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 20th 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 (t)dt

. All other dynamic properties of the system,

like momentum px( t)=mdxdt

=mv x, kinetic energy Ek=T=mv x

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 F x acting on a particle is proportional to the acceleration ax=d2 xd t 2

, where the constant is

the particle’s mass m. This can be written as: F x=max=md2 xd t 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 F x. 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 F x. 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

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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. 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.

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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 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=6.6 x 10−34(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= ωℏ , where f is the frequency of a photon, ℏ is the reduced Planck’s constant

ℏ= h2 π

=1.05x 10−34 J s=6.6 x10−16eV s(Called “hbar”) and ω=2πf is the angular frequency.

And momentump=Ec=hλ= kℏ , 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

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(due to discontinuity), assumptions: (1) The angular momentum of the electron is quantized 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=|E f−Ei|. The revolutionary idea here is that Bohr imagned certain allowable sationary orbits and energy levels for electrons. This 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=mc2 suggested an equivalence between energy and matter. He took this as a starting point for his theory. Based on Einstein’s photon theory, the

momentum of a particle is p=hλ . We can re-arrange this equation for the wavelength of a

particle ¿hp . Also, E=hf , which can be re-arranged for frequency f=

Eh .

Without taking into account relativistic effects, the de Broglie wavelength of a particle with mass m and velocity v can be obtained from λ:

λ= hmv

= h√2m Ek , where the momentum of a particle is p=mv and

Ek=12mv2is the kinetic energy of the particle. Taking into account

Einstein’s Special Theory of Relativity, light with energy E (E=hf=h cλ )

has momentum ¿Ec , where cis the speed of light. Therefore the momentum of a particle can

be expressed as p=Ec=hλ= kℏ . 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:

n λ=2πr→r= nλ2π and L=rp=r

hλ=( nλ2π )( hλ )= nh

2π=n h

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

with energy E=hf and momentum p=hλ . The key idea that de Broglie had proposed was that

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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= kℏ .

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. Schrodinger observed that de Broglie had extended Planck’s formula to express a relationship

between momentum and wavelength: pγ=hλ , 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 (point-mechanical 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

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according to E=12mv2= p2

2m. Considering de Broglie’s mathematical relations, energy can be

re-written as E=hf= ωℏ =( kℏ )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,

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 wavelength λ, within a potential V , the equation predicts that

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Etotal=KE+PE= p2

2m+V . Considering that E=hf= ωℏ , according to de Broglie, Schrodinger’s

wave equation translates into: E= ωℏ =( kℏ )2

2m+V=ℏ2k 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, 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.

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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 point-mechanical 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.

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

|ᴪ (x , t)|2dx={the probablity of findingthe particle betweenx∧( x+dx ) , at time t . }. Consider the following wave

function:

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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:

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

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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.

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 ∂ψ

∂ t=(−iα⋅∇+mβ )ψ

and in Schrodinger’s famous wave equation of an electron

. It is important to notice that Schrodinger’s wave function 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 20th 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

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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 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.

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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/born-lecture.pdf.1954 (Lecture).

2. Dubson,Michael.A Brief History of Modern Physics and the Development of the Schrodinger Equation.

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http://www.colorado.edu/physics/phys3220/phys3220_fa08/notes/notes/lecturenotes_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

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_excerpt.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%2Fusers.uoi.gr%2Fgtseper%2FCVenglish.pdf&usg=AFQjCNGnuW2BIh71UEUMaNSTgBTiV8J7Wg&sig2=4wA6YVupoOGPGu-5c4cKTg, 2001.