Nuclear Space in QM: Exploring & Integrating Finite Solutions

[TTT]

Homework Statement

See the attachment

Relevant Equations

integrate from negative infinity to infinity ∫|f(x)|[SUP]2[/SUP]dx is finite

If I understand this question correctly, I am supposed to prove an integrate from negative infinity to infinity ∫x|∅(x)|²(1+|x|)ⁿdx is finite. Sorry, but I have no idea.

 

[Cutter Ketch]

Your understanding of what you need to do is correct. (proving for all n, of course). Remember, the premise is that you already know

∫|∅(x)|^2(1+|x|)^n dx

is finite for all n. What can you do to to get that x out of the integral and relate it to the integrals you already know are finite?

 

[dextercioby]

What book is this?

 

[TTT]

Your understanding of what you need to do is correct. (proving for all n, of course). Remember, the premise is that you already know

∫|∅(x)|^2(1+|x|)^n dx

is finite for all n. What can you do to get that x out of the integral and relate it to the integrals you already know are finite?

I have tried to do this with integrating it by parts, but it did not go well. Are you suggesting something else?

 

[TTT]

What book is this?

This question is not quoted from the textbook I am using for this class, so I do not know where this question comes from

 

[PeroK]

I have tried to do this with integrating it by parts, but it did not go well. Are you suggesting something else?

This is more like a real analysis problem. Note that the integrand is non-negative in all cases. I suggest you look for a suitable inequality that bounds the integrand when you have the $x^2$ in it. In other words, you need to find an inequality:

$x^2|\phi(x)|^2(1 + |x|)^n \le f(x)$

Where $f(x)$ is some function you know is integrable. The result then follows from the properties of the integral.

Note: if you haven't studied any real analysis, then the approach to problems like this will be hard to find.

 

[Moolisa]

This is more like a real analysis problem. Note that the integrand is non-negative in all cases. I suggest you look for a suitable inequality that bounds the integrand when you have the $x^2$ in it. In other words, you need to find an inequality:

$x^2|\phi(x)|^2(1 + |x|)^n \le f(x)$

Where $f(x)$ is some function you know is integrable. The result then follows from the properties of the integral.

Note: if you haven't studied any real analysis, then the approach to problems like this will be hard to find.

I'm assuming this poster is in my class, we use the Griffiths Intro to QM book, but as stated, this isn't from it. We're not required to take real analysis at this point. Do you mind telling me what type of approach this is so I can study up on this?

 

[PeroK]

I'm assuming this poster is in my class, we use the Griffiths Intro to QM book, but as stated, this isn't from it. We're not required to take real analysis at this point. Do you mind telling me what type of approach this is so I can study up on this?

Personally, I don't think it's worth it. Real analysis is an art in itself. And I don't think you gain a lot of insight into QM from studying these sort of functional analytic properties.

I think it's worth noting that just because a function is square integrable doesn't mean it meets the requirements of QM. There are several examples:

In this case, multiplying by the function $x$ can make a function non-integrable (in terms of the integral becoming unbounded).

Differentiating the function (i.e. applying the momentum operator).

The momentum operator is not Hermitian on the space of all square-integrable functions.

But, generally, as long as a function tends to zero faster than any $x^n$, as $x \rightarrow \infty$, then you are okay.

In this case, rather than prove the above, I suggest it's sufficient to note that in QM you are dealing with something like $\Omega$, rather than the entire space of square-integrable functions.

 

[vela]

If I understand this question correctly, I am supposed to prove an integrate from negative infinity to infinity ∫x|∅(x)|²(1+|x|)ⁿdx is finite.

Almost. You're supposed to show that
$$\int_{-\infty}^\infty \lvert \hat x \phi \rvert^2 (1+\lvert x \rvert)^n\,dx = \int_{-\infty}^\infty |x|^2 \lvert \phi \rvert^2 (1+\lvert x \rvert)^n\,dx$$ is finite.

I suggest you try expressing $\lvert x \rvert^2$ as a linear combination of powers of $1+\lvert x \rvert$.

 

[martinbn]

$$\lvert x\rvert^2\le(1+\lvert x\rvert)^2$$
So
$$\int_{-\infty}^\infty \lvert x\rvert^2 \lvert \phi \rvert^2 (1+\lvert x \rvert)^n dx\le \int_{-\infty}^\infty \lvert \phi \rvert^2 (1+\lvert x \rvert)^{n+2}dx$$