Is the Integral of (psi_0 * d/dx psi_1) Mathematically Valid?

[Aziza]

I need help determining if the following statement is mathematically valid, where psi_0 and psi_1 are the first two stationary states of the SHO:


http://af10.mail.ru/cgi-bin/readmsg?id=13807553990000000798;0;1&mode=attachment&bs=2783&bl=281383&ct=image%2fjpeg&cn=mms_picture.jpeg&cte=base64

Basically what I did was cancel the dx's as if it was algebraic equation, and then since my integration became wrt psi_1, I changed my limits from 0->0, since at +/- infinity, psi_1 must be zero. And of course if your limits of integration are same, result must be zero. Specifically, I would like to know where exactly (at which of my above steps) is the flaw? (there must be a flaw since I do not get correct solution when I use this to calculate other stuff...this integral should in fact not be zero).

If I replace psi_1 by psi_0, for example, then I think this is correct, since I did it on my recent QM test and got full credit.

Any help with this is appreciated!








Any help is appreciated!

 

[PhysicsGente]

Hello,

$\psi_{1}(x) = \left(\frac{m\omega}{\pi\hbar}\right)^{1/4}\frac{1}{\sqrt{2^{n}n!}}2\xi e^{-\frac{\xi^{2}}{2}}$. And $\xi = \sqrt{\frac{m\omega}{\hbar}}x$

So it does not vanish at minus infinity.

 

[Aziza]

Hello,

$\psi_{1}(x) = \left(\frac{m\omega}{\pi\hbar}\right)^{1/4}\frac{1}{\sqrt{2^{n}n!}}2\xi e^{-\frac{\xi^{2}}{2}}$. And $\xi = \sqrt{\frac{m\omega}{\hbar}}x$

So it does not vanish at minus infinity.

How does it not vanish at minus infinity? first, it is a wavefunction, so by the QM postulates it must vanish at +/- inifinity for it to be a physical solution.

second, just looking at the formula above you can see it does vanish, since at -∞ you basically have the quantity (-∞)*exp(-∞)...this is equal to zero as you can prove with l'hopital rule

 

[Borek]

Aziza, we can't see the image attached to an email hosted at mail.ru. Please post it in the thread.

 

[jtbell]

If you need to learn how to use LaTeX to post equations, see here:

https://www.physicsforums.com/showpost.php?p=3977517&postcount=3

 

[Aziza]

sorry i did not realize that my picture could not be seen. Here is the equation:


∫ψ₀$\frac{d}{dx}$ψ₁dx = ∫ψ₀ dψ₁ = ψ₀ψ₁|$^{0}_{0}$ = 0

Where the first integral runs from -inf to +inf, and the second from 0 to 0, since my integration is now wrt dψ₁ and ψ₁ is zero at +/- inf

 

[vela]

Here's a similar specific example of what you did:
$$\int \cot\theta\,d\theta = \int \frac{1}{\sin\theta}\cos \theta\,d\theta = \int \frac{1}{\sin\theta}\,\frac{d}{d\theta}\sin \theta\,d\theta = \int \frac{1}{\sin\theta}\,d(\sin\theta).$$ So far so good, but from here, you claimed that
$$\int \frac{1}{\sin\theta}\,d(\sin\theta) = \frac{1}{\sin\theta}{\sin\theta}=1.$$

 

[Aziza]

Here's a similar specific example of what you did:
$$\int \cot\theta\,d\theta = \int \frac{1}{\sin\theta}\cos \theta\,d\theta = \int \frac{1}{\sin\theta}\,\frac{d}{d\theta}\sin \theta\,d\theta = \int \frac{1}{\sin\theta}\,d(\sin\theta).$$ So far so good, but from here, you claimed that
$$\int \frac{1}{\sin\theta}\,d(\sin\theta) = \frac{1}{\sin\theta}{\sin\theta}=1.$$

no...then using my method above i would say sinθ=u and so integrate wrt u...so the answer is ln(sinθ) by my method.


the reason i just pulled out the ψ₀ was because the integration is wrt ψ₁...so ψ₀ is treated as constant...is there an error here?

 

[vela]

I did exactly what you did where $\psi_0 = \frac{1}{\sin \theta}$ and $\psi_1 = \sin\theta$ and where $\theta$ plays the role of $x$.

 

[Aziza]

I did exactly what you did where $\psi_0 = \frac{1}{\sin \theta}$ and $\psi_1 = \sin\theta$ and where $\theta$ plays the role of $x$.

ohh I am sorry yes i see what you mean...so basically if i don't know (or actually just don't plug in, in this case) the actual psi's and their dependence on x, then there is no way to do this integral

 

[vela]

The Hermite polynomials have certain properties that might be helpful to evaluate the integral. You could try looking into that.

 

[PhysicsGente]

How does it not vanish at minus infinity? first, it is a wavefunction, so by the QM postulates it must vanish at +/- inifinity for it to be a physical solution.

second, just looking at the formula above you can see it does vanish, since at -∞ you basically have the quantity (-∞)*exp(-∞)...this is equal to zero as you can prove with l'hopital rule

Yes, I'm sorry I overlooked the $^2$. My bad.