Are Dot Products and Dirac Brackets Interchangeable in Vector Mathematics?

[sonnichs]

I am wondering about the formal definitions of the dot-product and dirac brackets <>. Of course <> brackets apply equally to functions as well as vectors (in Hilbert space).

Is it safe to assume that . and <> are equivalent?
Can one state that <a|b> equivalent to |a| |b| cosT where a and b are n dimensional vectors? (I assume that even in N space any pair of vectors has only 1 angle between them)

I see various material about this on the internet but not always consistent. Some of my texts ignore talking about it!

Thanks-fritz

 

[PeterDonis]

What specific sources have you looked at that have differing definitions?

 

[PeterDonis]

functions as well as vectors (in Hilbert space).

Square integrable functions, i.e., QM wave functions, are vectors in a Hilbert space.

Can one state that <a|b> equivalent to |a| |b| cosT where a and b are n dimensional vectors?

No, because in a Hilbert space inner products can be complex.

(I assume that even in N space any pair of vectors has only 1 angle between them)

The concept of "angle" isn't necessarily well defined in a Hilbert space, at least not unless you allow "angles" to be complex.

 

[sonnichs]

Peter
Sorry about my sources--they are diffuse, buried variously on the internet.

But moving on to your comments, I guess you are indicating that:
<> is not equivalent (a superset) of "the dot product"
since the later cannot have a complex angle.
I guess that is what is confusing me because I never have heard it stated outright that the angle cannot be complex. Probably complex angles are "outside my pay grade", but in proving for example, the Schwarz inequality that would come up.
Thanks
Fritz

 

[PeterDonis]

Sorry about my sources--they are diffuse, buried variously on the internet.

Then the first thing you need to do is fix that. Try an actual QM textbook. Ballentine, for example, in Chapter 1 talks about the basic math of Hilbert spaces, and in Chapter 2 applies it to the mathematical formulation of QM.

 

[PeterDonis]

I guess you are indicating that:
<> is not equivalent (a superset) of "the dot product"
since the later cannot have a complex angle.

No. I'm indicating that whatever vaguely specified "sources" you are looking at have not even given you enough of a basic knowledge to ask a well-defined question. Before you can even talk about things like "dot products" or Dirac brackets or "angles", you need to first decide what mathematical framework you are working in, and learn its basic axioms and theorems, and then look at how those apply to the specific concepts you are asking about. That is what taking the time to work through an actual QM textbook will give you.

I never have heard it stated outright that the angle cannot be complex.

Before you can even evaluate such a statement, you have to first look to see what an angle is. How is "angle" even defined in terms of the basic axioms and theorems of the appropriate mathematical framework. Again, that is what taking the time to work through an actual QM textbook will give you.

 

[sonnichs]

I will check Ballentine if I can get it from the library. I have "gone through" a few texts over the years, including Messiah, Schiff, Shankar and Griffiths. Can't recall any details on complex representation of angles. Perhaps I am reading too much into this.
I guess my usual approach to problems like this is to represent <a|b> as the integral of two sums. Since we can be sure to incorporate the complex numbers into the meaning of a and b. I believe this is "safe".

Thanks again for the reference
Fritz

 

[Vanadium 50]

If you've gone through 4 books and don't recall the basics and definitions, I suggest reviewing them. Perhaps more slowly.

 

[PeterDonis]

Can't recall any details on complex representation of angles.

That's because there aren't any. "Complex representation of angles" is not how any QM text I've ever seen even describes Dirac brackets or Hilbert space inner products. The closest you will get is something like the statement in Ballentine that the Hilbert space inner product "generalizes the notions of length and angle".

That was part of my point when I said that you are not even asking a well-defined question. You should not even be starting from the idea that "angle" is a well-defined concept and you just need a "complex representation" of it.

 

[sonnichs]

Yes. I see. My first approach would not usually involve and angle. (very Euclidean?)
As I mentioned I usually like to approach things without angles, using sums/linear algebra etc. "aT b" is safer for me.
The main cause of my initial question was that I was thinking of ways of visualizing the schwarz inequality, and it is very easy to just express the "lesser than" side as a dot-product/cos construct, or more inclusive, as a projection. But taking an absolute value of the cos becomes nebulous without a solid definition-or worse, trying to stretch the dot-product beyond its intentions.
So I think I am good with all this but the input you have provided is helpful. Not a lot of people to discuss and "bounce ideas" off of these days for me.

cheers
Fritz

 

[PeterDonis]

I was thinking of ways of visualizing the schwarz inequality

All the quantities involved in the Schwarz inequality are always real, so while visualizing it in terms of real dot products and angles is straightforward enough, it only is that way because you have thrown away all the extra information involved with complex numbers and Hilbert space vectors.