Zwiebach question with link to text

[ehrenfest]

Homework Statement

http://books.google.com/books?id=Xm...GCQ&sig=Eqnkxjj7B9gc4-nig1fgUkk7AqQ#PPA207,M1
As you can see on page 207, Zwiebach obtained 12.6 by simply demanding that it be true.
He does this also in chapter 11 with equations such as 11.27 (which I do not think is on the free preview). He just postulates that they are true and then he uses them in later calculations!
I have no idea what he is thinking! How can he just say for example. "equation 11.58 is reasonable" and then pretend like it is true without proving it!

Homework Equations

The Attempt at a Solution

 

[ehrenfest]

I keep asking this question but it seems to never get answered? Is it just that I am so lost that it would take to long to explain it to me?

This is a crucial concept and if I understood it I would probably have a ton fewer questions about other stuff...

 

[Jimmy Snyder]

I have been most reluctant to answer this question because I am not sure of myself and would prefer that nrqed, or someone else with more insight than I have, would tackle this. I hope that my poor attempt will not mislead and will not discourage anyone from providing a more insightful answer.

Chapter IV of "The Principles of Quantum Mechanics" by P.A.M. Dirac is a very lucid discussion of this topic. I'm sure that a similar discussion can be found in most introductory texts on QM. Of course, I cannot quote the chapter in its entirety, but here is a relevant excerpt (emphasis mine):

It follows that any function of $q_r$ and $p_r$ will commute with any function of $q_s$ and $p_s$ when s differs from r. Different values of r correspond to different degrees of freedom of the dynamical system, so we get the result that dynamical variables referring to different degres of freedom commute. This law, as we have derived it from (9), is proved only for dynamical systems with classical analogues, but we assume it to hold generally. In this way we can make a start on the problem of finding quantum conditions for dynamical systems for which canonical coordinates and momenta do not exist, provided we can give meaning to different degrees of freedom, as we may be able to to with the help of physical insight.

I think you will find that in the commutation relations that you have been quoting, that they consist of the quintessential pq - qp = i along with enough Kronecker deltas and Dirac delta functions to implement the assumption concerning 'different degrees of freedom'. If I am right, Zwiebach is 'assuming' the deltas as Dirac predicted he would.

 

[ehrenfest]

In the meantime, I found/thought of another answer.

The [p-hat,x-hat] = i h-bar commutator in regular quantum mechanics can be derived by using p-hat = - i h-bar d/dx and x-hat = x.

But where do those expressions for p-hat and x-hat come from? The Schrödinger equation.
And where does the Schrödinger equation come frome? Nowhere, it is an ansatz that seems to work.

So, in a sense that commutator is equivalent to the Schrödinger equation.

I think Zwiebach is doing the same thing. He is using those commutators as an ansatz and then deriving lots of equations that seem reasonable therefore justifying the original ansatz.