What Is the Variational Method's Role in Determining Energy Expectation Values?

[Ashkan95]

Homework Statement

Use variation method and Find approximate answer for enegies and coefficient c1 and c2

Relevant Equations

The all equations in the photo

 

[BvU]

Hello @Ashkan95 ,

$\qquad$ !​

Your 'please tell me' is NOT a problem statement. See PF guidelines , where we also 'ask' you to post an attempt at solution. Then we can help.

The $f$ you quote are for an unmodified box. Are they valid here ?

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[Ashkan95]

Hello @Ashkan95 ,

$\qquad$ !​

Your 'please tell me' is NOT a problem statement. See PF guidelines , where we also 'ask' you to post an attempt at solution. Then we can help.

The $f$ you quote are for an unmodified box. Are they valid here ?

But I don't know anything about this problem, phi is a function that we choose to guess the enegies and c1 and c3 and f1 and f3 because c2 is zero, in fact with changing in the shape of the box we want to find enegies E1 and E 3

 

[BvU]

Ah, I see. So there IS a problem statement . Read it to us, carefully.

Basically you haven't collected enough equations to start doing anything .
What equations do the unknown variables and the unknown function have to satisfy ?

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[Ashkan95]

Ah, I see. So there IS a problem statement. Read it to us, carefully.

Basically you haven't colected enough equations to start doing anything.

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What equations should I collect?my master said this is all we need.please help me

 

[BvU]

Your master thinks you know what to do. You don't. Apparently you have forgotten something.

You still haven't posted a problem statement. Do that and I'll ask you an 'oh, of course!' question.

 

[BvU]

I have to go. Ask yourself: is this an approximate $\phi$ we are looking for, or an exact one ?
Whatever, it has to satisfy the Schroedinger equation. And I suppose you are looking for a steady state ? Aha!

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[Ashkan95]

I have to go. Ask yourself: is this an approximate $\phi$ we are looking for, or an exact one ?
Whatever, it has to satisfy the Schroedinger equation. And I suppose you are looking for a steady state ? Aha!

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I posted a problem statement, now please help me , and yeah I want approximate answer for enegies and c1 and c3

 

[berkeman]

I posted a problem statement

Where? Which words are the problem statement?

 

[Ashkan95]

Where? Which words are the problem statement?

I can't understand what do you people want , what do you mean by problem statement?

 

[BvU]

Your modified version is a lot better already than "please tell me" !

Use variation method and Find approximate answer for enegies and coefficient c1 and c2

Even answers some of my questions ! (new: did c3 disappear? And you found c2 = 0; how?)

So it's an approximation we are looking for. Something gets varied and something gets minimized. You must have something in your notes(textbook, handout, ...) that should be added to the list of equations we are going to solve !

I find it an interesting exercise, but it's been a while since I did things like that, so you'll have to help me a little bit too.

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[Steve4Physics]

... what do you mean by problem statement?

Hi @Ashkan95. Maybe you can confirm (or otherwise) that the problem-statement is this:

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For the potential well and wave functions (f₁ and f₃) given in the Post #1 diagram, consider the wave function Φ = c₁f₁ + c₃f₃. Us the variational method to find (for Φ) the smallest possible expectation value of energy, <E>, and the corresponding values of c₁ and c₃.​

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Note:
It might be that the actual question is to estimate the ground-state energy, E₀. But the method is the same – we find <E>. Since <E> is equal to or larger than E₀, that means <E> is an upper limit on the value of E₀. And if Φ is a decent approximation to the true ground-state wave function, then <E> will be a decent approximation to E₀.