Solutions to a quantum harmonic oscillator - desperate for help

[Bowenwww]

I need to find the value σ for which:

ψ₀(x) = (2πσ)^-1/4 exp(-x²/4σ)

is a solution for the Schrödinger equation

I know the equation for the QHO is:

Eψ = (P²/2m)ψ + 1/2*mw²x²ψ

I've tried normalizing the wavefunction but I end up with a σ/σ term :(

Any help would be greatly appreciated :)

 

[Clever-Name]

You aren't using the right form of the Schrödinger Equation. Try plugging it into the form with the derivatives:

<br /> E\Psi = -\frac{\hbar^{2}}{2m}\frac{\partial^{2}}{\partial x^{2}}\Psi + \frac{1}{2}m\omega^{2}x^{2}\Psi

Solve the resulting equation

edit - wrote the wrong equation.

 

[Bowenwww]

that is the equation I was using I just put P as the momentum operator :( I'm confused! Do I enter the wavefuntion into Schrödinger and then solve for what?

<3 xoxo

 

[Clever-Name]

Are you given the general expression for E_n ?? You are given the expression for \psi_n for n = 0. So take the necessary derivatives, plug in the expression for E_0 and recall that this solution must be valid for all x. Do you know a specific value for x that might make this much easier?

edit - the general expression for E isn't really necessary now that I think about it. You'll still be able to solve for \sigma in terms of E

 

[Bowenwww]

I can find σ in terms of E but the question says give the value for it that makes ψ₀ (the one I originally posted) a solution to the Schrödinger

Then part II asks me to find the energy associated by putting the wavefunction ψ₀ into Schrödinger like you said, so I'm guessing there must be an actual value or σ?

I'm really confuzzled! :(

All I'm given is the function ψ₀ like I originally posted, nothing else.

<3 xoxo

 

[Clever-Name]

Huh, I'm confused then as well, typically when a question asks you what value for a constant makes it a solution to an equation the thing to do is plug it into said equation. Could you possibly write out the entire problem? Or is it in a certain textbook? If so, what text and what's the problem number.

 

[Bowenwww]

Part I

Find the value σ₀ for which:

ψ₀(x) = (2πσ)^-1/4 exp(-x²/4σ)

is a solution to the Schrödinger equation

Part II

What is the associated energy in part I? [hint: insert the given wave function into the Schrödinger equation]

That's it word for word.

Thank you so much for helping I really appreciate it! <3 xoxoxo

 

[qbert]

when you substitute ψ into schrodingers equation
Hψ=Eψ, you know E is a constant (independent of x)
but when you calculate Hψ you'll get terms proportional
to x^2. you need to pick σ so that those terms go away.

 

[Bowenwww]

I put the wavefunction in and took the 2nd derivative and all that I canceled a load of common terms and ended up with

E = (-h(bar)²/2m)*(x²)/(4σ²) + 1/2*(mω²x²)

Is this right? I don't get how to pick a value so all the x's disappear?

<3 Cheers guys xoxox

 

[Clever-Name]

qbert, that would mean \sigma is a function of x^2. This is definitely not the case.

 

[Clever-Name]

Bowenwww, You should end up with the following:

E = \frac{\hbar^{2}}{4m\sigma} - \frac{\hbar^{2}x^{2}}{8m\sigma} + \frac{1}{2}m\omega^{2}x^{2}

 

[qbert]

more like:
E = \frac{1}{4}\frac{\hbar^2}{m \sigma}-\frac{1}{8}\frac{\hbar^2x^2}{m\sigma^2}+\frac{1}{2} m \omega^2 x^2

Clever-Name, picking σ to make the x^2 terms 0 does not
mean that σ is a function of x. in fact, if you just do it
you'll see precisely what σ has to be!

 

[Bowenwww]

So that's how I find out the energy levels, right? But I still don't know how to get the sigma?

I really appreciate this guys I just need pointing in the right direction ^ _ ^<3 xoxoxox

 

[qbert]

i told you how to get σ.

 

[Bowenwww]

But there's still two unknowns E and sigma - sorry I know I'm being really stupid here; I know how to get to the form you've given I just don't know how to get from there to a value for sigma?

:( xoxox

 

[qbert]

pick σ to make the x^2 terms go away.

 

[gabbagabbahey]

If C_1 = (a-2)x^2 +C_2 where a, C_1 and C_2 are all constants, then a must equal 2 (otherwise C_1 would be a function of x, not a constant) and C_1 must equal C_2

 

[Clever-Name]

qbert you're right, my mistake, but your attitude is unnecessary. Bowen is obviously struggling to figure this out and your bitter responses aren't very helpful. I also made a typo in the equation you corrected. The appropriate thing to do is assume I made that typo instead of being so pompous in response.

Bowen, The last two terms (the ones with x^2) should cancel out based on what \sigma is. In other words, their sum should be zero! Work that out and tell me what your answer for sigma is! i.e. solve this equation:

<br /> -\frac{\hbar^{2}x^{2}}{8m\sigma^{2}} + \frac{1}{2}m\omega^{2}x^{2} = 0

 

[qbert]

qbert you're right, my mistake, but your attitude is unnecessary. Bowen is obviously struggling to figure this out and your bitter responses aren't very helpful. I also made a typo in the equation you corrected. The appropriate thing to do is assume I made that typo instead of being so pompous in response.

Bowen, The last two terms (the ones with x^2) should cancel out based on what \sigma is. In other words, their sum should be zero! Work that out and tell me what your answer for sigma is! i.e. solve this equation:

<br /> -\frac{\hbar^{2}x^{2}}{8m\sigma^{2}} + \frac{1}{2}m\omega^{2}x^{2} = 0

wow. bitter and pompous and here i thought i was being helpful.
i'm sorry if i offended anyone, that wasn't my intention.