Problem using Griffiths Intro to Quantum Mechanics

[jstrunk]

Summary:: The problem solutions contain a lot of unjustified steps, making them pointless.

I am trying to use Griffiths Introduction to Quantum Mechanics.
He states that the wave function $\psi$ approaches 0 as x approaches infinity to make normalization work.
I can accept that.
But then I can't solve the problems in the book so I peak at some of the solutions
that people have provided online and they are making assumptions like these:

$$x\left[ {{\psi ^*}\frac{{\partial \psi }}{{\partial x}} - \frac{{\partial {\psi ^*}}}{{\partial x}}\psi } \right]_{ - \infty }^\infty = 0$$

$$\left. {\frac{{\partial \psi }}{{\partial x}}\frac{{\partial {\psi ^*}}}{{\partial x}}} \right|_{ - \infty }^\infty = 0$$

Griffiths never mentioned these identities, much less justified them.
I don't see any point in trying to doing the exercises if you can just make up anything you
want and stick it in your solution so you get the answer you need.

I understand that in learning a new subject you sometimes have to take some things on faith
at the beginning, knowing they will be explained later. But Griffiths hasn't said these will
be explained later. In fact, I don't even know if he used these identities in his solutions. He doesn't
provide solutions. I found these in other people's solutions.

If there was a finite list of these identities that I was allowed to use in the exercises, and I was
sure that proofs exist for each of them, then I could continue trying to use Griffiths.
If not, I need to find a different book.

I know a lot of people learned quantum mechanics from this textbook.
How do people cope with this problem? Does the professor fill the gaps? That doesn't help me
because I am doing this on my own.

 

[hutchphd]

As I read your question you are annoyed with Griffiths because people online use identities (maybe?) he doesn't mention to solve his exercises. Is that really what you are saying?

 

[PeroK]

Summary:: The problem solutions contain a lot of unjustified steps, making them pointless.

I am trying to use Griffiths Introduction to Quantum Mechanics.
He states that the wave function $\psi$ approaches 0 as x approaches infinity to make normalization work.
I can accept that.
But then I can't solve the problems in the book so I peak at some of the solutions
that people have provided online and they are making assumptions like these:

$$x\left[ {{\psi ^*}\frac{{\partial \psi }}{{\partial x}} - \frac{{\partial {\psi ^*}}}{{\partial x}}\psi } \right]_{ - \infty }^\infty = 0$$

$$\left. {\frac{{\partial \psi }}{{\partial x}}\frac{{\partial {\psi ^*}}}{{\partial x}}} \right|_{ - \infty }^\infty = 0$$

Griffiths never mentioned these identities, much less justified them.
I don't see any point in trying to doing the exercises if you can just make up anything you
want and stick it in your solution so you get the answer you need.

I understand that in learning a new subject you sometimes have to take some things on faith
at the beginning, knowing they will be explained later. But Griffiths hasn't said these will
be explained later. In fact, I don't even know if he used these identities in his solutions. He doesn't
provide solutions. I found these in other people's solutions.

If there was a finite list of these identities that I was allowed to use in the exercises, and I was
sure that proofs exist for each of them, then I could continue trying to use Griffiths.
If not, I need to find a different book.

I know a lot of people learned quantum mechanics from this textbook.
How do people cope with this problem? Does the professor fill the gaps? That doesn't help me
because I am doing this on my own.

This is why there are mathematical prerequisites for a subject like QM - especially if you are self-learning. Generally, you are expected to be able to do some fancy mathematical footwork at this level.

Those steps are recognisable to a more experienced student as examples where the boundary terms in integration by parts vanish - which they almost always do!

I thought Griffiths mentioned this somewhere. See, for example, the footnote on page 97 (2nd Edition) relating to equation 3.19. "I threw away the boundary term for the usual reason", he says.

I managed Griffiths on my own when I learned QM. Even though I raised a pure mathematical eyebrow now and then.

 

[vela]

If you're looking for rigor in the sense of mathematical proofs of those types of statements, I think most physics textbooks will disappoint you. They are, after all, trying to teach you physics, not math. On a related note, I'd say it's generally safe to assume you're not going to run into some weird pathological function that a mathematician might worry about.

That said, you might find a book other than Griffiths (or in conjunction with Griffiths) might suit your needs better. There's no reason to think you have to stick with only Griffiths.

If you're struggling with understanding why those statements are generally true (in a physics context), I concur with @PeroK that you're expected to have the mathematical background to figure it out for yourself.

 

[hutchphd]

Can you direct us to the particular solutions on line that you reference? I wonder in what context these issues arise and what you should be expected to surmise..
This stuff is not easy you know.

 

[vanhees71]

Well, I think it's very good if you have a headache about these issues, and Griffiths is far from being very clear on these issues. I also think one shouldn't bother with too much mathematical rigor when starting to learn quantum theory, but one should be aware that there can be trouble leading to inconsistencies.

The important point is that the operators that represent observables in quantum theory have to be self-adjoint. In the wave-mechanics formulation in QM 1 you deal with the position and momentum operators and functions thereof. It's important that these operators must be self-adjoint on the corresponding Hilbert space of square-integrable functions $\text{L}^2$, and that as such the operators are only defined on a dense subspace, and that also the codomain must be within this subspace.

For a didactical introduction, see

https://arxiv.org/abs/quant-ph/9907069

 

[jstrunk]

vanhees71 stated:
I thought Griffiths mentioned this somewhere. See, for example, the footnote on page 97 (2nd Edition)
relating to equation 3.19. "I threw away the boundary term for the usual reason", he says.

I am using the 3rd Edition and I am only on page 18. So far, he hasn't given any explanation of how
to determine what you can through away. The only limitations that have been stated so far are that
the wave function approaches 0 as x approaches infinity and that velocity << c.

vanhees also stated:
The important point is that the operators that represent observables in quantum theory have to be
self-adjoint. In the wave-mechanics formulation in QM 1 you deal with the position and momentum
operators and functions thereof. It's important that these operators must be self-adjoint on the
corresponding Hilbert space of square-integrable functions $L^2$, and that as such the operators
are only defined on a dense subspace, and that also the codomain must be within this subspace.

That is good to know for later, but as of page 18, the terms self-adjoint and Hilbert space haven't
been introduced yet. Somehow, students are supposed to solve the problems in chapter 1 without
knowing that stuff.

I think the only thing I can do is try to guess and check other people's guesses until I can get to
the point where this is made more rigorous or I somehow develop a knack for it.

 

[PeroK]

vanhees71 stated:
I thought Griffiths mentioned this somewhere. See, for example, the footnote on page 97 (2nd Edition)
relating to equation 3.19. "I threw away the boundary term for the usual reason", he says.

I am using the 3rd Edition and I am only on page 18. So far, he hasn't given any explanation of how
to determine what you can through away.

On page 16 of the 2nd edition, re equation 1.30, he mentions throwing away the boundary term "on the grounds that $\Psi$ goes to zero at $\pm$ infinity.

He's not going to explain this further, because you are expected to know already that this is a common idea in physics. There can't be things happening at an infinite distance from a physical system, so boundary terms at infinity generally vanish.

This idea is used extensively in Electromagnetism as well.

That's really all there is to it. So, now you know.

 

[vanhees71]

That is good to know for later, but as of page 18, the terms self-adjoint and Hilbert space haven't
been introduced yet. Somehow, students are supposed to solve the problems in chapter 1 without
knowing that stuff.

I think the only thing I can do is try to guess and check other people's guesses until I can get to
the point where this is made more rigorous or I somehow develop a knack for it.

Well yes, one approach of textbook writers is to leave out the subtle mathematical points and just let you solve a lot of problems in a handwaving way. That has its merits when you like to get started with a subject and just concentrate on the physics. Building some intuition is also very important, particularly with a subject as unintuitive as quantum theory. You have to forget a lot of your intuitions getting used to in classical physics. So in the beginning it's maybe good not too present too much mathematical formalism but try to build the physical intuition first.

The great shortcoming of this method is that particularly the good students who try to really understand also the math behind the hand waving arguments of the physicists are left alone. In my opinion there should be a better compromise between mathematical rigorism and physicists' sloppyness than provided in the textbook by Griffiths. I don't know Griffiths's book too well, but from all the questions of confused readers of this book I conclude it's of the kind of QM textbooks which are too sloppy with the math. A good textbook should explain such more subtle mathematical issues in some appendices.

If you stumble over such mathematical questions, it's good to have another book, which is a bit more rigorous. For this I'd recommend Ballentine, Quantum mechanics (though it's at a pretty high level for beginners in QT).