Calculating the Commutator of x and p - Problem Discussion

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In summary, the conversation discusses a problem encountered while calculating <p|x|p> where the outcome is found to be 0. The conversation also mentions how many paradoxes can arise when assumptions are not taken into account in mathematical theorems. References are suggested for further understanding.
  • #1
lihurricane
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i have met a problem about the commutator of x and p.
[x,p]=ihbar

/p> is the eigenstate of momentum operator p.

<p/xp-px/p>

=<p/xp/p>-<p/px/p>

=p<p/x/p>-p<p/x/p> the second term is got by the momentum operator p acting on
the left state.

=0

so i get zero! is there anyone can point out where i am wrong?
 
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  • #2
Can you calculate <p|x|p>? Which number will you get? I can tell you: any number you want, depending on how you decide to calculate it. You are playing pseudo-math with formal expressions without really understanding their mathematical meaning. That is how many paradoxes appear.

Many theorems in mathematics are valid under certain assumptions. Skip the assumptions, apply the theorem, and you can well create a paradox.
 
  • #3
lihurricane said:
[...]
so i get zero! is there anyone can point out where i am wrong?

Take a look at Post #18 in this (very recent) thread:

https://www.physicsforums.com/showthread.php?t=426509&page=2

More importantly, check out the references I mentioned. :-)
 
  • #4

What is the commutator of x and p?

The commutator of x and p is a mathematical operation that involves taking the product of two operators and subtracting it in both orders. In this case, x and p represent the position and momentum operators, respectively.

Why is calculating the commutator of x and p important?

Calculating the commutator of x and p is important because it helps us better understand the behavior of quantum systems. It provides information about the uncertainty in the position and momentum of a particle, as well as how these quantities are related to each other.

How do you calculate the commutator of x and p?

The commutator of x and p can be calculated using the formula [x,p] = xp - px, where x and p are the operators and xp and px represent their product in both orders. This can also be written as [x,p] = iħ, where ħ is the reduced Planck's constant.

What does a non-zero commutator of x and p mean?

A non-zero commutator of x and p indicates that the position and momentum operators do not commute, meaning they cannot be measured simultaneously with perfect accuracy. This is a fundamental principle in quantum mechanics known as the Heisenberg uncertainty principle.

What are some applications of the commutator of x and p?

The commutator of x and p is used in various areas of quantum physics, including quantum mechanics, quantum field theory, and quantum information science. It is also used in the study of quantum systems such as atoms, molecules, and subatomic particles.

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