Comparing the Uncertainty Principle: Griffiths vs. Shankar

[Zerkor]

The general Uncertainty Principle is written in Griffiths' Intro to Quantum Mechanics 2nd Ed. Section 3.4, Page 109, Eq. (3.139) without dependence on the wave function itself. While it is written in R. Shankar's Principles of Quantum Mechanics 2nd Ed., Section 9.2, Page 239, Eq. (9.2.12) with a dependence on the wave function.

I can't understand the difference between the two equations. Is the one written in Shankar more general? Or they are the same equation but in a different formulation?

 

[Jazzdude]

Would it be too much to ask for you to actually show these two equations? No offense, but if you expect everyone to have those books and to look it up I'm not sure I even want to help you.

 

[Demystifier]

The general Uncertainty Principle is written in Griffiths' Intro to Quantum Mechanics 2nd Ed. Section 3.4, Page 109, Eq. (3.139) without dependence on the wave function itself.

A) Trivial stuff

First, the page and equation number you gave are from the 1st edition, not the 2nd.
Second, equation (3.139) does depend on the wave function, because the symbols <, > depend on the wave function. See eq. (3.116) where this dependence is more explicit.

 

[Demystifier]

While it is written in R. Shankar's Principles of Quantum Mechanics 2nd Ed., Section 9.2, Page 239, Eq. (9.2.12) with a dependence on the wave function.

I can't understand the difference between the two equations. Is the one written in Shankar more general? Or they are the same equation but in a different formulation?

B) Non-trivial stuff

The uncertainty relation (UR) in Shankar is not equivalent to the UR in Griffiths, even though they both depend on the wave function. The UR in Griffiths is what we usually call Heisenberg UR (even though he was not the first who derived it rigorously), while the UR in Shankar was first derived by Schrödinger. The Heisenberg UR follows from the Schrödinger UR, but the Schrödinger UR does not follow from the Heisenberg UR. In this sense the Schrödinger UR is more "general", but in most practical cases the Heisenberg UR is more useful.

See also
http://en.wikipedia.org/wiki/Uncert...2.80.93Schr.C3.B6dinger_uncertainty_relations
http://lanl.arxiv.org/abs/physics/0510275

 

[Zerkor]

B) Non-trivial stuff

The uncertainty relation (UR) in Shankar is not equivalent to the UR in Griffiths, even though they both depend on the wave function. The UR in Griffiths is what we usually call Heisenberg UR (even though he was not the first who derived it rigorously), while the UR in Shankar was first derived by Schrödinger. The Heisenberg UR follows from the Schrödinger UR, but the Schrödinger UR does not follow from the Heisenberg UR. In this sense the Schrödinger UR is more "general", but in most practical cases the Heisenberg UR is more useful.

See also
http://en.wikipedia.org/wiki/Uncert...2.80.93Schr.C3.B6dinger_uncertainty_relations
http://lanl.arxiv.org/abs/physics/0510275

Many Thanks :)
The confsion is solved. But, what kind of generality does the Schrödinger UR has? In other words, what distinguishes it from Heisenberg's UR?

 

[Demystifier]

Many Thanks :)
The confsion is solved. But, what kind of generality does the Schrödinger UR has? In other words, what distinguishes it from Heisenberg's UR?

I have put "general" in quotation marks. It is not really about generality, but about strength of an inequality. For instance, the inequality
$x\geq 2$
is stronger than
$x\geq 1$,
even if they are both simultaneously true. The stronger inequality implies the weaker inequality, but the weaker inequality does not imply the stronger inequality.

 

[Zerkor]

I have put "general" in quotation marks. It is not really about generality, but about strength of an inequality. For instance, the inequality
$x\geq 2$
is stronger than
$x\geq 1$,
even if they are both simultaneously true. The stronger inequality implies the weaker inequality, but the weaker inequality does not imply the stronger inequality.

Got it. Thank you for your help :)