Download Mathematics of Quantum Mechanics

Submitted by: Jeffrey Gallo
Math 5100 Fall/2015
The Time-Dependent Schrodinger Wave Equation
Introduction
Nobody has ever witnessed an electron orbiting around the nucleus of an atom
(borrowing the analogy from the planetary orbits), but our theories about the atom today have
widespread applications. How could this be possible? One need only turn to mathematics. The
abstractions of mathematics serve as a powerful tool to understand dynamical processes that
are invisible to the naked eye. Let’s consider electromagnetic waves, for example. Can we see
them with our eyes? No. But do they exist? Certainly! As a matter of fact, Clerk Maxwell had
created a set of equations that accurately describes the wave nature of electromagnetic
phenomena. Maxwell had to abandon common mechanical metaphors (i.e. the science of
wheels, pulleys, bars and levers, etc.) in order to understand the intangible nature of
electromagnetic fields. Let’s consider Isaac Newton and his derivation of the fundamental laws
of moving bodies. Here we see an almost similar scientific approach. Did Newton have direct
contact with the celestial realm? No. Equipped with accurate astronomical readings of the
motions of the planets, Newton asked one fundamental question that eventually opened the
floodgates of Physics: what keeps the moon from falling into the Earth? Using a mechanical
“pulling” metaphor, from direct experience, he was able to derive a mathematical equation
that described the force of universal attraction – a mysterious force that somehow pulled all
the planets and moons into their orbits around the sun. Newton, however, could not explain
what was doing the actual “pulling.” We would have to wait over 300 years to get an answer to
this problem. Then came Albert Einstein. Einstein, having no direct experience of space, used
his imagination, together with some powerful forms of mathematics, to derive his General
Theory of Relativity, which stated that the nature of gravity is a result of a curvature in the
space-time fabric. Nobody knew what this fabric looked like, but Einstein was able to imagine it
using mathematics! Einstein was able to transcend the limitations of the senses to reach new
heights in our understanding of the fundamental laws of the universe.
Physicists were forced to embrace mathematics because common sense and everyday
experience could not grasp these invisible, yet real phenomena. Descartes’ rational approach
to the sciences is echoed in this new way of “seeing”. Descartes believed that the fundamental
laws of nature could be discovered only by doubting all knowledge that was gained through
sense perception. The fundamental laws of nature, according to Descartes, transcend direct,
human sense perception. Isaac Newton, most famously known for his three laws of motion and
the universal law of gravitational attraction, exemplified this ideal. Ever since Newton first laid
the mathematical foundation for the study of dynamical systems (typically called Newtonian
mechanics or classical mechanics), scientists were able to make accurate predictions about
macro level phenomena (celestial dynamics is an example of this). Newton’s three
fundamental laws of physics, founded on some basic philosophical presuppositions about the
Submitted by: Jeffrey Gallo
Math 5100 Fall/2015
nature of space and time, prescribed how matter was to move under the influence of a known
force. The principles of Newtonian mechanics were proven to accurately describe the “laws of
nature” in the macroscopic world. This was the philosophical outlook that pervaded Physics, in
the years leading up to Schrodinger’s quantum mechanical wave equation, as the
interpretations became more mathematical.
When we turn the scientific lens towards the invisible phenomena of nature, an
important question arises: do the same principles that govern classical mechanics (i.e. classical
determinism) work on the atomic and subatomic scale? If not, what sort of mathematical
physics is needed to make any reasonable predictions at levels were electrons travel close to
the speed of light? With new experimental findings in the early 20 th century, a new
mathematical approach was born, namely quantum mechanics. Quantum mechanics dealt with
a world that was both discontinuous and counterintuitive. The quantum world required more
sophisticated forms of mathematics – a mathematics that would be able to solve the most
controversial results in modern science.
It has been the aim of science to discover the fundamental laws that govern various
phenomena. We do this to create order in the world. But we cannot establish any order unless
we are able to make predictions – this is the true test for science. Early forms of science,
formally introduced by Isaac Newton, relied on the principles of classical mechanics in order to
make predict the time evolution of any given dynamical system. Inherent within the classical
picture was the principle of causal determinism. It was the assumption that dynamic bodies
could be predictably traverse absolute space and time, while remaining unchanged. Newton
had created the Calculus for this very reason. Newton had showed that mathematics can
model the time evolution of dynamical systems – all that was needed was information of the
initial conditions of a system.
Newtonian (or Classical) Mechanics
According to classical mechanics, we can determine, with certainty, the future state of a
dynamic system given its initial conditions (that is, the speed and direction of travel). In other
words, all future states of a dynamic system can be determined by its present state. This is
what Newton’s mathematics had suggested. The effectiveness of his science had rested on the
principle of causal determinism, or predictive determinism. But do the physical laws of the
quantum level realm abide by the same principles? To answer this, we need to understand the
basic problem that both Newtonian mechanics and quantum mechanics seek to address: if the
state of a dynamic system is known initially and something is done to it (i.e. a force of influence
is applied), how will the state of the system change with time in response? In order to tackle
this question, we must first understand how Classical mechanics solves problems for systems in
the macroscopic world.
Submitted by: Jeffrey Gallo
Math 5100 Fall/2015
Classical mechanics is a branch of science that deals with physical laws that describe the
motion of bodies in fields of influence (i.e. gravitational fields, electromagnetic fields, etc). The
problem in classical physics is to determine the position of a particle at any given time, 𝑥(𝑡).
Let’s consider the simplest dynamic “system”: a single particle that is characterized by a static
property, its mass 𝑚. We can assume that its motion is confined to a one dimensional, x-axis.
According to Newtonian mechanics, the state of a particle at any given time 𝑡 is defined by its
values for position 𝑥(𝑡) and velocity 𝑣𝑥 (𝑡). Now 𝑣𝑥 (𝑡) is considered the rate of change of
position with respect to time, which is 𝑣𝑥 (𝑡) =
𝑑𝑥
𝑑𝑥(𝑡)
𝑑𝑡
. All other dynamic properties of the
system, like momentum 𝑝𝑥 (𝑡) = 𝑚 𝑑𝑡 = 𝑚𝑣𝑥 , kinetic energy 𝐸𝑘 = 𝑇 =
𝑚𝑣𝑥 2
2
, potential energy
𝑉(𝑥), total energy 𝐸 = 𝑇 + 𝑉, etc. can be determined with 𝑥 and 𝑣𝑥 . Hence, all dynamic
properties of a given system depend only on position and velocity. To know a dynamic system’s
initial state, the numerical values for 𝑥(0) and 𝑣(0) must be known. According to Newtonian
mechanics the force 𝐹𝑥 acting on a particle is proportional to the acceleration 𝑎𝑥 =
𝑑2 𝑥
𝑑2 𝑥
𝑑𝑡 2
, where
the constant is the particle’s mass 𝑚. This can be written as: 𝐹𝑥 = 𝑚𝑎𝑥 = 𝑚 𝑑𝑡 2 . Once the
force that acts on a particle (of known mass 𝑚) is known, the acceleration, which is the second
derivative of the particle’s position with respect to time, can be determined from 𝐹𝑥 . When
acceleration is known, then 𝑣𝑥 (𝑡) can be known at all times by integration (i.e. anti-derivative).
By integrating 𝑣𝑥 (𝑡), all values for the position 𝑥(𝑡) of a particle at various points in time can be
found. Here we see that if the initial conditions of a dynamic system (𝑥 and 𝑣𝑥 ) are known,
together with the force acting on the particle, we can predict the state of a particle for all times.
The important point to highlight here is that in order to find out how 𝑥 and 𝑣𝑥 change
with time, all that is required is either knowledge of the acceleration 𝑎𝑥 (the second derivative
of position with respect to time) or the force 𝐹𝑥 . These are basic Calculus concepts, invented by
Newton to solve problems in mechanics (Tang, ). However, most dynamical systems are much
more complicated than the idealized case mentioned above. Nevertheless, the dynamics of a
system, however complicated, can be predicted using the same deterministic principles of
Newtonian mechanics.
Science has evolved since the time of Newton, and, in recent years, much work has been
done to uncover the fundamental laws of nature at the other end of the physical, yet invisible
spectrum of the subatomic. Experimental evidence suggests that when we look more closely at
matter – into the very atoms that hold macro-level structures together – we observe
phenomena that runs counter to everyday intuition. It turns out that the mathematical
principles that lie at the heart of classical mechanics do not hold in the obscure world of
subatomic particles. Since we have no direct experience with the subatomic world, it is not
easy to determine how subatomic particles behave. If we wish to use mathematics to describe
the motions of particles we can only infer, through experiment, the rules that govern subatomic
realm. What we have discovered, looking back at earlier experiments, is that the physics of
subatomic particles (e.g. electrons) lie outside the scope of classical mechanics, but not entirely.
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Math 5100 Fall/2015
That is, we cannot know for certain the position and momentum of a particle simultaneously at
any given time. Werner Heisenberg’s uncertainty principle, or the Copenhagen interpretation
of quantum mechanics, encapsulates this idea:
One can never know with perfect accuracy both of those
two important factors which determine the movement of
one of the smallest particles—its position and its velocity. It
is impossible to determine accurately both the position and
the direction and speed of a particle at the same instant.
The laws of classical mechanics are constructed in a way such that if the variables, like position
and momentum, of a closed system were given, then they can be known for any other point in
time. This is often referred to as Newtonian mechanics. This implies that you can know, with
certainty, a particle’s position and momentum at any other point in time given the initial
conditions of the dynamic system. This presupposes a notion of continuity in both the physical
and mathematical structures of time, space and matter.
This was a very attractive way to understand physical phenomena, and it seemed to
have worked well on the macroscopic level (predicting large-scale planetary dynamics and, on a
relatively smaller scale, projectile motion). With new empirical discoveries into the nature of
light, scientists felt compelled to abandon determinism. On the microscopic level of atoms and
electrons, classical descriptions had failed in describing a world that had seemed discontinuous
and counterintuitive. More sophisticated forms of mathematics were needed to make sense of
phenomena on the quantum level. As a result, the methods and assumptions of Newtonian
mechanics had to be modified for quantum mechanical models.
Development of Schrodinger’s Equation
Quantum mechanics has its roots in the scientific study of the nature of light. The quest
to understand the nature of light begins with the fundamental old question, what is this world
made of? The ancient Greek philosophers and early scientists pondered these types of
questions – questions that would later summon the fundamental laws of nature. The ancient
Greek scientist-philosophers, such as Leucippus and Democritus had long ago posed atomic
theories of matter. Lucretius’s poem on nature, written in 50 BCE, says:
The bodies themselves are of two kinds: the particles
and complex bodies constructed of many of these;
Which particles are of an invincible hardness. So that
no force can alter or extinguish them.
We learn from the history of science that it is one thing to speculate (i.e. philosophize) about
the nature of matter and quite another to experiment and probe at the phenomena that we
wish to understand. Both scientific and philosophical approaches have been used by the
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Math 5100 Fall/2015
brightest minds to fully grasp the implications of newly proposed theories of matter as they
offer sing posts to the fundamental laws of nature.
Quantum mechanics, as we know it today, originated from Max Plank’s (1900)
experimental explanations for blackbody radiation. Using both Maxwell’s equations and
statistical mechanics, Planck discovered that the energy in an electromagnetic wave of
frequency 𝑓 is quantized according to the formula 𝐸𝐸𝑀 𝑊𝑎𝑣𝑒 = 𝑛ℎ𝑓, where 𝑛 = 1,2,3, … and
ℎ = 𝑃𝑙𝑎𝑛𝑐𝑘 ′ 𝑠𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 = 6.6𝑥10−34 (𝑆𝐼 𝑢𝑛𝑖𝑡𝑠). In fact, Quantum
mechanics is obtained with Planck’s constant ℎ. By this time, it was a
well-accepted fact that particles exhibited wave-like and particle-like
properties, the so-called wave-particle duality in Physics. It is
important to note that Planck’s constant is relatively small for the
macroscopic world, but, in the quantum world of subatomic
particles, ℎ is not that small.
In 1905, Einstein, motivated in part by Planck’s findings, invented the concept of a
photon to explain the photoelectric effect. According to Einstein, the photon was a particle, or
quantum packet of electromagnetic radiation, with energy 𝐸 = ℎ𝑓 = ℏ𝜔, where 𝑓 is the
ℎ
frequency of a photon, ℏ is the reduced Planck’s constant ℏ = 2𝜋 = 1.05𝑥10−34 𝐽 𝑠 =
6.6𝑥10−16 𝑒𝑉 𝑠 (Called “hbar”) and 𝜔 = 2𝜋𝑓 is the angular frequency. And
momentum
𝑝=
𝐸
𝑐
ℎ
= 𝜆 = ℏ𝑘, where 𝑘 = |𝒌| =
2𝜋
𝜆
is the wave number and 𝒌 the wave
vector. This was Einstein’s explanation for the particle nature of the electron.
Later in 1911, Ernest Rutherford proposed a planetary
model of the atom. He showed that the atom consists of a
small, heavy, positively changed nucleus, surrounded by small,
light electrons. There was a small problem with this model.
According to classical theory, the orbit of an electron around
the nucleus must be accelarting and, accroding to Maxwell’s
equations, an accelarating charge must radiate and give off
electromagnetic radiation. And if the charge continuosly gives of energy then it follows that the
electron should spiral and collapse into the nucleus. But this would be absurd because all
matter would then collapse into a soup of electons and hence there would be no structure to
matter.
In 1913 Neils Bohr invented what is known today as the Bohr model of the atom – a
model resembling a miniature solar system. Bohr’s atom is a classical model that treats the
electron as a particle with definite positon and momentum. His model makes two non-classical
(due to discontinuity), assumptions: (1) The angular momentum of the electron is quantized
𝐿 = 𝑛ℏ and (2) The electron orbits are stable and do not radiate, unless the atom emmits or
obsorbs a single photon of energy descibed by ℎ𝑓 = |𝐸𝑓 − 𝐸𝑖 |. The revolutionary idea here is
that Bohr imagned certain allowable sationary orbits and energy levels for electrons. This
Submitted by: Jeffrey Gallo
Math 5100 Fall/2015
implied that atoms, when left alone, are high stable structures. The most important of Bohr’s
assumptions was this: an atomic system cannot exist in a continuum of all mechanically
possible states, but in a series of discrete states (Born, 1954). Bohr’s model had successfully
predicted Hydrogen’s light spectrum. However, Bohr’s model had a problem. It was found that
the ground sate (n=1) of the Hydrogen has angular momentum 𝐿 = 0, but Bohr’s prediction
showed an angular momentum of 𝐿 = (1)ℏ = ℏ, which had contradicted experimeintal
evidence.
In 1923, Louis de Broglie proposed a wave-particle duality for the electron. He claimed
that all matter, not just photons, exhibit the wave-like nature (i.e. matter waves). De Broglie
noticed that Einstein’s energy equation 𝐸 = 𝑚𝑐 2 suggested an equivalence between energy
and matter. He took this as a starting point for his theory. Based on Einstein’s photon theory,
ℎ
the momentum of a particle is 𝑝 = 𝜆. We can re-arrange this equation for the wavelength of a
ℎ
𝐸
particle = 𝑝 . Also, 𝐸 = ℎ𝑓, which can be re-arranged for frequency 𝑓 = ℎ. Without taking
into account relativistic effects, the de Broglie wavelength of a particle
ℎ
with mass 𝑚 and velocity 𝑣 can be obtained from 𝜆: 𝜆 = 𝑚𝑣 =
ℎ
√2𝑚𝐸𝑘
,
1
where the momentum of a particle is 𝑝 = 𝑚𝑣 and 𝐸𝑘 = 2 𝑚𝑣 2 is the
kinetic energy of the particle. Taking into account Einstein’s Special
𝑐
Theory of Relativity, light with energy 𝐸 (𝐸 = ℎ𝑓 = ℎ 𝜆 ) has momentum
𝐸
= 𝑐 , where 𝑐 is the speed of light. Therefore the momentum of a particle
can be expressed as 𝑝 =
𝐸
𝑐
=
ℎ
𝜆
= ℏ𝑘. De Broglie’s hypothesis explained Bohr’s quantization of
energy levels inside the atom, satisfying the condition 𝐿 = 𝑛ℏ. Let’s assume, using Bohr’s
equation, that a integer number of wavelengths fit in one orbital circumference: 𝑛𝜆 = 2𝜋𝑟 →
𝑛𝜆
ℎ
𝑛𝜆
ℎ
𝑛ℎ
ℎ
𝑟 = 2𝜋 and 𝐿 = 𝑟𝑝 = 𝑟 𝜆 = (2𝜋) (𝜆) = 2𝜋 = 𝑛 2𝜋 = 𝑛ℏ.
In the following year, there was
experimental evidence that supported both the photon concept and de Broglie’s new
hypothesis. In the Compton Effect, observed by the American Arthur Holly Compton, gamma
rays were made to collide with electrons and the results from this experiment had confirmed
de Broglie’s mathematical relations. The change in wavelength of the gamma-rays after
colliding with the electrons could only be explained by assuming that gamma rays are photons
ℎ
with energy 𝐸 = ℎ𝑓 and momentum 𝑝 = 𝜆. The key idea that de Broglie had proposed was
that each electron, moving free of force, had its corresponding plane wave of definite
wavelength which was determined by Planck’s constant and its mass. In 1927, American
scientists Davisson and Germer diffracted a beam of electrons from a nickel crystal and
confirmed the momentum for electrons to follow the formula 𝑝 = ℏ𝑘.
Then in 1925, acting as Professor of Physics at Zurich University, Erwin Schrodinger had
presented a talk on de Broglie’s matter waves. Schrodinger, among other scientists, was
convinced that the subatomic world could be understood using a wave equation for matter.
Submitted by: Jeffrey Gallo
Math 5100 Fall/2015
Schrodinger observed that de Broglie had extended Planck’s formula to express a relationship
ℎ
between momentum and wavelength: 𝑝𝛾 = 𝜆 , where momentum 𝑝 is equal to Planck’s
constant ℎ divided by wavelength 𝜆. The dualism inherent in De Broglie’s equation is not so
obvious. Schrodinger was aware that de Broglie had connected momentum and energy (pointmechanical quantities) with frequency and wavelength (properties of a continuum) (Wick, 24).
Schrodinger came to the conclusion that matter might, in fact, be built out of waves. All
Schrodinger had to do now was write a mathematical wave equation to determine the time
evolution of an electron’s position in space. In 1927, Schrodinger proposed the first wave
equation that would confirm de Broglie’s matter waves.
Derivation of Schrodinger’s Equation
How did Schrodinger derive his wave equation? There is reason to believe that
Schrodinger had followed three main criteria for deriving his wave equation: (1) De Broglie’s
matter wave hypothesis, (2) the law of conservation of energy, (3) classical plane wave
equation (Huang, 2012).
Schrodinger first assumed that a free particle with potential energy 𝑉(𝑥) = 0 is a plane
wave described by:
, which can be written as,
And Euler’s relation states that,
Schrodinger chose to work with a complex wave function purely for mathematical convenience.
He planned to take the real part of his wave function Ψ to get the physically real matter wave
(Dubson, 2008). Schrodinger knew that in a conservative field, the total mechanical energy
𝐸𝑡𝑜𝑡𝑎𝑙 (𝐸𝑡𝑜𝑡𝑎𝑙 = 𝐸𝐾𝑖𝑛𝑒𝑡𝑖𝑐 + 𝐸𝑃𝑜𝑡𝑒𝑛𝑡𝑖𝑎𝑙 ) of a material particle is conserved (Huang, ). Since the
potential energy of the particle is considered to be zero, it will display only kinetic energy
1
𝑝2
2
2𝑚
according to 𝐸 = 𝑚𝑣 2 =
re-written as 𝐸 = ℎ𝑓 = ℏ𝜔 =
. Considering de Broglie’s mathematical relations, energy can be
(ℏ𝑘)2
2𝑚
. Schrodinger main goal was to find a wave equation that
can reproduce this energy relationship. He does the following:
and taking the first derivative of the wave function with respect to time,
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Math 5100 Fall/2015
and later taking the second derivative of the wave function,
So the trial equation yields:
(*)
which leads to:
→
Schrodinger had arrived at an equation that yielded de Broglie’s matter wave equation. He
knew that in order to describe a particle with both kinetic and potential energy, he had to add a
potential energy term 𝑉(𝑥) in his trial equation (*), yielding the time-dependent Schrodinger
equation:
(**)
Schrodinger’s wave equation is, fundamentally, an energy equation in disguise. Schrodinger’s
time-dependent equation (**), when read left to right, says: 𝐸𝑛𝑒𝑟𝑔𝑦 = 𝑘𝑖𝑛𝑒𝑡𝑖𝑐 𝑒𝑛𝑒𝑟𝑔𝑦 +
𝑝𝑜𝑡𝑒𝑛𝑡𝑖𝑎𝑙 𝑒𝑛𝑒𝑟𝑔𝑦 → 𝐸 = 𝐾𝐸 + 𝑃𝐸. Considering that a particles has a frequency 𝑓 and
𝑝2
wavelength λ, within a potential 𝑉, the equation predicts that 𝐸𝑡𝑜𝑡𝑎𝑙 = 𝐾𝐸 + 𝑃𝐸 = 2𝑚 + 𝑉.
Considering that 𝐸 = ℎ𝑓 = ℏ𝜔, according to de Broglie, Schrodinger’s wave equation translates
into: 𝐸 = ℏ𝜔 =
(ℏ𝑘)2
2𝑚
+𝑉 =
ℏ2 𝑘 2
2𝑚
+ 𝑉.
Schrodinger’s equation is a differential wave equation that describes material processes
as wave processes (Renn, 2013) where, instead of solving the Newtonian equations of motion,
we only need to solve the differential wave equation to find the wave packet that propagates
through space similar to that of a classical particle (Tsaparlis, 2001).
Excited with his new equation, Schrodinger tested his theory with the energies of the
Hydrogen atom. He found that his equation matched experimental readings of Hydrogen’s
energy levels. Schrodinger’s equation was so insightful that the famous physicist, Paul Dirac,
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Math 5100 Fall/2015
once said, “Schrodinger’s equation accounts for much of physics and all of chemistry” (Dubson,
2008). His wave equation is probably the most important equation of the 20th century.
Without it there would not have been as many technological advances that we see today.
Wave phenomena such as sound waves or light waves had been known for a long time
prior to Schrodinger’s quantum wave theory. The new idea that Schrodinger had proposed was
that matter allowed itself to be described as a wave phenomenon. The implication is that
different states of matter can overlap each other like vibrational states, and that matter can
exhibit diffraction (i.e. interference patterns), just like light could. For a deeper understanding
of Schrodinger’s wave equation, let’s look at Max Born’s interpretation of this famous formula.
Max Born’s Interpretation of Schrodinger’s Wave Function
Schrodinger had shared the Nobel prize with Paul Dirac for their formulation of
Quantum mechanics. However, Schrodinger himself was puzzled by his wave function. What
was the physical meaning of the wave function 𝛹(𝑥, 𝑡)? After all, a particle is localized at the
point, whereas the wave function in spread out in space. How can a wave function be said to
describe the state of a particle? Max Born held the answers in his palm. He proposed that the
wave equation was a probability distribution of an electron’s location in space. Let’s first look
at how Schrodinger understood his own wave theory.
Schrodinger had conceptualized the fundamental nature of matter as a wave packet
that carried energy. But this interpretation did not measure up with a series of experimental
results. What troubled Schrodinger was the result that a particle with an extended wave
function could only be found at one particular spot (𝑥 + 𝑑𝑥) once a measurement was made.
This is was referred to as the collapse of the wave function.
Schrodinger hinted at a probabilistic interpretation of his wave function but it was Max
Born who had advanced the concept. The reason for this is that Schrodinger thought of his
waves as a spread-out electron. In Schrodinger’s fourth major paper in quantum mechanics, he
writes:
My procedure is equivalent to the following interpretation,
which better shows the true significance of the wave function.
The square of the length of the wave variable is a kind of weight
function in the configuration space of the system…each pointmechanical configuration enters with a certain weight in the true
wave- mechanical configuration…If you like paradoxes, you can
say that the system is, as it were, simultaneously in all kinetically
conceivable positions, but not “equally strong” in all of them.
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Math 5100 Fall/2015
What did Schrodinger mean by “weights” and “equally strong”? It seems as though Schrodinger
was hinting at a probability distribution for the location of an electron, but he fails to state this.
Instead, he thinks that the electron is a wave in a strict sense.
In 1926, Max Born proposed a new interpretation of Schrodinger’s wave equation: (1)
The wave in Schrodinger’s equation is not a real wave but a wave probability. (2) The electron is
real but its position at any time is unknown. Instead, the square length of the wave function at
each point in space yields the probability that an electron is within a small volume (𝑥 + 𝑑𝑥, 𝑦 +
𝑑𝑦, 𝑧 + 𝑑𝑧) centered at that point. (3) Schrodinger’s wave equation is a deterministic wave
function that continuously evolves with time, but does not describe Bohr’s instantaneous
“quantum jumps” (jumps that occur unpredictably and discontinuously) from one quantum
state to another. (4) Following a quantum jump (by way of emission or absorption)
Schrodinger’s wave equation instantly collapses to one that describes the new state of the
particle (Wick, 1995).
Born’s statistical interpretation of the wave function, mathematically expressed as
𝑡ℎ𝑒 𝑝𝑟𝑜𝑏𝑎𝑏𝑙𝑖𝑡𝑦 𝑜𝑓 𝑓𝑖𝑛𝑑𝑖𝑛𝑔 𝑡ℎ𝑒 𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛
|ᴪ(𝑥, 𝑡) |2 𝑑𝑥 = {
}. Consider the following
𝑥 𝑎𝑛𝑑 (𝑥 + 𝑑𝑥), 𝑎𝑡 𝑡𝑖𝑚𝑒 𝑡.
wave function:
Once a measurement is made, it would seem more likely, looking at the square amplitude, to
find the electron in the vicinity of A, and unlikely in the vicinities of B and C. The probabilistic
interpretation suggests that we cannot know for certain the position of an electron at any given
point in time, even if we have the exact wave function for a particle! All quantum mechanics
can offer is statistical information about possible results (Griffiths, 1995). Now the question
becomes: is this a fundamental feature of nature, maybe a deficiency in the theory, or errors in
the measuring apparatus? Another controversial question that can be asked is this: if I happen
to find the position of a particle in space, where was the particle immediately before the
measurement was taken? It is hard to determine the location of a particle just before the
measurement because the wave function collapses a particular point in space at the moment
when the measurement is taken:
Submitted by: Jeffrey Gallo
Math 5100 Fall/2015
The particle appears at C but it is impossible to determine where the particle was located just
before the measurement. Hence there is reason to think that the wave function 𝛹(𝑥, 𝑡)
represents the state of a dynamical system (that of the electron).
Born’s probabilistic interpretation was quite different than Schrodinger’s original
intention. Born proposed that the wave function acted as an information wave – it held
information about the probability of the results of measurement, but did not carry any
information regarding the physical picture of what was really going on (Dubson, 2008). This
was later considered as the “Copenhagen interpretation” of quantum mechanics (since it was
developed at Neils Bohr’s research institute in Copenhagen). Bohr, Heisenberg and others
argued that it was unimportant to find out what was really going on because they knew that
the only way that we can understand the invisible realms is by applying the same principles that
govern classical mechanics – after all, it was the only way we could understand macro level
phenomena. And Bohr, along with other colleagues, realized that a classical lens could only
limit our knowledge of the subatomic world. All we can really know about the quantum world
are the results made with our macroscopic instruments. Einstein, de Broglie, Schrodinger and
other notables had felt uneasy with this claim. They believed that the quantum world was not
as obscure as Bohr’s school made it out to be. They believed that mathematics could describe
what was happening in the smallest dynamical levels and that the very nature of electrons
could be understood within a classical, deterministic framework. Einstein expresses this idea in
his comment: “God does not play dice.”
Contrary to Born’s perspective, Schrodinger had imagined a pure wave theory for the
electron, while falling short of a strictly probabilistic interpretation of his equation. Schrodinger
had visualized the electron as a wave packet that would travel through space without
dispersing, but experiment had shown that the electron had well defined tracks in cloud
chamber pictures/measurements. It was also shown that any size wave packet could disperse
to a size larger than a particle’s track and this had contradicted experimental observations. The
particle nature of the electron refused to go away.
Submitted by: Jeffrey Gallo
Math 5100 Fall/2015
Complex Numbers and the Wave Function
The equations of physics that deal with forces positions momentum, potentials, electric
and magnetic fields are all real mathematical quantities and the equations that describe them,
like Newton’s laws, Maxwell’s equations, etc. also involve real quantities. The mathematics of
the quantum world is a little different. The equations of quantum physics have a different
flavor in the sense that they express the fundamental laws of nature with more sophisticated
mathematics. Factors of 𝑖 = √−1 show up in the groundbreaking equations of quantum
mechanics, they are essential elements in both Dirac’s relativistic wave equation of an electron
i   (i    m ) and in Schrodinger’s famous wave equation of an electron
t
i

  r , t   H   r , t  . It is important to notice that Schrodinger’s wave function  ( r , t )
t
is
complex valued. How is it the case that an imaginary component shows up in a fundamental
law of nature? A common misconception about complex numbers is that they don’t represent
anything real in space and time. Schrodinger chose complex numbers as a useful tool for
combining two real equations into a single complex equation. The appearance of 𝑖 in quantum
mechanics is a mathematical convenience rather than a necessity.
Conclusion
What makes Quantum mechanics so revolutionary? Quantum mechanics was developed
to better understand the dualistic nature of light. In the early 20 th century, scientists were well
aware that the electron had exhibited both particle and a wave-like properties. Max Planck’s
studies on blackbody radiation has suggested that physical quantities, such as angular
momentum, electric charge and energy are all quantized phenomena. The implication of
quantization is that there is a smallest unit of energy for all of these quantities. What is
interesting and often misunderstood, however, is the behavior of these individual quanta. It
was soon realized that the behavior of an electron does not correspond to anything else we are
normally accustomed to in the macroscopic world.
Science has come a long way, from early atomic theories of matter to our modern day
notions of subatomic dynamics. With the discovery of quantized atomic energy in the early
1900s, a revolutionary shift in physics had occurred, namely the shift from classical mechanics
to modern, quantum mechanics. According to modern quantum physics, the concept of a
“particle” is somewhat outdated. Today, quantum physics deals with the mathematics of
quantum fields, where particles are only the local manifestations of this field. According to
Heisenberg, classical concepts such as momentum, position, time and energy had retained their
meaning in the new quantum mechanical descriptions, but certain pairs of quantities, such as
momentum and position, had lost their certainty. This was a very strange observation and one
Submitted by: Jeffrey Gallo
Math 5100 Fall/2015
that would challenge the entire scientific community. Neils Bohr, representing the Copenhagen
school of physics, proposed that the very words and macro-level methods that physicists use to
describe reality constrain their knowledge of it.
Schrodinger’s wave equation is the most important equation of modern physics.
Schrodinger’s equation is presented as an analogue of Newton’s second law of motion for a
classical system. It is commonly understood that Schrodinger’s wave equation does not suggest
a wave behavior for the electron. In fact, Schrodinger’s time-dependent wave function
behaves, in a qualitative sense, much like a wave. However, the wave function does not
represent a wave in physical space. Born’s interpretation of Schrodinger’s wave function came
more close to the true meaning of Schrodinger’s theory. He had proposed that if the square
modulus of the wave function is taken that one can find the probability distribution of locating
an electron at a particular point in space-time.
Re-imagining the subatomic as a collection of wave probabilities had changed the way in
which physicists view matter. The visual models of science had proven be limitations to our
understanding of Nature, and the only means of escaping this limitation was through
mathematical abstraction. This is the true genius of mathematics, it allows us to extend our
senses into realms unknown. In the case of particle physics, quantum mathematics is the key to
unlocking subatomic phenomena. The real mystery in this story is how the deterministic nature
of classical mechanics continues to play an important role in quantum physics today.
Submitted by: Jeffrey Gallo
Math 5100 Fall/2015
Sources Consulted
Books
1. Wick, David. The Infamous Boundary: Seven Decades of Controversy in Quantum
Physics. Birkhauser Boston, 1195.
2. Rae, Alastair. Quantum Mechanics 5th Edition. Taylor and Francis Group, LLC,
2008.
3. Schrodinger, Erwin. Wave Mechanics. Chelsea Publishing Company, New York.
1978.
4. Phillips, A.C. Introduction to Quantum Mechanics. Wiley, 2003.
Online Article/Websites
1. Born, Max. The Statistical Interpretation of Quantum Mechanics.
http://www.nobelprize.org/nobel_prizes/physics/laureates/1954/bornlecture.pdf.1954 (Lecture).
2. Dubson,Michael.A Brief History of Modern Physics and the Development of the
Schrodinger Equation.
http://www.colorado.edu/physics/phys3220/phys3220_fa08/notes/notes/l
ecturenotes_1_part2.pdf, 2008.
3. Freiberger, Marianne. Schrodinger Equation – What is it?
https://plus.maths.org/content/schrodinger-1
4. Ibid. Schrodinger’s Equation in Action.
https://plus.maths.org/content/schrodingers-equation-action
5. Ibid. Schrodinger’s Equation – What Does it Mean?
https://plus.maths.org/content/schrodingers-equation-what-does-it-mean
Submitted by: Jeffrey Gallo
Math 5100 Fall/2015
6. Huang, Xiuqing. How did Schrodinger Obtain the Schrodinger Equation?
http://vixra.org/pdf/1206.0055v2.pdf, 2001
7. Renn, Jugen. Schrodinger and the Genesis of Wave Mechanics
https://www.mpiwg-berlin.mpg.de/Preprints/P437.PDF, 2013.
8. Tang ,C. L. Classical Mechanics vs. Quantum Mechanics.
http://assets.cambridge.org/97805218/29526/excerpt/9780521829526_exc
erpt.pdf.
9. Tsaparlis, Georgios. Towards a Meaningful Introduction to the Schrodinger
equation Through Historical and Heuristic Approaches.
http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&ved
=0ahUKEwj49pavyNTJAhXLWD4KHcpqCc8QFggdMAA&url=http%3A%2F%2Fu
sers.uoi.gr%2Fgtseper%2FCVenglish.pdf&usg=AFQjCNGnuW2BIh71UEUMaNS
TgBTiV8J7Wg&sig2=4wA6YVupoOGPGu-5c4cKTg, 2001.