Prove P=h/i(d/dx) from Axioms: Get an Answer

  • Thread starter Palindrom
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In summary, the conversation revolves around proving the equation P=h/i(d/dx) using a list of axioms. The proof can be found in various books on QM and also in a document provided by Daniel. The concept of <p| representing linear momentum is also discussed.
  • #1
Palindrom
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O.K.

So we have a list of axioms.
Could someone please prove to me that these axioms dictate
P=h/i(d/dx)?
I'm interested in a proof, and if there isn't, why choosing this operator of all the operators that could keep the axioms?
Thanks!
 
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  • #2
The proof is pretty long,it can be found in any book on QM (Cohen-Tannoudji,Sakurai,...),i believe i typed it once (search for it),ain't going to do it again.It comes naturally,yes from the axioms and from the coordinate representation of the fundamental commutation relations:
[tex] [\hat{x}_{i},\hat{p}_{j}]_{-}=i\hbar \delta_{ij} \hat{1} [/tex]

Daniel.
 
  • #3
I looked in Cohen Tanoudji, didn't find it.

Could you give me a key word or phrase to find the proof you wrote?
 
  • #5
Tom Mattson said:
See pages 59-61 of the following document:

http://fafalone.hypermart.net/lectures.pdf

edit: Use the page numbers on the actual pages, not Adobe's page numbers.

First of all, thank you.

Now, we're getting to what's bothering me. In this doc., they simply define <p| as some kind of twisted Fourier Transform, which is what we did in our course as well.
Why? Why this? This is supposed to represent the old and familiar linear momentum.
 
  • #7
Thanks, I'll go over it all and come back if I have complains... :smile:
 

FAQ: Prove P=h/i(d/dx) from Axioms: Get an Answer

What are the axioms used to prove P=h/i(d/dx)?

The axioms used to prove P=h/i(d/dx) are the basic properties of derivatives, specifically the linearity, product, and quotient rules.

How does the linearity rule help in proving P=h/i(d/dx)?

The linearity rule states that the derivative of a sum is equal to the sum of the derivatives. This allows us to break down P into smaller, more manageable parts.

Can you explain how the product rule is applied in the proof of P=h/i(d/dx)?

The product rule states that the derivative of a product is equal to the first function times the derivative of the second function plus the second function times the derivative of the first function. This is used to simplify the expression for P and ultimately prove P=h/i(d/dx).

How does the quotient rule play a role in the proof of P=h/i(d/dx)?

The quotient rule states that the derivative of a quotient is equal to the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the square of the denominator. This rule is used to manipulate the expression for P and arrive at the desired result.

Are there any other key concepts or rules used in the proof of P=h/i(d/dx)?

Aside from the linearity, product, and quotient rules, the proof of P=h/i(d/dx) also relies on basic algebraic manipulations and the definition of a derivative. It is important to have a solid understanding of these concepts in order to fully understand the proof.

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