How to test multiple intelligence

this is supposed to be mathematical

physics and as you know there are

different styles of doing mathematical

physics one style is a very rigorous

approach very very rigorous where it's

all theorem and proof and I personally

find that less interesting okay let's be

polite so I don't like theorem and proof

type approaches to doing physics

especially mathematics that's going to

be used in physics and I prefer to talk

about what I regard is really useful and

powerful mathematical techniques and a

lot of what I'm going to be telling you

is in fact not rigorous at least not

rigorous yet and because of that it is

more powerful okay so the more rigorous

you get in fact the less powerful you

are and that's because the more general

you are and the more rigorous you are

the less you can use it to solve new

problems okay it's proving general

theorems is often not very useful and

I'm going to be showing you how to solve

problems the kind of problems that

you're going to encounter in physics and

that's going to be the purpose of this

course um so what I would like to do is

to develop the kind of mathematics that

you need to solve very very very

difficult problems okay that's that's

the objective here

um I don't know how much I'm going to

cover and that kind of depends on how

interested you are and how fast we can

go and I know how successful I am

because I can look at you while I teach

and if I see you drifting off then I'll

have to go slowly and if I see that

you're following me then we can go

faster okay so um let me begin

um by giving you a sort of partial

outline

my course that is an outline I guess of

this course okay why do the lights keep

going down you just have to savor with

it they ask you to say turn off the

light

okay so what I want to begin with is

talking to you about perturbation theory

if you have a very hard problem to solve

in fact let me put it this way if you

take most courses in in undergraduate or

graduate school they tend to concentrate

on problems that you can solve exactly

and it's very nice when you have a

problem that you can solve exactly but

almost no problems are exactly solvable

that's the problem

that's why it's difficult than all the

exactly solvable problems and there are

about three or four of them that's all

have already been solved unfortunately

and all the rest of the interesting

problems that need to be solved cannot

be solved exactly and you're going to

need to have tools for solving what I

will call generically a hard problem

that you can see this is an attack stick

okay so this is what we're going to try

to talk about hard problems and how you

can solve them okay and there are

basically only two approaches either you

can use numerical methods and this is

not a course on numerical methods okay

so one possibility is that you can use

numerical methods

I don't trust most numerical

calculations because I tend to think of

them as being like a sausage they're

just fine until you find out what went

into them but what we want to talk about

here is analytical approaches to very

very hard problems okay and there is one

standard analytical approach really only

one

and that's called perturbation theory

and most people when they hear the word

perturbation theory they begin to get

bored in fact perturbation theory is

amazing I'm going to show you that is

truly an amazing technique and it is

very powerful very beautiful and it

reveals all kinds of remarkable stuff

about quantum mechanics so let's begin

with a discussion of you know how do you

approach a hard problem one that you

can't solve exactly well there are

always three steps when you do

perturbation Theory and the first step

so here is a hard problem okay the first

step is to insert a small parameter

epsilon into the problem okay so instead

of solving one hard problem amazingly

now we're solving an infinite number of

hard problems that depend on a parameter

epsilon it's hard to believe you can't

solve one problem so instead you convert

it into an infinite number of problems

one for each value of Epsilon that

doesn't sound like progress but it is

you'll see the second step and this is a

very tricky and very subtle idea is to

assume that the answer to the hard

problem which of course depends on

epsilon because there's one answer for

each value of Epsilon we can assume that

the answer has the form of a

perturbation series okay okay that's a

perturbation series now in fact I have

assumed that it is a Taylor like series

and in general it isn't always a Taylor

like series but for now let's assume

that the perturbation series is in the

Taylor form okay a sub n epsilon to the

N and the objective now is to calculate

the coefficients in the series okay so

we calculate a zero and

and a-two and if you're you know if

you're in your professor mathematical

physics you could probably calculate

that much if you're a graduate student

you could probably calculate a few more

the most willing to work harder and if

you're an undergraduate student and if

you're in high school you can calculate

lots of coefficients and now we have the

many terms maybe even an infinite number

of terms if you're lucky in the

perturbation series and now what do we

do well the answer is you add up the

series and you calculate the answer okay

and every one of these steps is really

interesting it can be very subtle and

very tricky and it requires a lot of art

a lot of interesting mathematics but the

question the basic question is why does

this procedure work if it works why does

it work the answer is that you begin

with an extremely hard problem and you

reduce an infinitely hard problem to an

infinite sequence of very easy problems

or let's say relatively easy problems

okay because instead of trying to get

the whole answer in one shot now you

calculate a little bit of the answer and

a little bit more and a little bit more

in a little bit more and so on and you

get the complete answer to the problem

you add up the series and then at the

end you set epsilon equal one and that's

the final answer to the problem okay

that's the outline okay so let's make up

let's make up an interesting problem to

solve and let's see let's see how

powerful this tool really is so what's a

hard problem well how about solving this

equation

okay why is that hard you know why

that's a hard problem I mean why do i

why do i classify this as a as a hard

problem you know why that's right that's

right you know from high school you know

or junior high school you know quadratic

formula for quadratic equations and

there is some other formula you know for

the cubic equation cartoons that you can

you can write down out an exact solution

to a cubic equation same with a quartic

equation but that's it no more so I

wrote down a quintic equation and you

can't write down the answers so by

definition this is not an exactly

solvable problem can't write down the

answer okay now suppose your problem is

to find the real root of this by the way

what do you think the answer is going to

be

I mean roughly can you guess if we're

looking for the real root the problem is

to find the real not that big think

about it you have a number which you're

adding to another number and you're

getting one there's a positive root it's

about 0.7 roughly roughly seven five

five something like that

okay you can you know you can just make

a plot draw X to the five draw X add

this to this and ask where does it cross

one right and it's going to be about

seven it's about three quarters

something like that okay so

so we say fine we know we can't solve

this problem exactly let's try to use

perturbation theory so step one we have

to insert an epsilon now this requires a

lot of art and a lot of cleverness

because there are lots of places where I

can put an epsilon there are many

possible places to put in an epsilon

let's take one possibility let's put it

over here okay now if you are doing have

you had any quantum field theory yet

you've had some exposure so what you

should can you say what it is that I

just did this is a strong coupling

expansion if you had a quantum field

theory where the Lagrangian or the

Hamiltonian was something like grad Phi

squared plus M Squared v squared plus

say fight of the Ford does this have you

seen things like this okay so you have

some coupling constant over here okay

and we could treat this as epsilon that

would be called the weak coupling

expansion or in other words we could put

the small parameter in front of the

highest power okay in this case by or we

could put a small parameter in front of

the lower power in which in this case is

just what okay so this is called a

strong coupling expansion but never mind

that's just words okay great so now

we've finished step one now there's one

thing that's very important about step

one that I haven't explained to you yet

when we put the epsilon into the problem

we have to put it in in such a way that

when epsilon is equal to zero we can

solve the problem

because that when epsilon is equal to

zero that's the unperturbed problem okay

and it better be that that's a problem

we can solve exactly

okay so epsilon equals zero is the

unperturbed problem and we have to be

able to solve that problem exactly can

we solve that problem well if you set

epsilon equals zero you get equation X

to the 5 equals 1 and yes I can I can

solve that problem exactly and if we're

looking for the real root I see that the

answer is x equals 1 ok so that's

actually not bad

I mean you know as things go if we were

doing astrophysics or something you know

if the answer is 3/4 and we get one not

bad you know I mean if 23% of the

universe is dark matter and we come up

with say 18 percent 30 percent that's

pretty good um okay

good so we have solved the unperturbed

problem okay so that's as much as we can

say about step one and by the way what

is this one what is this one going to be

that is precisely the first term in this

perturbation series you understand that

so we have replaced one problem by an

infinite number of problems ok but now

we're going to look for an answer here

step two we're going to look for an

answer in the form of a series like this

so the first term in the series is 1

because that's what you get for epsilon

equals 0 and then we have a Upsilon plus

B epsilon squared plus C epsilon cubic

and so on like that ok great

okay so now the next thing we have to do

is to try to calculate these

coefficients in the perturbation series

okay so how are we going to do that well

what we have is this equation here and

what we have to do is to substitute this

series into that equation and there's

one technical problem and that is how do

we raise this series to the fifth power

okay so let's just do a side count can

everybody see if I work over here okay

so the question is how do we raise one

plus something s or something to the

fifth power okay well the first term is

1 and then we have five times that

something plus 10 all right 10 times

that something squared and and so on

okay so if we're if this something is a

epsilon plus B epsilon squared plus C

epsilon cubed and so on then 1 plus that

something to the fifth power is 1 plus 5

times all of this right which is 5 a

epsilon plus 5b epsilon squared plus 5 C

epsilon cubed and so on and then we have

plus 10 times the square of that okay so

if you square that guy what do you get

well first we'll square this term and

you get a squared epsilon squared right

and then you get to a B to a B epsilon

cubed and more terms and I'm tired right

okay

so if you write this out you get 1 plus

5a epsilon plus epsilon squared and

there are two terms there is a 5b

plus a 10 a squared okay and I don't

know whether we have the strength to

keep going but maybe okay then we have

five see we have epsilon cubed times 5 C

and then plus 20 a B ok great so now

this is the technical part of the

calculation it's kind of boring but

let's go ahead and do it

so this equation here the one we're

trying to solve says that 1 is equal to

first we have this X to the 5 term 1

plus 5 a 5 a epsilon plus epsilon

squared 5 B plus 10 a squared plus

epsilon Q 5 C plus 20 a be ok and then

we have epsilon times X ok so epsilon

times this series is just epsilon plus

epsilon squared a plus epsilon 2 B and

you can see that this is very long and

messy and busy but this is trivial okay

I mean you can do this if you're a

junior high school this is not hard

now what do we do next the left-hand

side is a series and powers of Epsilon

the right-hand side well that's a series

and powers of epsilon 2 but all the

coefficients in the series are 0 except

for the first one what am I going to do

right what right have you got to do that

why is it true that the coefficients on

the right hand side have to agree with

each power of Epsilon have to agree with

the coefficients on the left hand side

why is that true sure you could do that

and then equate them to 0 but why does

it have to be why does it have to vanish

term by term well it's because Taylor

series are unique okay so given a

function the Taylor coefficients of that

function are unique what assumption on

my am i making here right this is a very

deep and interesting assumption because

in order to make a statement like that

with the mathematics that you know from

first-year calculus basically you have

to know that the series converges we

don't know that the series converges

because we don't know what the series is

yet so that's why I love to do non

rigorous mathematics we're exploring

this problem maybe we'll find to our

delight that the series really does

converge but we don't know it yet so

we're proceeding boldly okay now term by

term they have to agree so what's the

coefficient of epsilon to the zero power

well that gives me the equation 1 is

equal to 1 it's a very difficult

equation and it's a true equation ok and

that's true because we already did this

calculation to 0th order in perturbation

theory we solve the unperturbed problem

ok now what about epsilon to the first

power that gives me the equation 5a ok

there's an epsilon plus plus 1 over here

because there's an epsilon is equal to 0

now you remember what I told you we do

perturbation theory because it reduces a

hard problem to a sequence of trivial

problem

that problem is pretty trivial you have

to agree this is a pretty trivial

problem okay what about the coefficient

of epsilon squared here's an epsilon

squared so that says 5v plus 10x squared

okay and here's an epsilon squared plus

a is equal to zero now this problem is a

little bit messier than this problem but

you have to agree that this is in the

category of tribute okay in fact if we

solve this problem you get a is equal to

minus one-fifth and if you solve this

problem you just plug in a okay so we

have five B plus ten a squared which is

two-fifths plus a which is minus

one-fifth is equal to zero so five B

plus 1/5 equals zero B is equal to minus

1 over 25 okay and I picked this problem

because without bothering to waste your

time to calculate the coefficient of

epsilon cubed well let me give you an IQ

test 1/5 or minus 1/5 minus 125th can

you guess what C is very good ok great

so now we have this term and this term

in this term so we now know that was

just the technical calculation but let's

write the answer down we now know that

the answer is a series that begins 1

minus 1/5 epsilon minus 1 over 25

epsilon squared minus 1 over 125 epsilon

cubed okay that's pretty cool we

finished step 2

there it is that's the end of step 2 we

have found

the perturbation series and what I said

before is really true if you were in

high school you could calculate lots of

these terms because you'd find this

interesting

okay now by now you'll find it so okay

but there we go now the question is how

do we do step three step three says you

have to add up the perturbation series

with epsilon equals one okay so we set

epsilon equal one so this is step three

we set epsilon equal one and the

perturbation series this is the answer

at epsilon equals one perturbation

series has the form 1 minus 1/5 minus

the 25th - 125 and so on which is 1

minus 0.2 minus 0.04 minus 0.08 right

which is 1 minus 0.2 for a which is 0.75

- ok that's the answer that we get 0.75

- so far okay and the correct answer is

0.75 five okay so with that amount of

work which is really trivial amount of

work very low brow it um we're off by

three parts the error is three parts in

about 750 okay which is a fraction of a

percent so we did a very small amount of

work and we got a very good answer we

didn't get the exact answer and we know

that we can't get the exact answer we

know we can't do that but we did get a

very accurate answer and if we do more

work we get paid for it the answer gets

more and more

more accurate okay so again why do we do

perturbation theory because it takes a

very very hard problem this very hard

problem and reduces it to an infinite

sequence of easy problems and we solve

them one at a time and we get an answer

that becomes increasingly accurate as we

do more and more work okay that's what

this course is going to be about now

this is deceptively simple this is

really simple

for example let's begin can you tell me

you you guessed the 1 over 125 can you

tell me the next term positive

at some point this custard has to rise

up

you're absolutely right whoops how do we

wait ok so the next term happens to be 0

it's not as simple as you think however

if there are people here who like to

solve interesting problems in fact you

can find a formula you can guess just by

calculating a bunch of terms in that

perturbation series you can guess a

formula great thank you you can guess a

formula for the nth term ok now I'm

going to show you some more terms in the

series this is the problem that we've

been talking about and here is the exact

answer you know you could put it on a

computer find the exact answer this is

the problem that we solved and if you

want to go through the trouble this is

what the sequence of easy problems looks

like this is what the coefficients look

like ok it turns out the fourth

coefficient is 0 but as you guessed

eventually you're going to have some

positive contributions and you will have

several positive contributions and then

two others zero and then several more

negative contributions and then another

zero and so on and it'll happen like

that again and if you sum up the series

you will find a very very very accurate

calculation of the exact answer okay and

in fact there is a formula you can guess

if you like to pets IQ test where you

guess the next number is equal there is

a formula for these numbers and you can

guess it if you just play and if you

look hard at those numbers you can guess

what the radius of convergence of the

perturbation series is um and the radius

of convergence is one point six four

nine now that's very very good and it's

very lucky because we were putting in

epsilon equals one so apparently epsilon

equals one is inside the radius of

convergence of a series okay however

what if you were trying to solve the

hard problem that looked like that okay

you would do exactly the same thing

except at the end of the you replace two

by epsilon right and at the end of the

calculation you would try to set epsilon

equals two but now you would have a

problem wouldn't you so in fact one of

the subtle things in perturbation theory

is this step you might think that the

first two steps are the subtle steps but

the really subtle and interesting step

in doing perturbation theory is step

number three that is plugging in the

value of epsilon and getting the right

answer and one of the things I'm going

to teach you in this course is

when you have a divergent series it's

not an obstacle I'm going to explain to

you how to sum a series that diverges

and how to obtain the correct finite

answer from a divergent series and in

doing this we're going to have to

develop a new kind of mathematics

because the kind of mathematics that

you've been taught until now is exact

mathematics okay

it is the mathematics of equal signs and

this is a problem because this

mathematics is very limited in power and

what we're going to do is replace this

mathematics by what is called

asymptotics instead of exact mathematics

we're going to be doing asymptotics and

we're going to replace the equal sign by

that symbol and this is a much more

powerful kind of mathematics because it

is not exactly correct okay so it's

going to be far more powerful of course

when I say it's not exactly right I want

to emphasize that it is arbitrarily

accurate so there's a difference between

saying that F is equal to G this means F

is exactly equal to G there's no

difference between F and G none

whatsoever or assume when I write down

an equation like this F is asymptotic to

G that means that F comes arbitrarily

close to G without actually being equal

to G okay now you may think what's the

difference and it turns out there's all

the difference in the world because this

kind of mathematics allows us to solve

incredibly hard problems okay and it's

based on the fact that having a

divergent series is no problem at all in

fact we're going to learn that divergent

series converge to the answers converge

to the things that they represent much

faster than convergent series generally

we're kind of disappointed if we get a

series like this that converges

okay it's much better and faster if the

series diverges because we can extract

the answer far more efficiently far more

rapidly okay I'm trying to entice you

into this

of course okay okay so let me let's look

at a slightly different way of solving

that problem because I think just will

emphasize what's going on someone else

comes across this problem here okay

someone else discovers this problem and

they have just taken a course in quantum

field theory so they say look I've

learned all about weak coupling

expansions and Fineman diagrams and

things like this so I'm not going to put

the epsilon over here I'm going to put

the epsilon over here seems like a

perfectly reasonable way to approach the

problem so step one I'm going to solve

the problem epsilon X to the five plus x

equals one okay why did I choose to

insert epsilon over here because once

again when epsilon is equal to zero the

unperturbed problem is really easy to

solve

in fact when epsilon is equal to zero

the unperturbed problem becomes x equals

one okay and the solution to that

problem is x equals one okay that sounds

pretty good hmm great so now let's do

perturbation theory okay let's assume

that the answer X as a function of

epsilon has the form 1 plus a epsilon

plus B epsilon squared and so on and we

plug it into this problem okay and let's

see what happens first of all we have to

we've already of course we've already

been raised we already know what happens

when you raise X to the fifth power but

now things are going to be a little bit

simpler because this thing is going to

be multiplied by epsilon so we don't

need if we're going to work to say order

epsilon squared we don't need so many

terms let's see we don't we don't need

this drug because all this is going to

be multiplied by epsilon right so when

we do perturbation theory we have

epsilon times m1 plus 5a epsilon plus

epsilon squared times 5b plus 10a

squared okay there's there's the epsilon

times X to the five term okay and then

there is a X term which is one plus a

epsilon plus B epsilon squared plus C

epsilon cubed okay and so on and all

this has to be equal to what okay now

you know what we do next

we have to look at the coefficients of

like powers of epsilon so epsilon to the

0 the coefficient of epsilon to the zero

on the left is 1 and on the right is 1

so the equation becomes 1 equals 1 and

that had to work because this is a this

is the unperturbed problem we've already

solved the unperturbed problem so this

is merely you know telling us we haven't

made a mistake yet okay and here's the

epsilon to the first power so let's see

there's one term over here which is just

1 that's epsilon squared X 1 Q up here's

an a epsilon 1 plus a is equal to zero

good

okay that's the next equation to solve

the epsilon squared equation reads from

here there's 5a and from here there's a

B and that's equal to zero these

equations actually look simpler than the

previous equations okay and epsilon cube

what about that oh let's see this says

5b plus 10a squared okay there's an

epsilon cube term here's one plus C is

equal to zero and so here are the

equations they're pretty simple so let's

see the solutions in this equation is a

equals minus one okay you put in minus

one here and you get what's beak five

okay

so b is equal to five now we go to this

equation okay and five times five is 25

plus 10 a squared 3 squared is 1 plus C

equals 0 so C is equal to minus 35 okay

so now the perturbation series looks

like this if you plug into the

perturbation series we know that X of

epsilon is 1 minus epsilon plus 5

epsilon squared minus 35 epsilon Q so on

ok now we get to the set the last step

the third step of the problem so this is

we're done with step two step three is

we set epsilon equals one and we sum the

series so our first approximation is 1

and if we take two terms in the series

we get zero and if we take three terms

in the series we get zero plus five okay

if we take four terms in the series

you get five - 35 - 30 and so on not

quite as effective as before

apparently hmm and so let's let's let me

just show you what happens if you

continue with the work and you will put

an epsilon over here this is what I call

a weak coupling approximation and you

work out the perturbation series this is

what the perturbation series looks like

and once again by the way if you like

interesting problems you can just by

staring at these numbers that's the best

way to do it but I don't know a better

way to it you can stare at these numbers

and if you work at it for a while you'll

find a formula there's there's a simple

closed formula for these numbers now if

you try to sum the series at I wrote x

equals 1 but I mean epsilon equals 1 um

the radius of convergence of the series

is 0.08 and you're plugging in epsilon

equals 1 so it looks like we have here a

disaster if you just take this series

just up to this point and you plug in

epsilon equals 1 the prediction is that

X of 1 is 20 1476 which is not a good

approximation to 0.75 5 not really very

good approximate and this is the problem

in quantum field theory okay which if we

have time we will talk about it of

course okay so in quantum field theory

this is exactly what happens in fact in

general in quantum mechanics not just

quantum field theory it is a fraud

because in a standard quantum mechanics

course you are taught perturbation

theory okay that's usually during the

sleepy part of the course

the second term when everybody is for it

your torque perturbation theory and

that's it okay and you think that you

know how to solve problems and that is

garbage because almost always if you

pick a series out of a hat it will be a

divergent series it is rare and unusual

for a series to converge

however we are delighted when we see a

divergent series like this and that is

because there are powerful although not

completely rigorous techniques in

mathematics that you can use to sum this

series and get an arbitrarily accurate

finite answer and not this answer of

course but the answer Oh point seven

five five okay and I'm going to be

teaching you those techniques it's hard

to believe that you can sum the

divergent series and get a finite answer

or meaningful finite answer but you can

one such technique is called pod a

summation we're going to talk about that

but there are lots of techniques such as

Borel summation um that we're going to

learn okay so we need to learn how to

sum a series okay and one thing that I

will be teaching you in this course is

how to sum a series if it converges

because the general rule is that if you

are given a series and you have to add

it up the dumbest thing that you could

possibly do is add it up okay if you

need to find the sum of a series the

worst possible technique is to add the

numbers in the series together one at a

time it's a terrible idea so much better

ways of summing a series okay and if the

series diverges it's not only a stupid

idea

it doesn't work because if the series

diverges you know what the answer is

it's infinity okay so this is the

outline of the course I was thinking

about

how to organize the material in this

course oh and this is what I came up

with on the airplane and I hope it works

way thank you what the last two times

it's it work we'll try it again um so

basically this is the beginning of what

I will be talking about um I will first

show you how to sum a convergent series

we're going to spend a day talking about

things like that I will then show you

how to sum a divergent series and we

will apply it to various interesting

series that we get solving hard problems

and what I want to do in this course

mainly is to teach you some very very

once you know how to some perturbation

series teaching perturbation theory is

no longer a fraud because it's something

that you can really use okay you do when

you get a divergent series you're going

to learn what to do with that theory and

I would like to teach you some very very

fancy techniques perturbative techniques

i'd like to teach you boundary layer

theory which is a very powerful

technique for solving very difficult

differential equations and boundary

layer theory in fact is a special case

of an even more powerful perturbation

theory called wkp wkb it how many how

many of you have seen wkb i've heard it

talked about good okay um I doubt that

you've seen it done correctly um it's a

technique that is usually just run over

very very quickly but it is an

incredibly beautiful idea and I would

like to talk about that and depending on

how strong you are maybe we'll do lots

more who knows maybe we'll do multiple

scale perturbation theory which includes

wkb as a special case we'll just have to

see how interested you are and how fast

we can go okay um the shanks transform

is what I'm going to be talking about

next time and I don't want to begin

talking about it now the shanks

transform is a technique that we're

going to use to solve converted series

okay but before we get to some of these

techniques what I want to do is talk a

little bit more about this problem

because I want to find out why it is

that this problem is a really this seems

to be a really nice problem the way we

solved it over here and the way we

solved it over here it wasn't such a

nice problem okay it seemed to give us

difficulty can you tell me why that is

why is it that solving the problem this

way was not didn't seem to be so

effective at least at a naive level it

gave rise to a divergent series why is

it that this method wasn't as useful

superficially at least as this method

what was the difference what

why is putting an epsilon over here

different from putting an epsilon over

here but in this case we put the epsilon

here in the second case we put the

epsilon in front of the X to the 5 term

what's the difference why is there a

difference yeah something about the

number of roots changing right very very

very good excellent because there you

always you have one root of n you have

some yeah excellent excellent you're

absolutely right ok so there's a

difference in this problem we put an

epsilon into the problem in such a way

that nothing abrupt happened at epsilon

equals 0 that is as you go down to the

unperturbed problem there were always

five roots but something very strange

happened over here when epsilon was

equal to zero there's only one root and

not five roots so the question is

where'd the other four roots go

okay so I would like to ask can we solve

this problem approximately we can't

solve this problem exactly but can we

solve this problem approximately when

epsilon is approaching zero now we don't

know the mathematics for doing this okay

of course we do we would know the

mathematics if we had a quintic formula

but we don't have a clicking formula so

I need to teach you new mathematics okay

that's the next step so let's let's

begin by talking a little bit about

asymptotics okay so I want to introduce

this is a very beautiful and a very

subtle idea and I'm just going to give

you a superficial introduction to the

notion of asymptotics okay

and remember asymptotics is in fact when

I explain this to classes that I teach

um I say you know the first amazing

breakthrough in mathematics you saw I

guess when you were in elementary school

and you learned this symbol okay and

then you learned and from there you

learned exact mathematics and what you

learned is that whatever you do to the

left side of this equation you have to

do to the right side of this equation

and it remains an equation okay and the

word equal an equation or the second

okay

what we're going to do is we're going to

replace this symbol by this symbol now

this symbol me if you see this symbol

this symbol is red is asymptotic to this

symbol means is equal to that simple

is asymptotic to let me tell you what

this symbol isn't this symbol don't make

a mistake that there are a lot of

symbols that are written down by sloppy

mathematicians or maybe there aren't any

sloppy method but sloppy physicists you

know you see things like this have you

seen that symbol or maybe this symbol or

I don't know you've seen various forget

these these are really these are

ridiculous symbols I mean you've seen I

mean is it

I think this symbol means is

approximately equal to but I don't know

what that means

is approximately equal to is 3

approximately equal to 4 that's a

ridiculous question of course it isn't 3

is 3 4 is 4 because it is true that the

square root of three approaches 2 for

large values of 3 but in general 3 is

not equal to 4 it's also not true that

99 is approximately 100 because it isn't

99 is 99 100 is 100 they're different so

you know you could try to introduce

fuzzy thinking into mathematics and say

99 is very much like 100 but it isn't

okay and this symbol here is a very

precise symbol okay

so what I want to define for you is what

it means to say that f of X is

asymptotic continue x okay it turns out

already this is not a correct statement

of anything okay this doesn't mean

anything okay or it's an incomplete

statement it's like the sentence that

reads the rest of the sentence is

written on a rock in

okay this is an incomplete sentence so

an asymptotic approximation okay or an

asset this is an absent product relation

okay

must be associated with a limit and a

correct statement would read f of X is

asymptotic to G of X as X approaches X

naught okay and I don't have to write as

this is a correct statement and let me

tell you what this means it means that

the limit as X approaches X naught of f

of X over G of X is what that's what it

means okay so let's write down a few

asymptotic approximations or add some

type of asymptotic relations sine of X

is asymptotic to X as X goes to 0 is

that true or not okay and by the way

this and let's write another one e to

the X is asymptotic to 1 as X goes to 0

that true

you notice something very interesting

happening this is a rather complicated

function and this is equated this is

quite a simple function okay same here

this is a rather complicated function

this is quite a simple function

asymptotics is a way of replacing very

complicated functions like Bessel

functions or functions that don't even

have names by functions that do have

names and that are very simple so

asymptotics is a way of simplifying

mathematics that's very very very

complicated okay so this is a this is a

correct statement how about the equation

X cubed is asymptotic to zero as X goes

to zero you say you say no that's right

this is a completely wrong statement so

let's remember that nothing is ever

asymptotic to zero you can never say

that something is asymptotic to zero

nothing is asymptotic to zero of course

X cubed is very very small as X

approaches zero but the best that I can

say is that X cubed is asymptotic to X

cubed as X goes to zero because I don't

know a simpler way of writing this of

course I could write it in a more

complicated way I could say X cubed is

asymptotic to X cubed plus X to the

fourth as X goes to zero that's true

okay

but if I'm trying to put a simpler thing

on the right-hand side I don't know a

simpler thing so remember that nothing

nothing is asymptotic to zero ever

nothing is ever asymptotic to zero oh

you are you feel something simple did

you muscle Thanks

what's that excuse its asymptotic tax no

it is not effective to zero no that's

not true

and cubed is not absent on ik to X as X

goes to zero okay because the ratio of X

cubed over X either approaches infinity

or 0 but not one okay so this this is a

quick definition of the asymptotic

symbol and there are two different

asymptotic symbols we're going to use in

class one of them is this symbol here

but there's another symbol that's also

very useful and that's this symbol okay

and this symbol is red is negligible is

negligible compared with okay so for

example X is negligible compared with 1

as X goes to 0 okay don't make a mistake

by the way this doesn't mean less than

and it doesn't mean much less than it

doesn't mean that okay because less than

is an order rate relation you know

positive negative you know minus 1 is

less than plus 1 okay so for example um

X is say x squared is 4 let's say minus

x squared is less than what that's

certainly true

if X is real certainly less than one but

one is negligible compared with x

squared or minus x squared okay as X

goes to infinity so what is negligible

compared with me it means when you write

it when you say f of X is negligible

compared with G of X as X approaches X

naught what this means is that the limit

as X approaches X naught of f of X over

G of X is zero so over here that limit

is 1 so they're approaching each other

okay over here 1 is becoming F is

becoming negligible compared with the

other because this ratio is going to

zero you understand that's a quick dose

of asymptotics okay so let's see how you

would use asymptotics to understand a

problem because good time is right good

okay so you pointed out that what we did

over there was we lost some roots and

the question is where did the other

roots go where are they okay we lost

four out of the five roots where'd they

go just disappeared out of the universe

or something like that

where'd they go so to understand where

those roots went let's go back and think

let's go back and think about what we

did in this problem as we approached the

unperturbed problem here that's that's

the question that we'd like to think

about all right so let's try to

understand the problem of X to the 5

plus x equals 1 when you put an epsilon

in front of here and when we examine in

this problem in the limit as epsilon

goes to 0 what happens okay now

asymptotics is fantastic because

problems that you cannot solve exactly

problems that you cannot solve when they

contain an equal sign suddenly become

very easy to solve when you have an

aesthetic sign ok so how do you solve

such a problem mmm well you notice that

there are three numbers in this equation

okay in fact let's make a general

statement before we approach that

suppose I have an equation that says f

of X plus G of X equals H of X suppose

you have such an equation okay and

suppose G of X is negligible compared

with f of X as X approaches X naught

suppose that's true okay

what can you then say say I heard you

say it exactly right somebody whispered

it with an emphasis of high pH exactly

if f of X is negligible compared with G

as X approaches X naught then okay let's

say if if this is true then I can throw

away f of X and I can conclude that G of

X is asymptotic to H of X as X

approaches X naught that's the

connection be f of X sorry

oh I'm sorry then f of X is asymptotic

to it G is negligible so f is asymptotic

to H of X as HS X approaches X naught

okay so this is the connection between

this symbol over here and this symbol

over here the asymptotic symbol and the

negligible simple okay so great here we

have an equation with three terms in it

and we're considering what happens as

epsilon goes to 0 now I think let's make

more space here

now typically not always but typically

what happens in a situation like this

where we have an asymptotic limit

typically what happens is that one term

becomes negligible and the other two

terms become asymptotic to one another

now it could be that all three terms are

exactly the same size but what's the

chance of that if you pick three numbers

out of a hat and all three numbers be

the same size very small okay of course

we have to consider that but there's

only a very small likelihood of that so

how do you solve or how do you leave an

attack such a hard problem as this well

we're going to use the method of

dominant balance okay the method of

dominant balance consists of saying it's

likely that one of these two terms

becomes negligible and the other two

terms are now asymptotic to one another

well which term becomes negligible well

there are three possibilities the first

possibility is that one becomes

negligible we could throw away one and

that epsilon X to the 5 a would now be

you know pluck plus X would be

asymptotic to zero oh wait a minute

excuse me

nothing is asymptotic to zero so let's

be more careful

if we can throw away one then it must be

that epsilon X to the five is asymptotic

to minus X that's one possibility we

don't know if that's a correct

possibility a second possibility is

maybe we can throw away X maybe X is the

neck is the thing that's negligible in

this equation in which case of epsilon X

to the five would be asymptotic to one

this is in the limit as epsilon goes to

0 this is as epsilon goes to 0 and a

third possible statement regime so we've

thrown away one we thrown away

X maybe epsilon X to the 5 is negligible

okay and we conclude that X is X and

producti 1 as epsilon goes to 0 so let's

look at each of these statements and

turn and see if any of these are correct

okay we don't know which of these might

be correct let's see um what about this

one

is this a poss is this a correct

statement

well absent products is self-referential

do you like self-referential statements

it's very hard to prove things with

asymptotics but it's very easy to verify

things okay so take this statement if

it's true that epsilon X to the 5 is

negligible then it must be true that X

is asymptotic to 1 as epsilon goes to 0

could X be asymptotic to 1 as epsilon

goes to 0 if it is would it then be true

that epsilon X to the 5 would be

negligible sure because epsilon is going

to 0 if X is asymptotic to 1 that means

it's very close to 1 you raise it to the

fifth power you multiply it by Epsilon

can we neglect this compared with the

other two terms in the equation yes this

is perfectly valid okay in fact

that is true about what we wrote down

here okay in fact there is a route that

is very near what okay in fact it's 3/4

that's quite one hmm okay so this is

okay there's nothing wrong with this but

we didn't learn anything we didn't know

what about this

well asymptotic equations are like

ordinary equations equations with equal

signs in the sense that you can

generally do the same thing to both

sides of the equation and it still

remains asymptotic so for example I have

an asymptotic equation here how would I

solve this equation

well I begin by dividing by epsilon X to

the 5 is asymptotic to 1 over epsilon as

epsilon goes to 0 take the 5th root so X

is asymptotic to 1 over epsilon to the

1/5 as epsilon goes to 0 okay and by the

way when I take the 5th root there are

five possible roots I get here

I really shouldn't write one I should

write Omega where Omega is one of the

fifth roots of one you know you know

some complex variables what are the

fifth roots of one what would Omega be e

to the 2 pi I over 5 or e to the 4/5

times

over five and so on those are the fifth

roots of one okay so so Omega is a

complex number and when you raise it to

the fifth power you get one so this is

the solution to this equation

that's the asymptotic solution to this

equation now was it valid to throw away

X well as epsilon goes to 0 you could

see that X is blowing up right because

there's an epsilon to the 1/5 in the

denominator you will see that so we

looked at this equation and we said

throw away X okay that's what we said it

throw away X keep these two terms so for

example we kept the term 1 and we threw

away X but you notice that X is gigantic

it's blowing up it's going to infinity

as epsilon goes to 0 and we throw it

away and we kept what was that valid no

not that contradiction okay so as some

products is self-correcting it says you

made a wrong assumption well there's

only one remaining possibility and I

sure hope this is going to be it okay

now if X is not equal to 0 we can divide

both sides of this equation this

asymptotic approximation by X and we get

epsilon X to the 4 is asymptotic to

minus 1 as epsilon goes to 0 and now

what do we do with this this is an

asymptotic equation for an asymptotic

relation right let's divide by epsilon X

to the 4 is asymptotic to minus 1 over

epsilon as epsilon goes to 0 now let's

take the fourth root X is asymptotic to

minus 1 over epsilon to the 1/4 as

epsilon goes to 0

but it's not really one here right it's

one of the fourth roots of one or minus

one

okay so I'll write here Omega where

Omega to the fourth is equal to minus

one some one of the four roots of minus

one okay now what did we do here we

threw away one but we kept X and epsilon

X to the five was that valid yes it is

true that one is negligible compared

with X okay which is of order you know 1

over epsilon to the 1/4 this is a valid

equation as epsilon goes to 0 okay

that's perfectly okay therefore now we

know that x has this form and now we

know what happened to the other four

roots we have just calculated

approximately using asymptotics the

other four roots and we know what

happened we know the solution to this

puzzle where do the other four roots go

the other four roots these are the four

roots of minus one the four roots of

minus one are distributed in in the

complex plane like this okay and the

distance from the origin in the complex

plane is going to infinity right it's

going like one over epsilon to the

one-fourth so the other four roots went

off to infinity or like this that's

where they went okay

do you see that we have cracked into

this very very hard problem by replacing

the equal sign by the approximation

santi asymptotic approximation sign okay

and in

fact this is just the first

approximation we can make we can develop

an entire an entire series where this is

the first term okay and we can find more

and more and more precise approximations

of the roots of this of this polynomial

equation we can find arbitrarily

accurate calculations of the solution to

this approximate equation arbitrarily

accurate there is no limit to the amount

of accuracy so if you walk into the room

and you say I demand 15 places 15

decimal places accuracy I can give you

15 decimal places okay and if you walk

in and you say no I want 30 decimal

places after C I can give you 30 no

problem okay but I can never tell you

what the answer is equal to that's

impossible because this is not a

solvable problem okay so I want to

emphasize in this course that there's a

very there's a giant goal in between an

exact approximation okay an exact

solution that is or an arbitrarily

accurate approximation okay now let's

see all right okay so let's stop here um

next time I want to show you that the

mathematics that I've taught you is so

powerful that we can solve problems like

that in a few minutes and it's entually

in under a minute that you have been

told your entire life are absolutely in

solvable problem cannot be saw that

what's an example of an infinitely hard

problem how about a Schrodinger equation

right nobody knows how to solve the

Schrodinger equation and you take

quantum mechanics you learn the harmonic

oscillator and you'll learn the hydrogen

atom and then you give up because there

really aren't any other

Skirving equations you know that's not

true we're going to solve them all if

they need more along the lines of G city

yeah we can then the next step is Q CP

which bitches okay um are there

questions before we click for comments

for something sorry

yeah well I have the recognize the world

what you haven't done

yes diamond this this word is dominant

balanced okay what I mean by dominant

okay is that all of the terms in the

equation dominate over one term in the

equation okay so here we have an

equation containing three terms and two

of these terms dominate over the third

term and what we found is that this is

the correct answer what we found is that

this term and this term are big and this

term is negligible so these two terms

here dominate over that term and there

but if this term is thrown away because

it's unimportant then these two terms

have to be balancing each other because

this very big term and this very big

term have to almost cancel each other

because this term is negligible you see

but it's fantastic because you see we

don't know how to solve an equation with

three terms in it but we do know how to

solve an equation with just two terms

okay so asymptotics allows us to

simplify equations we for example in

this course will never need to use the

quadratic equation because aesthetics

says you don't need to solve a three

term equation you can always replace it

by a two term equation which doesn't

require that you remember anything

so asymptotics is a fantastically

powerful and wonderful mathematical tool

if you don't know add some products

basically you are reduced to just

sitting in front of a computer and

generating numerical solutions to

problem and this course is about

generating analytical solutions okay

yes I'm with confused like in this kind

perturbation theory we introduce small

parameter yes then make sure zones as a

junior community but for example in

quantum mechanics is true we we usually

like have a small parameter we don't set

the unity like in the end so it's like

parameter and the problem okay so now

the parameter you're referring to here

is Oh example some relation of two like

physical LM symmetry okay so we are

going to talk about this in great detail

this is what this course is going to be

about often we treat H bar and this

being small but you and I know that H

bar is not small that's equal to what

okay because it depend if R is as a

number that contains units so depending

on what system of units you choose H bar

may be gigantic or it may be small you

can't say that H bar is a small number

or a big number in MKS units yeah it's

10 to the minus 34 but that's not a

small number because it contains

dimensions which make that number either

big or small okay but it's often very

very powerful to think of H bar at being

a small number and we have to make that

precise which we will and when you do

that is where wkb theory comes from

that's where that's where does it baby

if you put if you treat other numbers in

content items being small then you are

doing other kinds of perturbation theory

and we're going to do that we're going

to investigate that in great detail so

that's the objective of this course is

to take a very fancy equation like the

shorter human

and try to find a small parameter try to

find it and treat it consistently

through perturbation gear consult that's

what we're going to do there's a very

subtle problem and a very interesting

one of the things that you're going to

learn along the way is why it is that

quantum mechanics is quantized you know

why our energy levels to sweep

perturbation Theory will explain that

okay any other questions okay well I

hope I've got you interested in the idea

that

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