Confused about why wave function is from zero to infinity

[Justin LaRose]

Homework Statement

I am trying to solve a problem from a popular quantum mechanics text. I am learning on my own. I am trying to calculate the variance, which is <x^2>-<x>^2 = variance in x.

I posted a photo of the problem as a picture that is linked below as well as the solution, I simply do not understand the solution. It is on page 14 of the Griffith's QM text (2nd edition)

Homework Equations

The relevant equations are in the photographs, it would be very onerous and take a long time to type them here, I do not want to post the screen shots because then they would need to be downloaded, and I am not trying to waste anyones time.[/B]

The Attempt at a Solution

Here is what's confusing me a great deal. There are a lot of things that have nothing to do with the physics that are throwing me off...

In part a) when normalizing psi(x,0), it is a simple u substitution, and if you look at the solution, I have absolutely no idea where the heck this 2 is coming from. Maybe I brain fried from looking at it for too long. If someone could please explain where this 2 comes from or if this is a typo, that would be much appreciated, it's a simple calculus problem.

The next problem I am having is this.

in b) the solution to determine the expectation value of x

Ah! I've forgotten, the integral from - infinity to infinity = twice the integral from o to infinity, of course! Okay so I've answered my own question for part a), sorry about that!

Jeeze I guess I've answered my own questions here. I am going to keep this text because perhaps it will be educational to illustrate that sometimes by writing out a problem a light flickers in your head! (I won't be posting an online link to the photos of the solution because I've figured it out)

There is one thing I am kind of confused about, if someone could please refer me to some text or break it down for me, is it always permissible (excluding the much talked about non physical pathological functions) to have an integral like this...

Integral from - infinity to infinity (some function) = 2 integral 0 to infinity (same function), or does the function need to have an odd or even property, I simply forgot, I am kind of rusty?

Alright thank you.
Justin[/B]

 

[Justin LaRose]

Also, this is just really confusing...

What does it actually mean that the expectation value of x = 0.

What is this saying, the integrals are not important here, what does this actually mean, I do not have a good sense about what I am actually even doing.

Is this saying along a line a system of particles in the same quantum state are most likely to be found at the origin?

Thank you!

 

[PeroK]

Also, this is just really confusing...

What does it actually mean that the expectation value of x = 0.

What is this saying, the integrals are not important here, what does this actually mean, I do not have a good sense about what I am actually even doing.

Is this saying along a line a system of particles in the same quantum state are most likely to be found at the origin?

Thank you!

Your maths might be too rusty to get through Griffiths. You should brush up on probability theory and some calculus. You might want to leave linear algebra until you get to the chapter on formalism - but a solid background in linear algebra will be critical if you are doing this on your own.

Re an expected value of x = 0. This essentially means x is as likely to have a negative value as a positive value. It doesn't actually imply that $|\Psi(x)|^2$ is symmetrical, but that will usually be the case.

Saying x = 0 is the most likely value would equate to $|\Psi(x)|^2$ having a maximum at x = 0.

 

[Justin LaRose]

Right that's what I thought. thanks a lot.

I have a lot of confidence in my mathematical abilities, I took classes beyond linear algebra (my favorite math class ever). What I'll need to brush up on is Fourier series and the stuff I learned in my math methods class over at Wayne State University, partial differential equations and the Fourier transforms mainly. The cool thing about self study is I don't care about solving these integrals, it's pointless, I just put them in Mathematica.

I'll definitely have a lot of questions but I am thinking with the help of the forum I can figure them out. I took an intro class to QM, intro to modern physics, I'm not a complete rookie, but quantum mechanics is just not easy to understand. That's why I am here, I had a nervous breakdown because things were not making sense when we went to three dimensions recently and I just need to take some time off and regroup, but I love the physics so I am just doing what I love.

 

[PeroK]

Right that's what I thought. thanks a lot.

I have a lot of confidence in my mathematical abilities, I took classes beyond linear algebra (my favorite math class ever). What I'll need to brush up on is Fourier series and the stuff I learned in my math methods class over at Wayne State University, partial differential equations and the Fourier transforms mainly. The cool thing about self study is I don't care about solving these integrals, it's pointless, I just put them in Mathematica.

I'll definitely have a lot of questions but I am thinking with the help of the forum I can figure them out. I took an intro class to QM, intro to modern physics, I'm not a complete rookie, but quantum mechanics is just not easy to understand. That's why I am here, I had a nervous breakdown because things were not making sense when we went to three dimensions recently and I just need to take some time off and regroup, but I love the physics so I am just doing what I love.

QM is not the easiest subject, so as long as you're prepared for a hard fight! Good luck, anyway.

 

[Justin LaRose]

A fight it is, it takes about 20 minutes a page!

Since you are familiar with the book, one thing that is always tough to figure out, the decision on what problems to do at the end of the chapter. I have the instructor solution manual, it is just hard to tell what is most important. Any advice?

 

[PeroK]

Is this chapter 1?

 

[Justin LaRose]

Yeah

 

[PeroK]

I can't see the point of Problem 13 on Buffon's needle unless you want to work out a lovely mathematical problem. Question 18 seems (on the solids and gases) seems a bit ahead of its place.

The moral is probably to do more problems if you think you need to. Once you feel comfortable with the material in Chapter 1, you probably want to get started with Chapter 2. That's the main course.

 

[Justin LaRose]

Right. Yeah chapter 3 was when I just sort of lost touch, that was the first time I was like I have no clue wtf is going on. Maybe I'll just do the in book problems and then revisit the end of chapter exercises after I read part 1. I don't really know. I'm almost done with the chapter though, perhaps I will do all the no star and one star problems on my first go. Thanks for the advice.

 

[mfig]

The relevant equations are in the photographs

What photographs?

 

[PeroK]

Right. Yeah chapter 3 was when I just sort of lost touch, that was the first time I was like I have no clue wtf is going on. Maybe I'll just do the in book problems and then revisit the end of chapter exercises after I read part 1. I don't really know. I'm almost done with the chapter though, perhaps I will do all the no star and one star problems on my first go. Thanks for the advice.

There's linear algebra (finite dimensional vector spaces and nice matrices), then there's linear algebra (self-adjoint operators on infinite dimensional function spaces), then there's QM rigged linear algebra (continuous spectra of eigenfunctions that aren't really functions at all, and some fast and loose shenanigans going on). Chapter 3 involves "abstract" mathematics and that's a different ball-game. Things are what their properties say they are.

By the way, there's a website of solutions here:

http://www.physicspages.com/index-physics-quantum-mechanics/griffiths-introduction-to-quantum-mechanics-problems/

And I think this guy is happy to answer questions about the solutions as well.