three dimensional, time
dependent, complex field. To understand this, think about a temperature map for a minute.
Temperature is a real quantity (it has no imaginary part) but it can be different at every
position in space and it changes in time. A wavefunction is a complex number at every
position in space that changes as a function of time.
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If two particles interact with each other (like an electron and a proton in a hydrogen atom)
then there are not two wavefunctions (one for the electron and one for the proton) there is
just one wavefunction, Ψ(xe,ye,ze,xp,yp,zp,t). This wavefunction describes the joint
probability of finding an electron at position xe,ye,ze, and a proton at position xp,yp,zp. This
is a complex, time dependent field in six dimensions.
In a typical nanostructure, there are often millions of interacting particles. The
wavefunction in this case would be a complex, time-dependent field in millions of
dimensions. You could write down the Schrödinger equation for this case and try to solve
using a computer. However, it turns out that the Schrödinger equation is intractable for
more than about ten interacting particles. Intractable means that even though we know
exactly the equation that needs to be solved and we know exactly how to solve it, it would
take longer than the age of the universe to find the solution numerically on even the fastest
computer. For more than ten interacting particles, analytic solutions are known to the
Schrödinger equation for only a few special cases. For all other cases some kind of
approximation has to be used. The trick is finding an approximation that is simple enough
to solve but complex enough to describe the phenomena of interest. Albert Einstein
expressed this as, "Everything should be made as simple as possible -- but no simpler."
To understand why the Schrödinger equation is intractable, consider how it would be
solved numerically. For a numerical solution, the wavefunction could be approximated by
the value of the wavefunction on a grid of points. The number of points needed depends on
the number of particles and the desired precision of the solution. A wavefunction for N
particles moving in three dimensions would be a complex function of 3N spatial variables
plus time.
Ψ(x1, y1, z1, x2, y2, z2, ... ,xN, yN, zN, t)
For a numerical solution, assume that it is sufficient to consider 100 values for x1 and 100
values for y1 etc. Then the number of points of the grid that would be used to approximate
the wavefunction would be 1003N. Because N is in the exponent, the number of points
becomes very large for large N. For 10 interacting particles, the number of points would be
1060. This is more than the number of atoms on the earth. There is no way to numerically
solve the Schrödinger equation like this for a general problem involving 10 interacting
particles on a conventional computer. Simplifications always need to be made to make the
problem tractable.
This is a sobering result. The Schrödinger equation, the most important equation in
physics, is intractable. It is impossible in general to calculate the quantum dynamics of
many interacting particles on a conventional computer. Approximations must be made in
the calculations and then the validity of the approximations must be checked by carefully
preparing a quantum system of interacting particles and measuring what happens.
Quantum computing
It is sometimes possible to map one intractable onto another so that if you find the solution
to one of the intractable problems the solution to the other intractable problem will be
known. It has been shown that calculating the quantum dynamics of certain systems can be
mapped onto the problem of finding the factors of a large integer. Finding the factors of a
large integer is known to be an intractable problem. If the right quantum system is
prepared and then the evolution of that system is measured, the factors of the large integer
can be determined. When a quantum system is used to perform a calculation like this it is
called a quantum computer. These quantum computers can in principle solve certain
intractable problems that no conventional computer could. However, up until now only
very simple quantum computers have been made and the most powerful calculation that
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has been performed was the factoring of 15 into 3 and 5. Research is proceeding on
mapping intractable problems onto quantum dynamics (this is called finding quantum
algorithms) and building more complex quantum computers.
Uncertainty principle:
The uncertainty principle applies to all waves: water waves, light, sound, and the matter
waves described by quantum mechanics. It states that product of the variance in the width
of a pulse
times the variance in the wave numbers in the pulse
must be greater than or equal to 0.5.
ΔxΔk ≥ 0.5
The wave number is k = 2π/λ. Any wave pulse can be written as a superposition of
sinusoidal waves with a certain distribution of wavelengths. The narrower the wave pulse
is in position, the more wavelengths are needed to describe it. For instance, a wavepulse
with a Gaussian form can be written as a superposition of cosine waveforms with different
wavenumbers,
Here the width of the wavepulse, Δx = a/2 and the width of the distribution of
wavenumbers is Δk = 1/a and the product is ΔxΔk = 0.5. Any wavepulse can be written as
a superposition of sine and cosine waveforms like this and the product of the uncertainties
is always greater or equal to 0.5. A Gaussian wavepulse is a minimum uncertainty
wavepulse. The uncertainty relation is often applied to classical waves in
telecommunications, seismology, ultrasound imaging, or other disciplines where waves are
measured. In quantum mechanics it is common to multiply both sides of this equation by
h/(2π). Since p = hk/(2π),
ΔxΔp ≥ h/(4π).
This relation is often loosely interpreted as saying that it is impossible to know the position
and the momentum of a particle simultaneously.
Quantum dots:
Quantum dots are man-made structures that can be used to store a small amount of charge.
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Typically the structure is a piece of a semiconductor with dimensions of a few tens of
nanometers to a few microns. Quantum dots typically contain a charge somewhere
between a single electron and a few thousand electrons.
Fig. 4.1. A lateral quantum dot. In this case, gold gates are deposited on a layered
semiconductor. Mobile electrons exist at the interface between two kinds of
semiconductor about 100 nm beneath the surface. By apply a negative voltage to
the gold gates, the regions underneath the gates can be depleted of electrons. By
choosing appropriate gate voltages for all of the gates, isolated puddles of
electrons can be formed between the gates. These puddles are the quantum dots.
Fig. 4.2. A schematic of a vertical quantum dot.
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Fig. 4.3. Quantum dots sorted by size emitting light of differnt colors. Reference:
www.physik.uni-muenchen.de/sektion/feldmann/fieldsoi/qdot/qdhome_b.htm
A crude model for the quantum states in a quantum dot is the infinite potential well. This
problem is discussed in most physics textbooks and the results are summarized below.
1-d infinite square well
For a particle of mass m in an infinite square well potential, the potential energy of the
particle is infinite for x < 0 and x > L. For 0 < x < L, the potential energy of the particle is
zero. The wavefunctions for a one-dimensional potential well are:
The energies that correspond to these wavefunctions can be determined by substituting the
wavefunctions into the time-independent Schrödinger equation. The energies are:
En =
n²h²
= E1n²
8mL²
n = 1,2,3,...
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The energy states are plotted as a function of n.
Density of states
Sometimes it is useful to know the number of states in a certain energy range. If the 1-d
potential is very long (large L) then the states will be closely spaced and the number of
states in a certain energy range can be approximated as a continuous function. By looking
at the figure above it is clear that there are no states with energies below E1 = h²/(8mL²),
there are many states just above E1 at the bottom of the parabola, and there are then fewer
and fewer states as the energy increases.
This can be stated more mathematically by defining a function called the density of states
g(E). The number of states between energies EA and EB is,
EB
N = ∫ g(E)dE
EA
Using the relationship for the energy, E = E1n², the density of states for a one-dimensional
potential well can be determined to be,
Here a factor of 2 has been included to account for the two spins allowed for every value
of n.
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The density of states for a one-dimensional potential.
2-d infinite square well
For a particle of mass m in an infinite square well potential, the potential energy of the
particle is infinite for x < 0, y < 0, and x > Lx, or y > Ly. For 0 < x < Lx and 0 < y < Ly, the
potential energy of the particle is zero. The wavefunctions for a two-dimensional infinite
square well potential are:
The energies that correspond to these wavefunctions can be determined by substituting the
wavefunctions into the time independent Schrödinger equation. The energies are:
Enxny =
h² nx² ny²
(
+
)
8m Lx² Ly²
nx,ny = 1,2,3,...
Density of states
The density of states for a two-dimensional potential square well is zero for energies
below,
E1 =
h² 1
1
(
+
)
8m Lx² Ly²
and is constant for energies above this energy. Including spin degeneracy the density of
states is,
0
g(E) = π
2E1
for E < E1
for E > E1
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The density of states for a two-dimensional potential.
3-d infinite square well
For a particle of mass m in an infinite 3-d square well potential, the potential energy of the
particle is infinite for x < 0, y < 0, z < 0 and x > Lx, or y > Ly, z > Lz. For 0 < x < Lx,
0 < y < Ly, and 0 < z < Lz, the potential energy of the particle is zero. The wavefunctions
for a three-dimensional infinite square well potential are:
The energies that correspond to these wavefunctions can be determined by substituting the
wavefunctions into the time independent Schrödinger equation. The energies are:
Enxnynz =
h² nx² ny² nz²
(
+
+
)
8m Lx² Ly² Lz²
nx,ny,nz = 1,2,3,...
Density of states
The density of states for a two-dimensional potential square well is zero for energies
below,
E1 =
h² 1
1
1
(
+
+
)
8m Lx² Ly² Lz²
and is increases as a square root above this energy. Including spin degeneracy the density
of states is,
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The density of states for a three-dimensional potential.
Reading



Review the discussion of quantum mechanics in sections 13.6-13.10 in
Understanding Physics.
Review "The infinite square potential well", section 13.13 in Understanding Physics.
Review chapter 19, "Atomic physics" in Understanding Physics.
References


Modeling and Prospects for a Solid-State Quantum Computer, H. E. Ruda and B.
Qiao, Proceddings of the IEEE, vol 91, p. 1874 (2003).
Britney's guide to semiconductor physics: Density of states
Problems
1. In an electron microscope, electrons are accelerated through a potential of 70 kV. In the
normal operation mode of an electron microscope, it is not possible to image features
smaller than the wavelength of the electrons. What is the wavelength of these electrons?
2. A C60 is bound to a surface by van der Waals forces. The wavefunction that describes
the position of the C60 is ψ = (2/π)1/4exp(-(x-x0)2). What is the probability that the C60
molecule will be found between 0.8x0 and 1.2x0?
3. In practically every physics textbook that discusses wave-particle duality, there is a
calculation of the wavelength of the matter wave of a macroscopic object. In
Understanding Physics this discussion is on page 370. The relationship between the
momentum of an object and the wavelength of the object is p = h/λ. Since p is typically on
the order of 1 kg m/s for a macroscopic object and h is a very small number, the
wavelength of a macroscopic object is very small. The conclusion of this discussion is
always that it is impossible to observe quantum interference of a macroscopic object.
Waves can be scattered from a lattice. The smallest lattice that we can make is an atomic
crystal. Look at the description of Bragg scattering in Understanding Physics on page 335.
Then read about electron diffraction in the Davidson-Germer experiment on page 371. It is
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the Bragg scattering of electron waves from a crystal that was observed in the DavidsonGermer experiment. Imagine scattering a C60 molecule from a crystal in the same way.
Estimate the largest angle of diffraction that can be achieved (Consider only the first order
maximum n = 1). Choose a reasonable spacing between the atoms in the crystal. You will
need to know the minimum momentum that a particle can be given. Assume that thermal
fluctuations limit the resolution of the momentum i.e. 0.5kBT = 0.5mv2. Consider
temperatures down to 1 K.
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