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Extended Essay on Mathematics

Quantum Theory: The Schrodinger Equation

Research Question:

How did Schrodinger's mathematical wave equation contribute to the current quantum mechanics' model of waves?

Subject: Mathematics

Word Count: 3771

Table of Contents

I. Introduction 3

II. Introduction to Quantum Theory 4

III. The Schrodinger Equation 7

IV. Interpreting the Schrodinger Equation 10

V. Particle in a 1-Dimensional Box 12

VI. Different Variations of the Schrodinger Equation 18

VII. Linearity and Time in Quantum Mechanics 20

VIII. Schrodinger vs. Einstein 23

IX. Space and Time in the Quantum World 24

X. Conclusion 26

Bibliography 27

I. Introduction:

Humans have been fascinated about the world and its components for centuries. More and more scientists and physicists have stood up and have made huge contributions to help explain to society how and why are world is what it is. But only a few have made it into the history and science books that children study in school. One of these contributors to the world of physics is none other than Erwin Schrodinger, an Austrian physicist who made revolutionary findings about atomic behavior and furthering quantum theory built up upon past physicists such as Ernest Rutherford and Niels Bohr. While it is still a mysterious topic that is being heavily studied every day, Schrodinger created a mathematical equation that describes the probability of an electron being found in an energy level in an atom – the Schrodinger Equation. Through extensive research and the utilization of a multitude of college lectures and reliable scientific websites, we will introduce basic quantum theory and explore answers to the following question: How did Schrodinger's mathematical wave equation contribute to the current quantum mechanics' model of waves?

II. Introduction to Quantum Theory:

Before we dive into the Schrodinger equation and how it contributed to the quantum model of waves as well as quantum theory, it would be best to learn what exactly quantum theory and the history behind it. Going back to the 1900s, Max Planck hypothesized that all energy is quantized and that light exhibits wave-particle duality. By saying something like energy can be “quantized”, it is simply referring to the existence of electrons on specific energy levels. More importantly, this means that energy can only be absorbed and emitted with specific values, and not just any possible value. The quantum model below shows these quantum levels that exist at the subatomic level.

http://www.thestargarden.co.uk/Bohrs-atom.html

Planck also came up with an equation that is still constantly used today in modern physics called the Planck postulate:

E = hv (or) E = hf

This equation restates that “photon energy (E) is proportional to its frequency (v/f) multiplied by Planck’s constant (h) which is equal to 6.626 x 10-34 Js” (Weisstein 2). This is important because if this quantum energy has a frequency, it also means that it must have a wavelength:

E = hf

f =

E =

This postulation of wave-duality for all particles was hypothesized by the 20th century French physicist Louis de Broglie. De Broglie extended the theorem by stating all matter also possessed wave-duality attributes, meaning that all matter is the world possesses a wavelength of sorts. The reason humans don’t see wave-like characteristics in everyday objects in the world is due to inverse relationship between wavelength (Λ) and mass (m).

E = mc2

E = hc/Λ

Setting these 2 formulas equal to each other and solving for Λ:

mc2 = hc/Λ

Λ = h/mc

Any object bigger than a molecule have such large masses that their wavelength is negligible, but an electron’s wavelength is not due to its mass being so tiny. Therefore, electrons are viewed both as a particle as well as a wave, but the wave characteristics of an electron are not linear. The kind of wave that an electron has is called a circular standing wave, which means that in order for this kind of wave to exist, it must have a certain number of wavelengths. Referring back to the quantization of energy, “an electron must have a discrete amount of energy that it can absorb and emit in the quantum model” (Blamire 112). When a photon hits an electron, this electron increases in the number of wavelengths for a short duration, which leads the electron to a higher energy level in the atom. After this discovery, scientists and physicists set out to try and find a mathematical equation to help describe this unusual movement of subatomic particles, which is why the Schrodinger equation is one of the biggest mathematical and scientific breakthroughs in history.

III. The Schrodinger Equation

Similar in classical mechanics, this formula is used to predict certain physical outcomes in a given system:

F = ma

As well as another formula that is used to determine the energy for an object in a given system:

Etotal = Ekinetic + Epotential

While the Schrodinger equation is written in different ways for certain reason, this form is consistent with the conservation of energy and is considered the “F=ma” and of quantum models:

EΨ(x) = + VΨ(x)

The V in VΨ(x) is sometimes written as the potential energy of a function. The Ψ is the Greek symbol psi and known as the wave function, “which describes the probability of the quantum state of a particle in a system” (Cresser 171). So, if VΨ(x) is the potential energy of an electron, then that means = must be the kinetic energy of the system. Deriving the formula for kinetic energy from the classical physics energy formula will give us this result:

Et = Ek + Ep

Et = mv2 + V

For substituting momentum (p) in the equation:

p = mv

E = + V

The general form of the wave equation where e is a mathematical constant, k is the wave vector, ω is the angular frequency (since this is a three-dimensional plane/system):

Ψ =

Differentiating the equation gives us:

= ik = ikΨ

We differentiate here because the derivative of the wave function can help describe the change in the state of the system as it evolves over time.

= i2k2 = -k2Ψ

The De Broglie relationship, where ħ is the reduced Planck’s constant, is:

k =

And by substituting k into -k2Ψ:

= Ψ

Multiplying on both sides:

-ħ = p2Ψ

Now multiplying Ψ into the energy formula where E = + V, p2 will turn into p2Ψ and then setting that term equal to -ħ :

EΨ = + VΨ

EΨ = + VΨ

IV. Interpreting the Schrodinger Equation

The Schrodinger equation is not easy to interpret, but by breaking it down by comparing it to the classical physics energy equation, it might help us understand what this equation’s purpose is:

Et = Ek + Ep

EΨ = + VΨ

I put the 2 equations on top of each other to help with picturing the distinct similarities they have. Notice how both equations solve for the total energy of something, that something being in this case a particle in space. Also take note of how both equation of have 2 terms that add up to the total energy. Both of the 2 terms in the equations parallel to kinetic and potential energy of a system.

The wave function Ψ describes the probability of the quantum state of a particle in a system. In a 1-dimensional spinless particle, it’s state in the system is described by the wave function as:

Ψ(x,t)

Where x is the position and t is the time. Since the wave function only describes the probability of the position of the particle, it has all possibilities of being anywhere in the system at a given time. The probability density is another way of describing the relative likelihood that a value at any given sample in a space will be there. Similarly, in quantum mechanics, this function will give the probability amplitude of the particle from the wave function where p(x) is a probability density function:

|Ψ(x,t)|2 = Ψ(x,t)* Ψ(x,t) = p(x,t)

The probability that this particle will be at position x in the interval a ≤ x ≤ b is in the integral of the density:

Pa≤x≤b(t) =

A probability is a real number between 0 and 1, 0 meaning it has a 0% chance of happening and 1 meaning it has a 100% chance of happening. Since there is a 100% that if the particle is observed and measured, it will be always be somewhere. This also means that a measurement of x must give a value between -∞ and +∞, and P-∞

Take this graph to be a 1-dimensional space for example. By taking the integral of the equation, we can therefore say that there is a 100% chance that a particle will be found in the area under the graph (space) from intervals a ≤ x ≤ b.

V. Particle in a 1-Dimensional Box

Imagine an electron inside a box that is continually bouncing in between the walls of that box. But this box is 1-dimensional, and the position of this electron falls on the x-axis. The left wall is at x = 0 and the right wall is at x = L. The electron has zero potential energy inside the box since the walls are the forces causing the particle to bounce back and forth, so:

V(x,t) = 0 for 0 < x < L

There are infinitely large forces causing the electron to be bounced back and forth inside the box from the walls, therefore:

V(x,t) = ∞ for 0 ≥ x ≥ L

Since an electron can only have discrete levels of energy, we have to find an equation that will give us only these discrete levels. We will refer to this picture again later.

We are now going to introduce the Hamiltonian operator:

Ĥ = T̂ + V̂

where V̂ = V = V(x,t) is the potential energy operator and T̂ = = ∇2 is the kinetic energy operator. Notice how these terms have the name “operator”, which means that they won’t always give an exact value or numeric answer. Rather by finding out the operators in an equation, it will help us determine the end result, which in this case is the energy of a particle. Also notice how both the Hamiltonian operator and the Schrodinger equation have paralleling terms:

EΨ(x) = + VΨ(x)

Ĥ = T̂ + V̂

If we set these equal to each other:

Ĥ = EΨ

Putting the Schrodinger wave equation in operator form:

Ĥ = = + V

Notice how in this equation, we are going to use Planck’s constant and not h-bar. The reason for this is because whenever we typically use h-bar, our problem usually includes some sort of angular momentum or angular frequency. But in a 1-dimensional box, this won’t be necessary. Also, H-bar is the equivalent of writing .

And then putting that value into our first equation:

EΨ = + VΨ

Subtract EΨ on both sides:

0 = + VΨ – EΨ

0 = + (E – V)Ψ

Multiplying out by :

0 = + (E – V)Ψ

Since there is an infinite amount of potential energy outside the box (V = ∞)

0 = + (E – ∞)Ψ

And since there is no amount of potential energy inside the box (V = 0)

0 = + (E – 0)Ψ

0 = + EΨ

We can replace the E in the equation with k2 where k2 is equal to and ħ is equal to .

0 = + k2Ψ

From this point, we have to know that the general solution to this equation is written as:

Ψ(x) = Asin(kx) + Bcos(kx)

A and B in this equation are just arbitrary constants used to solve for Ψ(x). Recall that in this 1-dimensional box, when x = 0 (the very left wall of the box), Ψ0 = 0:

Ψ(0) = Asin(0) + Bcos(0)

For sin(0) = 0 and cos(0) = 1:

0 = 0 + B

B = 0

Now recall that when x = L (the very right wall of the box), ΨL = 0

0 = Asin(kL) + 0

0 = Asin(kL)

kL = nπ

k =

But k2 = , therefore by combining both equations to equal k2:

= x 4π2

There is no h-bar in the equation above because by squaring Plank’s constant in the denominator, it is the same as x 4π2 if ħ = .

Solving for E in = x 4π2, we get:

E =

The energy of a particle is quantized, “meaning that it can absorb and emit only discrete levels of energy” (Blair 55). Hence why we have the n variable in our problem.

For example, when the particle is in the second shell (n = 2):

E2 =

Earlier we figured out that Ψ(x) = Asin(kx), therefore:

Ψ(x) = Asin(x)

The probability of finding the particle in a space between x and x + dx is given by Ψ2(x)dx:

Remember that in the integral , there was a time variable represented as t, and by setting the integral equal to 1, we are saying that there is a 100% chance that the particle is somewhere in the space if we choose the observe/measure it. The difference in the 2 integrals is that one is time-dependent and the other is time-independent:

A2

A2 =

Ψ(x) = sin (

And this is the equation for the wave function for a particle in a 1-dimensional box.

VI. Different variations of the Schrodinger Equation

Before we dive into why and how the Schrodinger Equation was such a huge breakthrough in quantum physics, we have to realize that there are different variations of the equation. The most common form of the time-dependent equation is given by the Hamiltonian operation:

ĤΨ = iħΨ

Ĥ|Ψ(r, t)〉= iħ|Ψ(r, t)〉

We also mentioned the time-independent equation:

0 = + (E – V)Ψ

Ĥ|Ψ〉 = E|Ψ〉

The reason the Hamiltonian is used here is because of how this operator provides a spectrum of outcomes when measuring the total energy of a system, “the sum of the kinetic and potential energies” (Branson 1).

Here is an example of a three-dimensional Schrodinger Equation:

iħΨ = ∇2Ψ + VΨ

There is even a variation of the equation for a single nonrelativistic particle known as the “time-dependent Schrodinger equation in position basis”:

iħΨ(r,t) = [∇2 + V(r,t)] Ψ(r,t)

Where μ is the particle’s “reduced mass”, V is the particle’s potential energy, and ∇2 is the Laplacian differential operator.

All of these variations of the Schrodinger equation help in understanding the different kinds of scenarios and systems this mathematical concept is practical in. Whether time is considered or not, or if electron spin, fine structure, magnetic moments, or antimatter have to be taken into account (relativistic quantum mechanics), another brilliance behind Erwin Schroinger’s equation.

VII. Linearity and Time in Quantum Mechanics:

A principle characteristic of quantum theory is linearity. Linearity is used throughout the quantum world and throughout Schrodinger’s Equation. An example of this is superposition. “The superposition principle states that all linear systems that contain two or more inputs will produce the sum of its outputs” (Smith III 6-7). In other words, if input A produces X and input B produces Y, the product of the sum of inputs (A + B) will be the sum of the outputs (X + Y). To put it simply, the art of being superposed is the idea that two or more states of being exist instantaneously. Quantum superposition is present in Schrodinger’s equation:

Ψ(r, t) =

A linear combination of plane waves can be allowed in quantum physics if the equation is linear. Similar to classical physics, quantum states can be superposed to produce an allowed quantum state and conversely, any quantum state could be viewed as the sum of two or more distinct states. We talked about how probability is all quantum equations calculate, as there is no real or true value that these equations tell us since it is all probabilistic. For the probability of any state, and given any two different states, there is another state which is partly this and partly that, with positive real number coefficients, the probabilities, which say how much of each there is. For example, if we are given a probability distribution for where a particle is:

ρ is the probability density function. But in quantum mechanics, this number can be negative. In a three-state probability system, the chances of these states can be multiplied by a positive number to make their sum equal to 1:

x|1〉 + y|2〉 + z|3〉

x + y + z = 1

Where |1〉 would be the Dirac notation for the quantum state that always covert to 1. However, the same system in quantum mechanics would look a little different since there the probability density function value can be positive or negative, giving twice as many quantities:

A|1〉 + B|2〉 + C|3〉 = (Ar + iAi)|1〉 + (Br + iBi)|2〉 + (Cr + iCi)|3〉

+ + + + + = 1

The wave-function in Schrodinger’s equation is the probability of a certain particle being in a space. This means that the quantum wave-function could be a superposition of states that add up to one specific state, and vice versa. This is why the geometry of space is a high-dimensional sphere represented by the equation above since the sum of the squares is equal to 1. The same way a sphere has symmetry in different coordinate systems, quantum superposition has varying bases in which it could be symmetrical. Going back to the role of operations such as the wave-function (Ψ), the value/quantity that Schrodinger’s equation gives us is the absolute square of the coefficient of the superposition. If the superposition principle states that multiple solutions to a liner equation is a solution of itself. And since the Schrodinger Equation is a linear equation, then it would follow that:

Ψ = c1f1 + c2f2 + c3f3

Where c is a coefficient and f are different quantum states. A mixture of quantum states is equal to another quantum state. You could also interpret this theorem by imagining 1 input leading to multiple outputs. This idea of quantum superposition was one of many things that inflicted confusion on in the science world at the time, and was also just a fraction of what Schrodinger provided to science. In fact, the current quantum mechanics model was a product of the contrasting ideas and theories between Schrodinger and one of the most notable scientists ever: Albert Einstein.

VIII. Schrodinger vs Einstein

It’s also no surprise that Schrodinger’s findings were also very unusual and not widely accepted in the science world by many. The most notable physicist at the time who was not in tune with Quantum Theory was Albert Einstein, who postulated an idea of how the world works that clashed heads with Schrodinger’s at the time – the Theory of Relativity. Einstein’s famous equation, E = mc2, translates how any object with mass has an equal amount of energy. The energy of an object is found by its mass multiplied by the speed of light (3.00 x 108 m/s) squared. The Theory of Relativity and Quantum Theory are the two sides of the physics coin, where one describes the world at the subatomic level, and the other at astronomical sizes. The reason Einstein’s equation doesn’t align well with Quantum Theory is because General Relativity is all deterministic, whereas Quantum Theory deals is all probabilistic. The wave-equation and the entirety of Schrodinger’s Equation tells us probable values about subatomic particles and their nature’s, where General Relativity and Special Relativity discusses a non-linear explanation of nature itself. The idea to create one theory that could encompass both of these polar ideas was needed. Oddly enough, the math behind the combination of both theories actually worked to create an equation that conformed to both Einstein and Schrodinger’s ideas. So how could Einstein’s explanation of space and gravity implement Schrodinger’s quantum mechanics?

IX. Space and Time in the Quantum World

If there was any way to combine General and Special Relativity into the quantum world, then we would have to use the general form of the time-dependent Schrodinger Equation:

ĤΨ = iħΨ

Combining multiple equations into one relativistic wave equation was difficult, drawing in multiple scientists’ equations for the purpose of combining special relativity and quantum mechanics. The special relativistic energy-momentum relation of an object with mass m at rest, energy E, and with a 3-momentum p with its value in terms of p = is presented as:

E2 = c2p * p + (mc2)2

Using the energy and momentum operators, respectively:

Ê = iħ

p = -iħ∇

A partial differential equation consistent with the energy–momentum relation and solved for the wave-function (Ψ) to predict a particle in a system could construct the relativistic wave equation. Using the relativistic energy-momentum relation that we got earlier, we get the Klein-Gordon equation:

Ψ - ∇2Ψ + Ψ = 0

And substituting E for Einstein’s energy equation E = mc2:

Ψ - ∇2Ψ + Ψ = 0

Where both equations are consistent with the use of conservation of energy in the Schrodinger Equation. The Klein-Gordon equation applies to massive spinless particles, which aligns with both theories of relativity and quantum mechanics. Schrodinger’s math wasn’t necessarily wrong in the eyes of Einstein, but rather the thought that everything in the world was based off probability dumbfounded Einstein. Out of Schrodinger and Einstein’s equations was born new relativistic quantum mechanical equations that combined special relativity and quantum theory simultaneously.

X. Conclusion

Going back to the question: “How did Schrodinger's mathematical wave equation contribute to the current quantum mechanics' model of waves?” From a mathematical standpoint, Schrodinger’s use of the wave-function and conservation of energy to introduce a probabilistic viewpoint on atomic structure is revolutionary. With the help of past physicist such as Niels Bohr, Albert Einstein, and Louis de Broglie, Erwin Schrodinger changed the game of quantum physics by addressing a complex and intricate theory that used probability to find an electron at a certain energy level in an atom. Whether it be in a 1-dimensional or a 3-dimensional box, the profound mathematics that back up the Schrodinger Equation help conclude that probability can even be found with the smallest particles on Earth. Implementing Classic Physics’ conservation of energy with Atomic Physics’ structure of an atom and the behavior of electrons in their discrete levels do not fail to prove that the Schrodinger Equation is consistent and reliable to this day. There may be some technicalities about the Schrodinger Equation that are worth looking into, such as the intricate mathematics behind the probability of finding an electron is a multi-dimensional box that could be greater than 3, or if there is an outside force that is unknown that could explain the behavior of an atom better than Schrodinger can. With more research and time, these postulations could all be solved and are beyond the scope of this essay.

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