The h Bar

0celo7

20:00

Also that seems pretty hard. You'd want a graph theorist for that, not a topologist.

loltospoon

@0celo7 sorry to bother you again about the same thing. I'm still working on getting the total energy of a numerically wavefunction, but now I need to decide on the method of integration. The standard method that I used simply did $\psi (x)^* \frac{\hbar ^2}{2m}\frac{\partial ^2}{\partial x^2}dx = \frac{\hbar ^2}{2m}(\psi (x) \psi (x) '' $(step_size). But this gave me 10000% error! Should I be using some other method of integration??

0celo7

10000%??

loltospoon

Yes lol

dmckee

@DanielSank There are umptidy-ump nusiance games for phones that work on the same principle. They're fun for a while, but eventually you get to the point where it's too easy unless it is too hard.

Then you de-install the game.

0celo7

Are you sure it's not 100%?

loltospoon

20:02

Its either that Euler Method is really bad or my method of integration was wrong

0celo7

did you divide by 100

loltospoon

Percent Error = $\frac{\text{computed - analytic}}{\text{analytic}} * 100 $%

DanielSank

@dmckee Hmmm, are you suggesting I de-install the project I'm working on?

heheheheh

loltospoon

@0celo7 that's what I did

0celo7

er, multiply

well, idk

I'm studying for an exam rn

so I can't help, sorry

loltospoon

20:05

Oh ok, no worries

0celo7

ask @ACuriousMind

loltospoon

Can anyone else suggest a method of integration for computing the energies of a wavefunction?

@ACuriousMind

0celo7

did you remember to normalize $\psi$

AccidentalFourierTransform

@loltospoon can you use Mathematica?

loltospoon

No, unfortunately :( I can only use it to check that my integration is accurate

AccidentalFourierTransform

20:08

so what are you using? c++, matlab?

loltospoon

@0celo7 wait......wait a minute....

@AccidentalFourierTransform I am using Swift >.>

AccidentalFourierTransform

wat

loltospoon

...because my professor is trying to be hip and trendy.

AccidentalFourierTransform

really?

loltospoon

sigh yea....

We already tried to talk him out of it but he wouldnt budge

so it's Swift for an entire semester

AccidentalFourierTransform

20:10

so, you probably want to represent the second derivative as a matrix

loltospoon

of what size? I am currently storing second derivative values in a 1-D array

AccidentalFourierTransform

let me see if I can find something online

loltospoon

THANK YOU!!! This assignment is already late and I'm just trying to make something that works to send it to him. I thought I was done last night until I checked the percent error and was like whaa........

AccidentalFourierTransform

read this:

5

Q: Numerical solution to Schrödinger equation - eigenvalues

This is my first question on here. I'm trying to numerically solve the Schrödinger equation for the Woods-Saxon Potential and find the energy eigenvalues and eigenfunctions but I am confused about how exactly this should be done. I've solved some initial value problems in the past using iterativ...

the first answer explains how to represent the second derivative as a matrix

that is what you want to do

loltospoon

Ok I'll give it a go

0celo7

20:15

D I D Y O U N O R M A L I Z E

AccidentalFourierTransform

no need to normalise

you are solving an eigenvalue equation

the norm of the vector is irrelevant

0celo7

if he's computing $\int \psi^* H\psi$ he needs to

AccidentalFourierTransform

but why would he!?

0celo7

because he's doing the variational method

AccidentalFourierTransform

is he?

lol

0celo7

20:16

That's what I've been assuming

AccidentalFourierTransform

@loltospoon are you?

0celo7

@loltospoon what is the truth

AccidentalFourierTransform

if you are, forget everything I said

0celo7

He asked me a few days ago how to compute the energy of a wave function

I said $\langle H\rangle$

So I'm assuming he wants a variational method

AccidentalFourierTransform

damn I seriously misread his first message

0celo7

20:18

honestly why do I bother with you

you don't smell good, can't read, and think I'm a failure

loltospoon

What? This isn't the variational method

0celo7

wtf

loltospoon

I need to generate a wavefunction for any given potential

So I started with the infinite square well

because $V=0$

0celo7

that's what the variational method gives...

it gives you an energy that lets you know how good your approximation is

AccidentalFourierTransform

no, wait

ACuriousMind

20:19

@loltospoon Could you maybe lay out what exactly the question you're trying to answer is? What are your givens, and what is your goal?

loltospoon

Variational method gives an upper bound on the energy of ground state, I'm trying to get eigenvalues and eigenfunctions

AccidentalFourierTransform

what is exactly what you want?

loltospoon

Sure @ACuriousMind

Given: Create a program that can take in data about any potential $V(x)$ and return the eigenvalues to the time-independent Schrodinger equation.

So, in my simple case of the infinite square well, $V(x)=0$ for $0<x<a$.

0celo7

Eigenvalues.

So variational method.

AccidentalFourierTransform

ok so back to where we were

loltospoon

20:21

I'm confused

I'm also new to QM, so please bear with me.

AccidentalFourierTransform

no, no variational method

you want to numerically diagonalise a matrix

0celo7

@AccidentalFourierTransform Dude it's not a matrix

it's a potential depending on $x$

AccidentalFourierTransform

even though I misunderstood your initial question, my answer was correct

loltospoon

The program needs to tell me $E_1$, $E_2$, ...$E_n$ where these energies are allowed energies for the system.

AccidentalFourierTransform

you want to represent the Hamiltonian as a matrix, and numerically diagonalise it

@loltospoon yeah, did you read the post I linked before?

loltospoon

20:22

@AccidentalFourierTransform finishing it now

AccidentalFourierTransform

ok

the last paragraph is irrelevant

loltospoon

Do I need to use Numerov's method or nah?

AccidentalFourierTransform

nope

\begin{align}
-\frac{\hbar}{2md^2}&\left(\begin{array}{cccccccc}
-2 & 1 & & & \\
1 & -2 & 1 & & \\
&\ddots&\ddots&\ddots& \\
& & 1 & -2 & 1 \\
& & & 1 & -2 \\
\end{array}\right)
\left(\begin{array}{c}
\psi_1 \\
\psi_2 \\
\vdots \\
\psi_{N-1} \\
\psi_N
\end{array}\right) +\\
&\qquad\qquad\quad\left(\begin{array}{cccccccc}
V_1 & & & & \\
& V_2 & & & \\
& & \ddots & & \\
& & & V_{N-1} & \\

this is what you need

loltospoon

Ok

AccidentalFourierTransform

do you understand what that formula means?

loltospoon

20:24

so now pretend

that I am completely incompetent

How do I use that?

Remember, we are just pretending :p

AccidentalFourierTransform

lol

so, again, do you understand what that formula above means?

loltospoon

It's the Schrodinger equation but with diagonal matrices

AccidentalFourierTransform

(BTW you know how to user mathjax, right?)

loltospoon

Yup

AccidentalFourierTransform

@loltospoon yes. It is an approximation to the Schrödinger equation using a discretisation of the interval

DanielSank

20:25

@dmckee Routing this board just got hard. I've heard that simulated annealing is used to do this kind of thing automatically. If only I had a quantum computer.

AccidentalFourierTransform

it is not exact, but you can get arbitrarily close to the true energies by letting $N\to\infty$

0celo7

@DanielSank those don't actually exist

AccidentalFourierTransform

so you want to take a very large $N$, say, $N=100$

0celo7

It's a ploy by various groups to get funding

loltospoon

@AccidentalFourierTransform ok so then what does the left most matrix represent?

0celo7

20:26

Like global warming, string theory

DanielSank

@0celo7 It's possible that the CIA has a quantum computer and the government funds other projects to keep us all of their scent.

AccidentalFourierTransform

@loltospoon it represents the second derivative of the wave function

you know how to approximate derivatives, right?

via Taylor, etc

0celo7

@DanielSank doubtful. I don't really believe in quantum mechanics

loltospoon

@AccidentalFourierTransform no.

DanielSank

@0celo7 Lol, see JS Bell.

loltospoon

20:27

For my second derivatives I was using $\psi _n '' = -k^2 \psi _n$

AccidentalFourierTransform

@loltospoon basically, $f'(x)\approx \frac{f(x+h)-f(x)}{h}$

that expression becomes exact if you let $h\to 0$

and it is a good approximation if $h$ is small and $f$ is not wild around $x$

@loltospoon nope, that is wrong here

not really

it is wrong in general, here it is correct because your $V$ is flat

loltospoon

Ohhhhhhhhhhh geez, then that's where my error was coming from?

AccidentalFourierTransform

but what you want is a piece of code that works for any $V$, not only for flat potentials

loltospoon

Right

AccidentalFourierTransform

you want something much more general

loltospoon

20:30

Ok quick side question - how long will you be in the chat room?

AccidentalFourierTransform

15 more minutes, maybe a lil bit more

loltospoon

Ok in that case let me plan all this out on paper and then I'll try to make it so that I fully understand where to go with this

0celo7

@DanielSank Just Shitting Bell?

AccidentalFourierTransform

@loltospoon im not sure what you mean. Do you want me to try to explain it to you now, or do you prefer to think about it and discuss it all again some other time?

0celo7

@DanielSank So since you don't care about uniform convergence, I'm assuming you wouldn't want to see how to solve that integral equation?

loltospoon

20:33

@AccidentalFourierTransform do I specifically need to use matrices or can I simply use arrays?

AccidentalFourierTransform

you definitely need matrices

loltospoon

Oh I just meant that if you were going to leave soon I'll try to get all my questions in quickly before you go

AccidentalFourierTransform

ok, as you wish

DanielSank

@0celo7 I'd be interested to know the solution.

AccidentalFourierTransform

in a nutshell, what you want to do is the following:

DanielSank

20:34

I didn't actually think about it though.

Also, I never said I don't care about uniform convergence, did I?

AccidentalFourierTransform

first, construct an $N\times N$ matrix, your Hamiltonian $H$

that matrix is constructed using the formula I copied above

dmckee

@DanielSank I saw a paper once characterizing the real difficulty of presumed NP problems in terms of their size. The message was small enough NP problems are actually easy, slightly bigger ones are tractable if you accept approximate solutions by using a heuristic then improve approach, and there is a size at which they suddenly become just plain hard.

AccidentalFourierTransform

you pick some $N$, eg $N=100$, and construct your matrix $H$

this matrix is the sum of two matrices, the first one representing the kinetic energy (the one with $2$'s and $\pm1$'s)

and the second one representing the potential energy (the one with $V_i$'s on the diagonal)

once you have the $H$ matrix, you numerically diagonalise it

with any built-in function that your program has

DanielSank

@dmckee Sounds right.

AccidentalFourierTransform

(you really do not want to code your own diagonalisation algorithm, it is a pain in the neck)

tge eigenvalues of $H$ are the energies of your system $E_1,E_2,\cdots$

(youll only get the first $N$ of them)

loltospoon

20:39

Ok so bad news....Swift can't do matrices....

facepalm.jpeg

AccidentalFourierTransform

that makes no sense

DanielSank

@loltospoon Swift the programming language?

loltospoon

@DanielSank yes

AccidentalFourierTransform

you cannot solve the problem without diagonalisation

loltospoon

@AccidentalFourierTransform wait, there's a glimmer of hope. I've found something about multi-dimensional arrays. One sec.

AccidentalFourierTransform

20:41

not really

you need to diagonalise the matrix

if Swift doesnt do matrices, it wont diagonalise them

even if you work with multi-dimensional arrays

if "matrix" is not a natural concept in that programming language, then neither is "diagonalisation"

and you wont be able to code your own diagonalisation algorithm, it takes a lot of time

loltospoon

Oh gosh

I'm so screwed

AccidentalFourierTransform

tell your professor that Swift is crap

loltospoon

Lol we went to the Department Chair and he sided with our professor....

AccidentalFourierTransform

tell him that you want to use Mathematica or c++

loltospoon

We've spent so much time just figuring things about the language rather than the physics, which is ironic because he stresses that he wants us to focus on the physics behind the problems.....all we can really do is waste time dealing with Swift syntax

AccidentalFourierTransform

20:47

oh I've been there

0celo7

@AccidentalFourierTransform I love Tay Tay, you take that back

AccidentalFourierTransform

oh no too late, cant delete it

oh no oh no

loltospoon

I posted in SO to see if someone can help with Swift matrices/diagonalization

AccidentalFourierTransform

@mod delete it for me, will you hun?

@loltospoon ok, but really, talk to your professor and tell him that your assignment requires diagonalisation, which cannot be done in Swift

loltospoon

@AccidentalFourierTransform I'm gonna ask a friend to see how he did it

AccidentalFourierTransform

20:50

oh look at me the popular guy who has friends

loltospoon

"friend" i.e. one friend (see: singular)

0celo7

@DanielSank I can write a proof later. It involves uniform convergence :)

AccidentalFourierTransform

lol :-P

loltospoon

I'm so screwed. This assignment was due this past Friday and last night I had to send the pity email to my professor apologizing for the tardiness...

DanielSank

@0celo7 Is it a constructive proof or not?

AccidentalFourierTransform

20:57

anywayz, see you later

loltospoon

See you later, thanks for the help!1

0celo7

@DanielSank Sort of.

It uses a very powerful theorem from functional analysis that I can teach easily. I think the proof of that is what one would call "constructive."

DanielSank

@0celo7 Can I evaluate the function that satisfies the equation?

0celo7

@DanielSank Good question. I'll have to think about it. I can give you a sequence for it I bet.

@DanielSank I think one could find a uniformly convergent sequence. But I'm not sure what the terms are. Maybe it's not constructive :(

DanielSank

@0celo7 nonconstrutive is fine. It's just fun to plot the solutions.

0celo7

21:08

@DanielSank It's constructive. You pick an arbitrary function, then apply an operator a bunch of times and in the limit you get the solution.

DanielSank

@0celo7 Oh neat.

I remember stuff like that.

Isn't that how you prove the uniqueness of differential equations in general?

ODE's, I mean.

0celo7

Yeah, same argument

DanielSank

I remember liking the proof that the sequence of $\{f, Df, D^2f, \dots\}$ is Cauchy.

Here $D$ is some differential operator.

I forget all the assumptions.

0celo7

@DanielSank Not quite. I can tell you later.

Slereah

22:07

@DanielSank $f = \cos(x)$ doesn't sound very Cauchy for that :p

0celo7

Yeah, idk what he's trying to say there

DanielSank

22:29

@0celo7 Picard iterates.

0celo7

@DanielSank yes, but that's not what you wrote.

You want to apply an integral operator

@DanielSank Ok, so $g(x)-\int_0^x f(x-t)e^{-t^2}\,dt=f(x)$.

I actually already wrote it in the suggestive form.

Define $F:C[0,1]\to C[0,1]$ by $F(f)(x)=g(x)-\int_0^x f(x-t)e^{-t^2}\,dt$.

Then check that it's a contraction: $$||F(f)-F(h)||=\max \left|\int_0^x[f(x-t)-h(x-t)]e^{-t^2}\,dt \right|$$

$$\le ||f-h||\int_0^1e^{-t^2}\,dt<||f-h||$$

Slereah

23:37

you know a few years back, when I was young, I tried to derive the Lorentz magnetic mass by hand

and couldn't do it

I wonder if I'd do okay these days

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