Understanding Quantum Mechanics: An Introduction to Quantum Theory

welcome to quantum mechanics my name is brent carlson since this is the first lecture on

quantum mechanics um we ought to have some sort of an introduction and what i want to do to

introduce quantum mechanics is to explain first of all why it's necessary and second of all to put it in

historical context to well i'll show one of the most famous photographs in all of physics

that really gives you a feel for the brain power that went into the construction of

this theory and hopefully we'll put it in some historical context as well so you can

understand where it fits in the broader philosophy of science but the the main goal of this lecture is

about the need for quantum mechanics which i really ought to just have called why do we need quantum mechanics

uh this subject has a reputation for being a little bit annoying so why do we

bother with it well first off for some historical context

imagine yourself back in 1900 turn of the century science has really advanced a lot we

have electricity we have all this fabulous stuff that electricity can do

and even almost 100 years before that physicists thought they had things

figured out there's a famous quote from laplace given for one instant an intelligence which could comprehend all

the forces by which nature is animated and the respective position of the beings which compose it nothing would be

uncertain in the future as the past would be present to its eyes now

maybe you think intelligence which can comprehend all the forces of nature is a bit of a stretch

and maybe such a being which can know all the respective positions of everything in the universe is a bit of a

stretch as well but the feeling at the time was that if you could do that you would know

everything if you had perfect knowledge of the present you could predict the future

and of course you can infer what happened in the past and everything is connected by one

unbroken chain of causality now in 1903 albert michaelson another famous quote from that time period

said the more important fundamental laws and facts of physical science have all been

discovered our future discoveries must be looked for in the sixth place of decimals

now this sounds rather audacious this is 1903 and he thought that the only thing that we had left to nail down was the

part in a million level precision well to be fair to him he wasn't talking about never discovering new fundamental

laws of physics he was talking about really astonishing discoveries like the discovery of uranus on the basis of

orbital perturbations of neptune never having seen the planet uranus before they figured out that it

had to exist just by looking at things that they had seen that's pretty impressive

and michaelson was really on to something precision measurements are really really useful especially today

but back in 1903 it wasn't quite so simple and michaelson probably regretted that remark for the rest of his life

the attitude that i want you guys to take when you approach quantum mechanics though is not this

sort of 1900s notion that everything is predicted it comes from shakespeare horatio says one

oh day and night but this is wondrous strange to which hamlet replies one of the most

famous lines in all of shakespeare and therefore as a stranger give it welcome there are more things in heaven and

earth horatio than are dreamt of in your philosophy so that's the attitude i want you guys

to take when you approach quantum mechanics it is wondrous strange

and we should give it welcome there are some things in quantum mechanics that are deeply non-intuitive

but if you approach them with an open mind quantum mechanics is a fascinating

subject there's a lot of really fun stuff that goes on now to move on to the necessity for

quantum mechanics there were some dark clouds on the horizon even at the early 20th century

michelson wasn't quite having a big enough picture in his mind when he said that everything was down to the sixth

place of decimals the dark clouds on the horizon at least according to kelvin here

were a couple of unexplainable experiments one the black body spectrum now a black

body you can just think of as a hot object and a hot object like for example the

coils on an electrical stove when they get hot will glow and the question is what color do they

glow do they glow red they go blue what is the distribution of radiation that is emitted by a hot object

another difficult to explain experiment is the photoelectric effect if you have some light

and it strikes a material electrons will be ejected from the surface

and as we'll discuss in a minute the properties of this experiment do not fit

what we think we know about or at least what physicists thought they knew about the physics of light in the

physics of electrons at the turn of the 20th century the final difficult experiment to

explain is bright line spectra for example if i have a flame coming from say a bunsen burner

and i put a chunk of something perhaps sodium in that flame

it will emit a very particular set of frequencies that looks absolutely nothing like a black body

we'll talk about all these experiments in general or in a little bit more detail in a minute or two but

just looking at these experiments now these are all experiments that are very difficult to explain

knowing what we knew at the turn of the 20th century about classical physics they're also also experiments that

involve light and matter so we're really getting down to the

details of what stuff is really made of and how it interacts with the things around it

so these are some pretty fundamental notions and that's where quantum mechanics really got its start

so let's pick apart these experiments in a little more detail the black body spectrum as i mentioned

you can think of as the light that's emitted just by a hot object

and while hot objects have some temperature associated with them let's call that t

the plot here on the right is showing very qualitatively i'll just call it the intensity of the

light emitted as a function of the wavelength of that light

so short wavelengths high energy long wavelengths low energy now if you look at t equals 3 500 kelvin

curve here it has a long tail to long wavelengths and it cuts off pretty quickly as you go

to short wavelength so it doesn't emit very much high energy light whereas if you have a much hotter object

5500 kelvin it emits a lot more high energy light the red curve here is much higher than the black curve

now if you try to explain this knowing what early 20th century physicists knew

about radiation and about electrons and about atoms and how they could possibly emit

light you get a prediction and it works wonderfully well up until about here at which point it

blows up to infinity um infinities are bad in physics this is the the rayleigh jeans law and

it works wonderfully well for long wavelengths but does not work at all for short wavelengths that's called the

ultraviolet catastrophe if you've heard that term on the other end of things

if you look at what happens down here well it's not so much a prediction but an observation

but there's a nice formula that fits here so on one side we have a prediction

that works well on one end but doesn't work on the other and on the other hand we have a sort of

empirical formula called veen's law that works really well at the short wavelengths but well also blows up to

infinity at the long wavelengths both of these blowing up things are a problem the question is how do you get something

that explains both of them this is the essence of the the black body spectrum and how it was difficult

to interpret in the context of classical physics the next experiment i mentioned is the

photoelectric effect this is sort of the opposite

problem it's not how a material emits light it's how light interacts with the material

so you have light coming in and the experiment is usually done like this

you have your chunk of material typically a metal and when light hits it electrons are

ejected from the surface hence the electric part of the photoelectric effect

and you do all this in a vacuum and the electrons are then allowed to go across a gap to some other material

another chunk of metal where they strike this metal and the experiment is usually done like

this you connect it up to a battery so you have your material on one side

and your material on the other you have light hitting one of these materials and ejecting electrons

and you tune the voltage on this battery such that your electrons when they're ejected never quite make it

so the electric field produced by this voltage is opposing the motion of the electrons

when that voltage is just high enough to stop the motion of the electrons keep them from completely making it all the

way across we'll call that the stopping voltage now

it turns out that what classical e m predicts as i mentioned doesn't match what

actually happens in reality but let's think about what does classical e m predict here

well classical electricity and magnetism says that

electromagnetic waves here have electric fields and magnetic fields associated with them

and these are propagating waves if i increase the intensity

of the electromagnetic wave that means the magnitude of the electric field

involved in the electromagnetic wave is going to increase and if i'm an electron

sitting in that electric field the energy i acquire is going to increase

that means the stop is going to increase because i'll have to have more voltage to stop a higher

energy electron as would be produced by higher intensity beam of light the other parameter of this incoming

light is its frequency so we can think about varying the frequency if i increase the frequency i have more

intense light now that doesn't say anything about the string sorry if i increase the frequency

i don't necessarily have more intense light the electric field magnitude

is going to be the same which means the energy and the stopping voltage

will also be the same now it turns out what actually happens in reality

does not match this at all in reality when the intensity increases the energy

which i should really write as v stop the stopping voltage necessary doesn't change

and when i increase the frequency the voltage necessary to stop those electrons increases

so this is sort of exactly the opposite what's going on here that's the puzzle in explaining the

photoelectric effect just to briefly check your understanding consider these plots of stopping voltage

as a function of the parameters of the incident light and check off which you think shows the

classical prediction for the photoelectric effect the third experiment that i mentioned is

bright line spectra and as i mentioned this is what happens if you take a flame

or some other means of heating a material like the bar of sodium i mentioned

earlier this will emit light

and uh in this case the spectrum of light from red to blue of sodium

looks like this oh actually i'm sorry that's not sodium that's mercury

uh the these are four different elements hydrogen mercury neon

and xenon and instead of getting a broad continuous distribution like you would

from a black body under these circumstances where you're talking about gases you get these very

bright regions it's the spectrum instead of looking like a smooth curve like this

looks like spikes those bright lines are extraordinarily

difficult to explain with classical physics and this is really the straw that broke the camel's back broke

classical physics is back that really kicked off quantum mechanics how do you explain this

this is that famous photograph that i mentioned this is really the group of people who first built quantum mechanics

now i mentioned three key experiments the

black body spectrum this guy figured that out this is plonk the photoelectric effect

this guy who i hope needs no introduction this is einstein that out

this is the paper that won einstein the nobel prize and

as far as the brightline spectra of atoms it took a much longer time to figure out

how all of that fit together and it took a much larger group of people but they all happen to be present

in this photograph there's this guy and this guy

and these two guys and this guy this photograph is famous because

these guys worked out quantum mechanics but that's not the only these aren't the only famous people in

this photograph you know this lady as well this is marie curie this is lorenz

which if you studied special relativity you know einstein used the lorentz transformations

pretty much everyone in this photograph is a name that you know i went through and looked up who these people were

these were all of the names that i recognized which doesn't mean that the people whose names i didn't recognize

weren't also excellent scientists for example ctr wilson here one of my personal favorites inventor of the cloud

chamber this is the brain trust that gave birth to quantum mechanics and it was quite a

brain trust you had some of the most brilliant minds of the century working on some of the

most difficult problems of the century and what's astonishing is they didn't really like what they found they

discovered explanations that made astonishingly accurate predictions but throughout the history you keep seeing

them disagreeing like no that can't possibly be right not necessarily because

the predictions were wrong or they thought there was a mistake somewhere but because they just disliked the

nature of what they were doing they were upending their view of reality einstein in particular really disliked quantum

mechanics to the day that he died just because it was so counter-intuitive and so with that introduction

to a counter-intuitive subject i'd like to remind you again of that shakespeare quote

there are more things in heaven and earth horatio than are dreamt of in your philosophy

uh try to keep an open mind and hopefully we'll have some fun at this

knowing that quantum mechanics has something to do with explaining the interactions of light and matter for

instance in the context of the photoelectric effect or

black body radiation or bright line spectra of atoms and molecules one might be led to the question of when

is quantum mechanics actually relevant the domain of quantum mechanics is unfortunately not a particularly simple

question when does it apply well on the one hand you have classical physics

and on the other hand you have quantum physics and

the boundary between them is not really all that clear on the classical side you have things that are certain

whereas on the quantum side you have things that are uncertain what that means in the context of

physics is that on the classical side things are predictable they may be chaotic and difficult to

predict but in principle they can be predicted well on the quantum side things are

predictable too but with a caveat in the classical side you determine

everything basically every property of the system can be

known with perfect precision whereas in quantum mechanics what you predict are probabilities

and learning to work with probabilities is going to be the first step to getting comfortable with quantum mechanics

the boundary between these two realms when the uncertain and probabilistic effects of quantum mechanics start to

become relevant is really a dividing line between things that are large

and things that are small and that's not a particularly precise way of stating things

doing things more mathematically quantum mechanics applies for instance when angular momentum

l is on the scale of planck's constant or the reduced flux constant h bar now

h bar is the fundamental scale of quantum mechanics and it appears not only in the context of angular momentum

planck's constant has units of angular momentum so if your angular momentum is of order planck's constant or smaller

you're in the domain of quantum mechanics we'll learn more about uncertainty

principles later as well but uncertainties in this context have to do with

products of uncertainties for instance the uncertainty in the momentum of a particle times the

uncertainty in the position of the particle this if it's comparable to planck's

constant is also going to give you the realm of quantum mechanics energy and time also have an uncertainty

relation again approximately equal to planck's constant most fundamentally

the classical action when you get into more advanced studies of classical mechanics you'll learn

about a quantity called the action which has to do with the path the system takes as it evolves in space and time

if the action of the system is of order planck's constant then you're in the quantum mechanical

domain now klonk's constant is a really small number it's 1.05

times 10 to the negative 34 kilogram meters squared per second times 10 to the negative 34 is a small

number so if we have really small numbers then

we're in the domain of quantum mechanics in practice these guys are the most useful

whereas this is the most fundamental but we're more interested in useful things than we are in fundamental things

after all for example the electron in the hydrogen atom now

you know from looking at the bright line spectra that this should be in the domain of quantum mechanics

but how can we tell well to use one of the uncertainty principles

as a calculation consider the energy the energy

of an electron in a hydrogen atom is you know let's say

about 10 electron volts if we say that's p squared over 2m using the classical kinetic energy

relation between momentum and kinetic energy that tells us

that the momentum p is going to be about 1.7 times 10 to the

minus 24th kilogram meter square sorry kilogram where'd it go where's my eraser

kilogram meters per second now this suggests that the momentum of

the electron is you know non-zero but if the hydrogen atom itself is not moving we know the average

momentum of the electron is zero so if the momentum of the electron is going to be zero with

still some momentum being given to the electron this is more the uncertainty in

the electron momentum than the electron momentum itself the next quantity if we're looking at

the uncertainty relation between momentum and position is we need to know the size of or the uncertainty in the

position of the electron which has to do with the size of the atom now the size of the atom

that's about 0.1 nanometers which if you don't remember the conversion from nanometers is 10 to the

minus 10th meters so let's treat this as delta x our uncertainty in position

because we don't really know where the electron is within the atom so this is a reasonable guess at the uncertainty

now if we calculate these two things together delta p delta x you get something

i should say this is approximate because this is very approximate 1.7 times 10 to the negative 34th

and if you plug through the units it's kilogram meters squared per second this

is about equal to h-bar so this tells us that quantum mechanics is definitely important here

we have to do some quantum in order to understand this system as an example

of another small object that might have quantum mechanics relevant to it this is one that we would

actually have to do a calculation i don't know intuitively whether a speck of dust in a light breeze is in the

realm of quantum mechanics or classical physics now

i went online and looked up some numbers for a speck of dust let's say the mass is about 10 to the minus sixth

kilograms a microgram uh has a velocity in this light breeze

of let's say one meter per second and let me make myself some more space

here um the size of this speck of dust

is going to be about 10 to the minus 5 meters

so these are the basic parameters of this speck of dust in a light breeze now we can do some calculations with

this for instance momentum well

in order to understand quantum mechanics there's some basic vocabulary that needs to that i need to go over so let's talk

about the key concepts in quantum mechanics thankfully there are only a few there's

really only three and the first is the wave function the wave function is and always has been

written as psi the greek letter my handwriting gets a little lazy sometimes and it'll end up just looking

like this but technically it's supposed to look something like that

details are important provided you recognize the symbol psi

is a function of position potentially in three dimensions x y and z

and time and the key facts here is that psi is a complex function which means that while x y z and t here

are real numbers psi evaluated at a particular point in space will potentially be a complex number

with both real and imaginary part what is subtle about the wave function and we'll talk about this in great

detail later is that it while it represents the state of the system

it doesn't tell you with any certainty what the observable properties of the system are it really only gives you

probabilities so for instance if i have

coordinate system something like this where say this is position in the x direction

psi with both real and imaginary parts might look something like this this could be the real part of psi

and this could be say the complex or the imaginary part of psi

what is physically meaningful is the squared magnitude of psi which might look something like this

in this particular case and that is related to the probability of finding the particle at a particular

point in space as i said we'll talk about this later but the key facts that you need to know

about the wave function is that it's complex and it describes the state of the system but not with certainty

the next key concept in quantum mechanics is that of an operator

now operators are what

connect psi to observable quantities that is one thing operators can do

just a bit of notation usually we use hats for operators for instance x hat

or p-hat our operators that you'll encounter shortly operators

act on psi so if you want to apply for instance the x-hat operator to psi you would write x

hat psi as if this were something that were as it appears on the left of psi the

assumption is that x acts on psi if i write psi x hat doesn't necessarily mean that x

hat acts on psi you assume operators act on whatever lies to the right likewise of course p hat psi

now we'll talk about this in more detail later but x hat

the operator can be thought of as just multiplying by x

so if i have psi as a function of x x hat psi is just going to be x times psi of x

so if psi was a polynomial you could multiply x by that polynomial the

the p operator p hat is another example is a little bit more complicated this is

just an example now and technically this is the momentum operator but we'll talk more about that later it's equal to

minus h bar times the derivative with respect to x so this is again something that

needs a function needs the wave function to actually give you anything meaningful

now the important thing to note about the operators is that they don't give you the observable quantities either

but in quantum mechanics you can't really say the momentum of the

wave function for instance p hat psi is not

and i'll put this in quotes because you won't hear this phrase very often momentum

of psi it's the momentum operator acting on psi and that's not the same thing as the

momentum of psi the final key concept in quantum mechanics is the schrodinger

equation and this is really the big equation so i'll write it big

i h bar partial derivative of psi

with respect to time is equal to h hat that's an operator

acting on psi now h hat here is the hamiltonian

which you can think of as the energy operator so the property of the physical system

that h is associated with is the energy of the system and the energy of the system

can be thought of as a kinetic energy so we can write a kinetic energy operator plus a potential energy

operator together acting on psi and it turns out the kinetic energy operator can be written down this is

going to end up looking like minus h bar squared over 2m

partial derivative of psi with respect to sorry second partial derivative of side with respect to

position plus and then the potential energy operator is going to look like the

potential energy as a function of position just multiplied by psi so this is the schrodinger equation

typically you'll be working with it in this form so i h bar times the partial derivative

with respect to time is related to the partial derivative with respect to space and then multiply multiplied by some

function the basic quantum mechanics that we're going to learn in this course mostly

revolves around solving this function and interpreting the results so to put these in a bit of a roadmap

we have operators we have the schrodinger equation and we have the wave function

now operators act on the wave function and operators are

used in the schrodinger equation now the wave function that actually describes the state of the system is

going to be the solution to the schrodinger equation now i mentioned operators acting on the

wave function what they give you when they act on the wave function is some property of the system

some observable perhaps and the other key fact that i mentioned so far is that the wave function doesn't

describe the system perfectly it only gives you probabilities so that's our overall concept map

to put this in the context of the course outline the probabilities are really the key feature of quantum mechanics and

we're going to start this course with the discussion of probabilities we'll talk about the wave function after

that and how the wave function is related to those probabilities and

we'll end up talking about operators and how those operators and the wave functions together give you

probabilities associated with observable quantities that will lead us into a discussion of

the schrodinger equation which will be most of the course really the bulk of the material before the

first exam will be considered with very concerned with various examples a solution to the schrodinger equation

under various circumstances this is really the main meat of quantum mechanics in the beginning

after that we'll do some formalism and what that means is we'll learn about some advanced mathematical tools that

make keeping track of all the details of how all of this fits together a lot more straightforward

and then we'll finish up the course by doing some applications so those are our key concepts and a

general road map through the course hopefully now you have the basic vocabulary necessary to understand

phrases like the momentum operator acts on the wave function or the solution to the schrodinger equation describes the

state of the system and that sort of thing don't worry too much if these concepts

haven't quite clicked in order to really understand quantum mechanics you have to get experience

with them these are not things that you really have any intuition for based on anything you've seen in physics so far

so bear with me and this will all make sense in the end i promise

complex numbers or numbers involving conceptually you can think about it as the square root of negative one

i are essential to understanding quantum mechanics since some of the most

fundamental concepts in quantum mechanics for instance the wave function are expressed in terms of complex

numbers complex analysis is also one of the most beautiful subjects in all of mathematics

but unfortunately in this course i don't have the time to go into the details lucky you perhaps

here's what i think you absolutely need to know to understand quantum mechanics from the perspective of complex analysis

first of all there's basic definition i squared is equal to negative 1 which you can think of also as i equals the square

root of negative 1. a in general a complex number z then can be written as a the sum of a

purely real part x and a purely imaginary part i times y

note in this expression z is complex x and y are real where i times y is purely imaginary

the terms purely real or purely imaginary in the context of this expression like

this x plus i y something is purely real if y is zero something is purely imaginary if x is zero

as far as some notation for extracting the real and imaginary parts typically mathematicians will use this funny

calligraphic font to indicate the real part of x plus iy or the imaginary part of x plus iy and

that just pulls out x and y note that both of these are real numbers when you pull out the

imaginary part you get x and y you don't get i y for instance another one of the most beautiful

results in mathematics is e to the i pi plus one equals zero

this formula kind of astonished me when i first encountered it but it is a logical extension of this

more general formula that e raised to a purely imaginary power i y is equal to the cosine of y plus i times

the sine of y this can be shown in a variety of ways in particular involving the taylor

series if you know the taylor series for the exponential the taylor series for cosine of y and the taylor series for

sine of y you can show quite readily that the taylor series for complex exponential is the taylor series of

cosine plus the taylor series of sine and while that might not necessarily constitute a rigorous proof

it's really quite fun if you get the chance to go through it at any rate the trigonometric functions

here cosine and sine should should be suggestive and there is a

geometric interpretation of complex numbers that we'll come back to in a minute

but for now know that while we have rectangular forms like this x plus i y where x and y the

nomenclature there is chosen on purpose you can also express this in terms of r e to the i theta where you have now a

radius and an angle the angle here by the way is going to be the

arc tangent of y over x and we'll see why that is in uh in a moment when we talk about the geometric

interpretation but given these rectangular and polar forms

of complex numbers what do the basic operations look like how do we manipulate these things

well addition and subtraction in rectangular form is straightforward if we have two complex numbers a plus ib

plus and we want to add to that the second complex number c plus id we just add the

real parts a and c and we add the imaginary parts b and d this is just like adding in any other

sort of algebraic expression multiplication is a little bit more complicated you have to distribute

and you distribute in the usual sort of draw smiley face kind of way a times c and b times d are going to end

up together in the real part the reason for that is well a times c a and c both being real numbers a times c will be

real whereas ib times id both being purely complex numbers

you'll end up with b times d times i squared and i squared is minus 1. so you just

end up with minus bd which is what we see here otherwise the complex part is perhaps a

little more easy to understand you have i times b times c and you have a times i times d both of which end up with plus

signs in the complex part division in this case

is like rationalizing the denominator except instead of involving radicals you have complex numbers

if i have some number a plus i b divided by c plus id i can simplify this by both multiplying

and dividing by c minus id note the sign change in the denominator here c plus id is then

prompting me to multiply by c minus id over c minus id now when you do the distribution there

for instance let's just do it in the denominator c plus

id times c minus id my top eyebrows here of the smiley face c squared

minus sorry c squared times id c squared plus now id

times minus id which is well i'll just write it out i times minus id

which is going to be d squared times i times minus i so i squared times minus one and i

squared is minus one so i have minus one times minus one which is just one so i can ignore that

i've just got d squared so what i end up with in the denominator is just c squared plus d

squared what i end up with in the numerator well that's the same sort of multiplication thing that we just

discussed so the simplified form of this has no complex part in the denominator

which helps keep things a little simple and a little easier to interpret now in polar form addition and

subtraction while they're complicated under most circumstances if you have two complex numbers given in polar form it's

easiest just to convert to rectangular form and add them together there multiplication and division though in

polar form have very nice expressions q e to the i theta times r e to the i phi

well these are just all real numbers multiplying together and then i can use the rules regarding multiplication of

exponentials meaning if i have two things like e to the i theta and e to the iv i can just add

the exponents together it's like taking x squared times x to the fourth and getting x to the sixth

but q are e to the i theta plus v so that was easy we didn't have to do any distribution at all

the key factor is that you add the angles together in the case of division it's also quite

easy you simply divide the radii q over r and instead of adding you subtract the angles

so polar complex numbers expressed in polar form

are much easier to manipulate in multiplication and division while complex numbers represented in

rectangular form are much easier to manipulate for addition and subtraction taking the magnitude of complex number

usually we'll write that as something like z if z is a complex number just using the same notation for

absolute value of a real number usually is expressed in terms of the complex conjugate the complex conjugate

notationally speaking is usually written by whatever complex number you have here in

this case x plus iy with a star after it and what that signifies is you flip the sign

on the complex part on the imaginary part x plus iy becomes x minus iy the squared magnitude then which is

always going to be a real and positive number this

absolute value squared notation is what you get for multiplying a number by its complex conjugate and that's what

we saw earlier with c plus id say i take the complex conjugate of c plus id and multiply it by c plus i d

well the complex conjugate of c plus id is c minus id times

c plus id and doing the distribution like we did when we calculated the denominator when

we were simplifying the division of complex numbers in rectangular form just gave us c squared

plus d squared this should be suggestive if you have something like

x plus i y that's really messy x plus i y and i want to know the squared absolute

magnitude thinking about this as a position in cartesian space

should make this formula c squared plus d squared in this case just make uh make a little more sense

you can also of course write that in terms of real and imaginary parts but let's do an example

if w is 3 plus 4i and z is -1 plus 2i first of all let's find w plus z well w

plus z is three plus four i plus minus one plus two i that's straightforward if you can keep

track of your terms 3 minus 1 is going to be our real part so that's 2 and 4i plus 2i which is plus 6i is going

to be our complex part sorry our imaginary part now w times z

3 plus 4 i times minus 1 plus 2i for this we have to distribute

like usual so from our top eyebrow terms here we've got three

times minus one which is minus three and four i times 2i both positive so i

have 4 times 2 which is 8 and i times i which is minus 1 minus 8.

then for my imaginary part the i guess the mouth and the chin if you want to think about it that way i

have 4i times minus 1 minus 4 with the i out front will just be minus 4 inside the parentheses here

and 3 times 2i is going to give me 6i plus 6 inside the end result you get here is 8 or

minus 8 minus 3 is minus 11 and minus 4 plus 6 is going to be

2. so i get minus 11 plus 2i for my multiplication here i guess i'm going to circle that answer i should

circle this answer as well now slightly more complicatedly w over z w is three plus four i

and z is minus one plus two i and you know when you want to simplify an expression like this you multiply by

the complex conjugate of the denominator divided by the complex conjugate of the denominator so minus 1 minus 2i divided

by -1 minus 2i and if we continue the

same sort of distribution i'll do the numerator first same sort of multiplication we just did

here only the signs will be flipped a little bit we'll end up with minus three plus eight instead of minus three minus

eight and for the complex sorry for the imaginary part we'll end up with minus 4

minus 6 instead of minus 4 plus 6 and you can work out the details of that distribution on your own if you want

the denominator is not terribly complicated since we know we're taking the absolute magnitude of a complex

number by multiplying a complex number by its complex conjugate we can just write this out as the square

of the real part 1 plus the square of the imaginary part

minus 2 which squared is 4. so if i continue this final step

this is going to be 5 this is going to be minus 10 i and our denominator here is just going to be

5. so in the end what i'll end up with is going to be 1

minus 2 i so it actually ended up being pretty simple in this case now for the absolute magnitude of w

3 plus 4 i you can think of this as w times w star

square root you can think of this as the square root of the real part of w plus the imaginary

part of w sorry square root of the squared of the real real part plus the square of

the imaginary part which is perhaps a little easier to work with in this case so you don't have to

distribute out complex numbers in that in that way real part is three imaginary part is

four so we end up with the square root of three squared plus four squared

which is five now this was all in rectangular form let me

move this stuff out of the way a little bit and let's do it again at least a subset

of it in polar form in polar form w

three plus four i we know the magnitude of w that's five so that's going to be our radius 5

and our e to the i theta where theta is like i said the arctan since complex numbers are so important

to quantum mechanics let's do a few more examples in this case i'm going to demonstrate how to manipulate complex

numbers in a more general way not so much just doing examples with numbers first example simplify this expression

you have two complex numbers multiplied in the numerator and then a division

first of all the first thing to simplify is this multiplication you have x plus iy times

ic this is pretty easy it's a simple sort of distribution

we're going to have x times ic that's going to be a complex part so i'm going to write that down a

little bit to the right i x c and then we're going to have i y times i c which is going to be minus

y c that's going to be real we also have a real part in the numerator from d here so i'm going to write this as d minus y

c plus i c that's the

result of multiplying this out that's then going to be divided by f plus i g

now in order to simplify this we have a complex number in the denominator you know you need to

multiply by the complex conjugate and divide by the complex conjugate so f minus i g

divided by f minus ig now expanding this out is a little bit messier

but fundamentally you've seen this sort of thing before

you have real part real part an imaginary part imaginary part in the numerator

and then you're going to have imaginary part real part and real part imaginary part

and what you're going to end up with from this first term you get f times d minus yc

from the second term you have minus ig times ixc which is going to give you xcg

we have a minus i times an i which is going to give us a plus incidentally if you're having trouble

figuring out something like minus i times i think about it in the geometric

interpretation this is i in the complex plane this is minus i in the complex plane

so i have one angle going up one angle going down if i'm multiplying them together i'm adding the angles together

so i essentially go up and back down and i just end up with 1 equals i times minus i

otherwise you can keep track of i squared equals minus 1s and just count up your minus signs

this then is the real part

suppose i should write that in green unless my fonts get too confusing excuse me

so that's the real part the imaginary part then is what you get from these terms here

i'm going to write an i out front and now we have x c times f so x c f with an i from here

and then we have d minus y yc times ig which i'll just write as g

d minus yc in the denominator we're now multiplying a number by its

complex conjugate you know what to do there f squared plus g squared

this is just the magnitude of this complex number sorry squared magnitude

now this doesn't necessarily look more simple than what we started with but this is effectively fully simplified you

could further distribute this and distribute this but it's not really going to help you very much

the thing to notice about this is that the denominator is purely real

we've also separated out the real part of the numerator and the imaginary part

of the numerator my handwriting is getting messier as i go

imaginary part of the numerator so we can look at this numerator now and

say ah this is the complex number real part imaginary part and then it's just divided by this real number which

effectively is just a scaling it's it's a relatively simple thing to do to divide by a real number

as a second example consider solving this equation for x now this is the same expression that we had

in the last problem only now we're solving it for equal to zero

so from the last page i'm going to borrow that first simplification step we did distributing

this through we had d minus y c for the real part

plus i x c for the imaginary part and that was divided by f plus i g if we're setting this equal to zero

the nice part about dealing with complex expressions like this is that 0 treated as a complex number is 0 plus

0 i it has a real part and an imaginary part as well it's just kind of trivial

and in order for this complex number to be equal to zero the real part must be zero and the imaginary part must be zero

so we can think of this as d minus y c plus i x c this has to equal zero and this has to

equal 0 separately so we effectively have two equations here not just 1 which is nice we have d

minus yc equals 0 and xc equals 0 which unless c equals 0 just means x equals zero

that's the only way that this equation can hold is if x equals zero

the key factor is to keep in mind that the in order for two complex numbers to be

equal both the real parts and the imaginary parts have to be equal as a slightly more involved example

consider finding this the cubed roots of one now you know one cubed is one that's a

good place to start we'll see that fall out of the algebra pretty quickly what we're trying to do is solve the

equation z cubed equals one

which you can think of as x plus i y where x and y are real numbers

cubed equals one now if we expand out this cubic you get

x cubed plus three x squared times i y plus 3 x times i y squared

plus i y cubed and this is going to have to equal 1.

excuse me equal 1. now

looking at these expressions here we have an i y here we have an i y squared this is going to give me an i squared

which is going to be a minus sign and here i have an i y cubed this is going to give me an i cubed which is

going to be minus i so i have two complex parts and two real parts

so i'm going to rewrite that x cubed and then now a minus sign from the i squared

3 x y squared plus pulling an i out front the imaginary part then is going to come

from this 3x squared y and this y cubed so i've got a 3 x squared y here and then a minus y cubed minus coming from

the i squared and this is also going to have to equal 1.

now in order for this complex number to equal this complex number both the real parts and the imaginary parts have to be

equal so let's write those two separate equations x cubed minus three x y

squared equals the real part of this is the real part of the left hand side has to equal

the real part of the right hand side one and the imaginary part of the left hand side three x squared y minus y cubed has

to equal the imaginary part of the right hand side zero so those are our two equations

this one in particular is pretty easy to work with um we can simplify this

this is you know we can factor a y out this is y times three x squared minus y squared

equals zero one possible solution then is going to come from this

you know you have a product like this equals zero either this is equal to zero or this is equal to zero and saying y

equals to zero is rather straightforward so let's say y equals zero and let's substitute that into this

expression that's going to give us x cubed equals 1

which might look a lot like the equation we started with z cubed equals 1 but it's subtly different because z is a

general complex number whereas our assumption in starting the problem this way is that x is a purely real number

so a purely real number which when cubed gives you 1 that means x equals 1. so x equals one y equals zero that's one

of our solutions z equals one plus zero i or just zero z equals

one now we could have told me that right off the bat z

z cubed equals one well z one possible solution is that z equals one since one cubed is one

the other thing we can do here is we can say three x squared minus y squared

is equal to zero this means that i'll just cheat a little bit and

simplify this 3x squared equals y squared now i can substitute

this in this y squared into this expression as well

and what you end up with is x cubed minus 3x and then y squared was equal to 3x squared so 3x squared is going to go

in there that has to equal 1. now let's move up here what does that leave us with that says

x cubed minus nine x cubed equals one so minus

eight x cubed equals one this means x again being a purely real

number is equal to minus one-half minus one-half times minus one-half times minus one-half

times eight times minus one is equal to one you can check that pretty easily now

where does that leave us where do they go that leaves us substituting this back in to this

expression which tells us that three x squared

equals y squared x equals minus one half so three minus one half squared equals y

squared which tells you that y equals plus or minus the square root of three fourths

if you finish your solution so now we have two solutions for y here coming from one value for x and that

gives us our other two solutions to this cubic we have a cubic equation we would expect there to be three solutions

especially when we're working with complex numbers like this and this is our other solution

z equals minus one half plus or minus the square root of three fourths

i so those are our three solutions now

finding the cubed roots of one to be these complex numbers is not necessarily particularly instructive

however there's a nice geometric interpretation the cubed roots of unity like this

the nth roots of unity doesn't have to be a cubed root all lie on a circle of radius 1 in the

complex plane and if you check the complex magnitude of this number the complex magnitude of

this number you will find that it is indeed unity to check your understanding of this

slightly simpler problem is to find the square roots of i um put another way you've got z some

generic complex number here equals to x squared plus x plus i y quantity squared if that's going to

equal y we'll expand this out solve for x and y in the two equations that will result

from setting real and imaginary parts equal to each other same as with the cubed roots of one

the square roots of i will also fall on a circle of radius one in the complex plane

so those are a few examples of how complex numbers can actually be manipulated

in particular finding the roots of unity there are better formulas for that than the approach that we took here

but i feel this was hopefully instructive if probability is at the heart of

quantum mechanics what does that actually mean well the fundamental source of

probability in quantum mechanics is the wave function psi psi tells you everything that you can in

principle know about the state of the system but it doesn't tell you everything with perfect precision

how that actually gives rise to probability distributions in observable quantities like position or energy or

momentum is something that we'll talk more about later but from the most basic perspective

psi can be thought of as related to a probability distribution

but let's take a step back and talk about probabilistic measurements in general first

if i have some space let's say it's position space