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# So You Want to Find an Electron?

A Class Lecture on How to Find an Electron Using Quantum Mechanics
by

## Aabid Patel

on 8 December 2015

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#### Transcript of So You Want to Find an Electron?

So You Wanted to FInd an Electron?
Alright, first things first, do you know what an electron is?
FIRST, YOU NEED SOME BASIC KNOWLEDGE
IN MATH!
Trigonometry
sinx = O/H
cosx = A/H
tanx =O/A
Derivatives
Integrals
Special Numbers
i^2 = -1
e = 2.71828183...
TEST TIME, KIDDOS!
Next, we need to understand what we're dealING WITH.
I MEAN WE'RE SEARCHING FOR A CLOUD?
Δ
Δ
Δ
Δ
VS
NEWTON
HUYGENS
HIt the Waves!
Wave-PartIcle HIstory
So what Is LIGHT? A PartIcle or Wave?
1.) Young's Double Slit Experiment
Light is shot through two parallel slits and projected on to a screen. If it were a particle, you would see random spots on the screen due to the collisions off of the sides of the slit. Yet, one sees diffraction (spreading of a wave past an obstruction) where the "wave" interferes showing arranged and organized light and dark spots on the screen
2.) Ether Non-Existing
For light to be a wave, it needs to have a medium to move around in (ex. ripples in water). Many attempts were made to find the medium through which it was proposed light propagates through, but no medium was ever found. Imagine believing something that was there, and understanding it to its complexity, only to realize it was not there and you don't know anything! A bunch of pissed off scientists...
3.) Photoelectic Effect
In 1905, Einstein published his work on the photoelectric effect. He discussed the ability of a metal to absorb energy and release electrons. The energy absorbed were quanta of light energy, later called photons.

So in one sense, light acts as a wave but in another it acted as a particle. After more experimental and theoretical work, it was decided that light functions as both a wave and a particle depending on the manner of observation and measurement
ITS A DRAW!
Louis de Broglie took a bold step in 1923 by saying that Einstein's relation with light can be used to determing the wavelength of any matter.

Wavelength = Planck's Constant (h)/Momentum of Matter

This wavelength (de Broglie Wavelength) was experimentally confirmed by Davisson and Germer at Bell Labs by firing electrons at a Nickel target and noticing the diffraction. Later on, the double slit experiment was redone for electrons, also showing the diffraction.

This was the first time a Ph.D. thesis received the Nobel Prize!
SO WHAT'S A PartIcle WAVE?
The idea that the electron we're looking for can be explained as a particle and a wave makes things difficult to understand. When we say the wave nature of an electron, its not that it forms the shape of a wave, but it can be described as a wave.
Ψ(x, t) = The wavefunction as a function of space and time describes the quantum state of a particle or system of particles ...this thing holds everything!
Now why can't I just look at an atom and look for It? Why do I need math?
Werner Heisenberg in 1925 formulated the Heisenberg Uncertainty Principle. It is impossible to simultaneously both measure the present position while determining the future momentum of an electron or any other particle with 100% accuracy.
ΔxΔp ≥h/4π
So we need a method of calculation that is able to determine the electron's quantum system as it varies in time?
SchrodINGER'S eQUATION
In 1927, Erwin Schrodinger developed an equation that was able to describe the change in time of the quantum state of any physical system. He wanted to find the wave equation of an electron based upon the previous Quantum Theory developed.
1.) Conservation of Energy
Energy in = Energy Out

E=T+V

2.) Einstein's Quanta of Energy

3.) De Broglie Wavelength
T = Kinetic Energy (The energy associated with
a moving object
p = Momentum = mass of object times velocity
V = Potential Energy = Energy stored in a body or system due to its position in a certain arrangement or force field
THe DerIvatION
1.) Make an AssumptIon

2.) Let's Try MessIng Around WIth It a BIT, See What We Get

3.) Sub IT INTO THE CONSERVATION EQUATION
NOW THAT WE HAVE THE BACKGROUND, ITS TIME TO GET LOST...
ERR...
I MEAN ITS TIME TO FIND AN ELECTRON!
FIND AN ELECTRON IN THE HYDROGEN ATOM,
IN THE FIRST ORBITAL
1.) wRITE DOWN THE EQUATION THEN
SEPARATE THE WAVEFUNCTION
2.) sUB IN THE pOTENTIAL AND wORK IT baby!
3.) nEEDS TO BE ABLE TO COVER ALL OF ITS BASES
4.) TAKE THAT POWER SERIES TO THE MAX!
5.) wHAT THE EFF IS F?
6.) nOW THE ELECTRON, WHERE IS THAT SUCKA?
LOCATIONS OF ELECTRON IN HYDROGEN!
Sub in and divide by Ψ(r,t)
If r is large
Sub in original equation, collect terms with r, setting coefficients to 0 and solve
In order to satisfy non-infinite condition
Discrete levels of energy of Bohr atom
Simplify
Normalization!
a FEW nOTES TO CONCLUDE WITH
Choice of potential
Gets more complicated with larger atoms