Prev: Three Old Men
Next: Flame photometry, gas discharge and absurdities of modern science
From: Jay R. Yablon on 22 Feb 2010 16:33
In his development of the path integral at:

http://jayryablon.files.wordpress.com/2009/12/sakurai-and-zee.pdf

professor A. Zee has two errata which he has corrected at:
http://www.kitp.ucsb.edu/~zee/nuts.html. One is a missing factor of
2pi, the other a wrong range in the product, which should start at j=1
not j=0.

But as pointed out at
http://en.wikipedia.org/wiki/Relation_between_Schr%C3%B6dinger%27s_equation_and_the_path_integral_formulation_of_quantum_mechanics#From_Schr.C3.B6dinger.27s_equation_to_the_path_integral_formulation,
there appears to be another error as well:

At the bottom of page 11, Zee says "As an exercise you should
convince yourself that had we started with the Hamiltonian for a
particle in a potential H=p-hat/2m +V(q-hat) . . . the final result
would have been:

<q_F|exp[piHt]|q_I>
= $Dd(t) exp i ${0 to T} dt [.5 m d-dot^2 -V(q)] (Zee-5)

Is that really so? Or, is he assuming that the commutator:

[V(q-hat), p-hat] = 0 (1) ?

It seems to me that the first calculation at the top of page 11 does not
succeed if (1) is not satisfied. This is because:

e^(a+b) = e^a e^b (3)

if and only if [a,b]=0. Otherwise, once needs the
Baker-Campbell-Hausdorff formula and one cannot effectively get the
exp(-i V(q)) out of the integral.

Further since the canonical relationship:

[q-hat, p-hat] = i h-bar (2)

seems to suggest that (2) is not satisfied, this looks like a glaring
oversight by Zee, or at the very least, he is making an a non-physical
assumption / approximation and not saying so.

Questions:

1) Is it correct that Zee's calculation only works by assuming (1)?

2) Is it correct that (1) is not a good assumption?

3) If so, then (Zee-5) is only an approximation, or worse. Under what
*physical* circumstances is this a valid approximation, and under what
circumstances is it not? In other words, would a calculation proceeding
from (Zee-5) and the related assumption that (1) is true give us good
approximate results that need some sort of "fine" adjustment afterwards,
at least under some reasonable range of physical conditions? Or, is
(Zee-5) just plain wrong and unusable anywhere, any time?

Jay.
____________________________
Jay R. Yablon
Email: jyablon(a)nycap.rr.com
co-moderator: sci.physics.foundations
Weblog: http://jayryablon.wordpress.com/
Web Site: http://home.roadrunner.com/~jry/FermionMass.htm

From: Jay R. Yablon on 23 Feb 2010 15:37
As regards the topic of this thread:

Is Zee relying on http://en.wikipedia.org/wiki/Lie_product_formula to
get the result in (Zee-5)? Looks to me like he is.

Jay
 | 
Pages: 1
Prev: Three Old Men
Next: Flame photometry, gas discharge and absurdities of modern science