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Cell[CellGroupData[{
Cell["Quantum Oscillations", "Title"],

Cell["AP1603", "Subsubtitle"],

Cell["\<\
Quantum mechanics describes equations of motion of atomic and subatomic \
particles. At the small size of atoms, electrons and ions behave both as \
individual particles and as waves. Like marbles and balls, we can count atoms \
and electrons one-by-one. Like waves crashing onto the beach, electrons can \
constructively and destructively interfere with each other forming \"crests \
and valleys\" or interference patterns of particle current. \
\>", "Text"],

Cell[TextData[{
 "Although it is not possible to learn and to ",
 StyleBox["understand",
  FontSlant->"Italic"],
 " quantum mechanics in a few lectures, it is very easy to ",
 StyleBox["experiment",
  FontSlant->"Italic"],
 " with quantum mechanics using ",
 StyleBox["Mathematica",
  FontSlant->"Italic"],
 ". In order to proceed, you will have to accept a few (unusual) rules about \
the quantum nature of particles. This is not surprising. We have absolutely \
no first-hand experience with atomic particles. But, if you accept these few \
rules about quantum mechanics, you can use ",
 StyleBox["Mathematica",
  FontSlant->"Italic"],
 " to begin a very useful study of quantum behavior brought to you by \
computer simulation."
}], "Text"],

Cell[TextData[{
 "I have included below a short introduction to quantum mechanics adapted \
from an article written by the great Prof. Gerald Feinberg of Columbia's \
Physics Department. Following this introduction, I will review complex \
arithmetic, present the ",
 StyleBox["rules for the interpretation of quantum wave mechanics",
  FontSlant->"Italic"],
 ", and, finally, introduce a method for simulating quantum fluctuations with \
",
 StyleBox["Mathematica",
  FontSlant->"Italic"],
 ". "
}], "Text"],

Cell[CellGroupData[{

Cell["A Brief Introduction to Quantum Mechanics", "Section"],

Cell[TextData[{
 "(Adapted from ",
 StyleBox["Quantum Mechanics",
  FontSlant->"Italic"],
 " by G. Feinberg.)"
}], "Text"],

Cell["\<\
Quantum mechanics is the fundamental theory used by 20th- century physicists \
to describe atomic and subatomic phenomena. It has been successful in tying \
together a wide range of observations into a coherent picture of the \
universe.
While quantum mechanics uses some of the concepts of Newtonian mechanics, the \
previous description of physical phenomena, it differs fundamentally from the \
Newtonian description. For example, in Newtonian physics, quantities were \
believed to be continuously variable, able to take on any value in some \
range. An example is angular momentum, which, for a particle revolving in a \
circular orbit about some center of attraction, is proportional to the speed \
multiplied by the distance from the center. Because that distance could have \
any value in Newtonian mechanics, so could the angular momentum. In quantum \
mechanics, on the other hand, angular momentum is always restricted to \
certain discrete values, whose ratios are simple rational numbers.
An even more fundamental difference between quantum mechanics and previous \
physical theories is that probability enters in a basic way into how quantum \
mechanics describes the world. This is made evident by the ways that quantum \
mechanics and Newtonian mechanics deal with predictions of the future. For \
something described by Newtonian mechanics, such as the motion of planets \
about our Sun, it is possible, if sufficiently accurate measurements are made \
at one time, to predict the future behavior of the system to arbitrarily \
great accuracy. For systems described by quantum mechanics, even one as \
simple as an atom with a single electron, precise prediction of future \
behavior is usually impossible. Instead, only predictions of the probability \
of various behaviors can be made. \
\>", "Text"],

Cell[CellGroupData[{

Cell["Quanta", "Subsection"],

Cell[TextData[{
 "Quantum mechanics was developed over a period of 30 years, during which it \
was successively applied to several physical phenomena. \nThe first use of \
quantum ideas was made in the analysis of how electromagnetic radiation is \
produced. This was done by the German physicist Max Planck in 1900. Planck \
was trying to account for the distribution, among different frequencies, of \
the radiation emitted by a hot object, such as the surface of the Sun. He \
found that to obtain results in agreement with observation, he had to assume \
that the radiation was not emitted continuously, as was previously believed. \
Instead, it was emitted in discrete amounts, which he called quanta. For \
these quanta, there was always a relation between the frequency ",
 StyleBox["f",
  FontSlant->"Italic"],
 ", and the amount of energy emitted ",
 StyleBox["E",
  FontSlant->"Italic"],
 ", of the form ",
 StyleBox["E",
  FontSlant->"Italic"],
 " = ",
 StyleBox["hf",
  FontSlant->"Italic"],
 ". Here ",
 StyleBox["h",
  FontSlant->"Italic"],
 " is a universal constant introduced by Planck, and now named after him. \
Planck's constant has the units of energy multiplied by time, known as \
action. Its numerical value is approximately 6.63 ",
 Cell[BoxData[
  FormBox[
   SuperscriptBox["10", 
    RowBox[{"-", "34"}]], TraditionalForm]]],
 " joule-seconds. The specific result of Planck's analysis was a formula \
expressing the amount of energy radiated at any frequency as a function of \
the temperature of the emitting object. This relation, the blackbody \
distribution, agrees accurately with observation."
}], "Text"],

Cell["\<\
The development of actual quantum mechanics--the mathematical theory--took \
place in the years 1924-27. Initially there were two seemingly different \
approaches: matrix mechanics, invented by Werner Heisenberg, and wave \
mechanics, invented by Erwin Schrodinger. However, it was soon shown that \
these were distinct aspects of a single theory, which came to be known as \
quantum mechanics. This unified version was invented by Paul Dirac. In matrix \
mechanics, physical quantities such as the position of a particle, are \
represented not by numbers, but by mathematical quantities known as matrices. \
Matrix mechanics is most useful in dealing with situations in which there is \
a small number of relevant energy levels, such as a particle with definite \
angular momentum in a magnetic field.\
\>", "Text"],

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Cell["\<\
The German physicist Max Karl Ernst Ludwig Planck, b. Apr. 23, 1858, d. Oct. \
3, 1947, developed the concept of the quantum, or fundamental increment of \
energy--basic to quantum mechanics, and a cornerstone of modern physics. \
After receiving his Ph.D. from the University of Munich in 1879, Planck \
taught at the University of Kiel (1885-89) and the University of Berlin \
(1889-1926). His appointment at the latter institution included the \
directorship of the Institute of Theoretical Physics that was newly founded \
for him.
Planck began studying blackbody radiation in 1897 and discovered that at long \
wavelengths it did not obey the distribution laws given by Wilhelm Wien. This \
discovery led him to announce (1900) his revolutionary idea that an \
oscillator could emit energy only in discrete quanta, contrary to classical \
physical theory. The quantum theory--which gained Planck the Nobel Prize for \
physics in 1918--was used by Albert Einstein to explain (1905) the \
photoelectric effect and by Niels Bohr to propose (1913) a model of the atom \
with quantized electronic states; the theory was later developed into quantum \
mechanics. Planck mastered every aspect of physics from thermodynamics and \
electrodynamics to relativity and also wrote extensively on the philosophy of \
science.\
\>", "Text"]
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Cell[CellGroupData[{

Cell["The Wave Function", "Subsection"],

Cell[TextData[{
 "Wave mechanics is more useful than matrix mechanics in a situation where \
the number of energy levels is infinite, as with an electron in an atom. It \
is based on the idea originally suggested by Louis de Broglie, that particles \
such as electrons have waves associated with them. The wavelength, \[Gamma], \
of the wave is related to the mass, ",
 StyleBox["m",
  FontSlant->"Italic"],
 ", and speed, ",
 StyleBox["v",
  FontSlant->"Italic"],
 ", of the particle by the relation \[Gamma] = ",
 StyleBox["h/mv",
  FontSlant->"Italic"],
 ". This implies that for electrons moving at 10 percent the speed of light, \
such as those produced by some television tubes, the wavelengths are about ",
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 " meters, or about the distance between atoms in a crystalline solid. This \
prediction of deBroglie was verified by Clinton Davisson and by George \
Thomson, who were able to pass the electron waves through metallic crystals, \
and so produce diffraction patterns similar to those produced by X rays."
}], "Text"],

Cell["\<\
In 1925, Erwin Schrodinger developed an equation, now bearing his name, which \
describes how a wave associated with an electron or other subatomic particle \
varies in space and time as the particle moves under the influence of various \
forces. This equation has many types of solutions, and Schrodinger imposed \
the condition that for a particle bound in an atom, the solution should be \
mathematically well defined everywhere. When applied to the case of an \
electron in a hydrogen atom, Schrodinger's equation immediately gave the \
correct energy levels previously calculated by Bohr. However, the equation \
could also be applied to more complicated atoms, and even to particles not \
bound in atoms at all. It was soon found that in every case, Schrodinger's \
equation gave a correct description of a particle's behavior, provided that \
the particle was not moving at a speed near that of light.
In spite of this success, Schrodinger initially misunderstood the meaning of \
the quantum wave function and believed that the intensity of the wave at a \
point in space represented the \"amount\" of the electron that was present at \
that point. That is, the electron was spread out, not concentrated at a \
point. However, this interpretation was wrong, because even if a particle was \
originally concentrated on a small region, in most cases it would soon spread \
over an increasingly larger region, in contradiction to what was the observed \
behavior of particles.\
\>", "Text"],

Cell["\<\
The correct interpretation of the waves was proposed by Max Born. While \
studying how quantum mechanics describes collisions between particles, he \
realized that the intensity of the deBroglie-Schrodinger wave was a measure \
of the probability of finding the particle at each point in space. In other \
words, a measurement would always find a whole particle, rather than a \
fraction of one, but in regions where the wave intensity was low, the \
particle would rarely be found, whereas in regions of high intensity, the \
particle would often be found.\
\>", "Text"],

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Cell["\<\
The Austrian theoretical physicist Erwin Schrodinger, b. Aug. 12, 1887, d. \
Jan. 4, 1961, published (1926) four papers that laid the foundation of the \
wave-mechanics approach to quantum theory and set forth his now-famous wave \
equation. Schrodinger earned a doctorate at the University of Vienna in 1910. \
He succeeded (1927) Max Planck in the chair of theoretical physics at the \
University of Berlin but left Germany in 1933 because of Nazi threats--the \
same year he shared the Nobel Prize for physics with Paul Dirac for his \
contributions to atomic theory. In 1939 he joined the newly formed Institute \
for Advanced Studies in Dublin. There he continued his studies of the \
application and statistical interpretation of wave mechanics, the \
mathematical character of the new statistics, and the relationship of these \
statistics to statistical thermodynamics.\
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Cell["The Schrodinger Equation", "Section"],

Cell[TextData[{
 "For our ",
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Schrodinger's wave function. \[Psi](",
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value at any point has both real and imaginary parts. We will solve \
Schrodinger's equation to see how the complex function \[Psi](",
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what gives \[Psi] its usefulness. As Schrodinger wrote it, \[Psi] is a \
function of space, or ",
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conjugate is proportional to a particle's ",
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at all possible locations, this sum must equal to 1, or 100%."
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Cell["Momentum and Energy Representations", "Subsubsection"],

Cell[TextData[{
 "In order to determine the probability distribution functions for momentum \
or energy, we need to transform \[Psi] into a different coordinate system \
(sometimes called a different \"representation\"). For momentum, the \
appropriate coordinate system is the wavenumber of the spatial oscillations \
of the \[Psi] (",
 StyleBox["i.e. ",
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oscillations of \[Psi], the higher is the particle's momentum. For energy, \
the appropriate coordinate system (or \"representation\") is the frequency of \
the temporal oscillations. The more quickly \[Psi] oscillates in time, the \
higher the energy of the particle. These transformations can be performed by \
Fourier (or Laplace) transforms. If we were to Fourier transform the spatial \
coordinate of \[Psi](",
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momentum, ",
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probability of measuring a particle at a particular energy, ",
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Cell[TextData[{
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1. The wavefunction, \[Psi](",
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the measurements of particles. \n2. The probability of finding a particle at \
a location, ",
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We write this probability as ",
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Cell[CellGroupData[{

Cell["A very small ball in a quantum box", "Section"],

Cell[TextData[{
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Cell[TextData[{
 "If our box were the size of a billiards table, the ball's motion could be \
accurately described without the need to use quantum mechanics. However, if \
the box were the size of an atom and if the ball were an electron, then \
quantum mechanics would be ",
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compute the probabilities for measurements of the particle's position, ",
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Cell[CellGroupData[{

Cell["Schrodinger's equation on a grid", "Subsection"],

Cell[TextData[{
 "In general, the simulation of quantum behavior on a computer is difficult. \
Nevertheless, the same techniques developed for simulations of the \
oscillations of a string can be used for a quantum ball in a box.\nOur first \
step is to define a grid on which to represent the wavefunction, \[Psi]. For \
real particles, the coordinate system used to represent quantum oscillations \
is continuous. (The same can be said for oscillations on a string.) However, \
we can represent quantum oscillations ",
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spacing between grid points."
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Cell[TextData[{
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 " + ",
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 ".\nWhat does this equation mean? \nSchrodinger's equation defines the \
relationship between the spatial and temporal oscillations of the \
wavefunction. Recall our quantum rules: the spatial wavelength is inversely \
proportional to the momentum and the temporal frequency is proportional to \
the energy. If we were to substitute for the wavefunction, \n\t\[Psi](",
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We choose an initial condition with a Gaussian spatial extent and a complex \
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Since we will be working with \[Psi](x, t) on a grid, we define a useful \
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