Proof of the generalized Uncertainty Principle?

In summary, the conversation discusses a proof of the generalized uncertainty principle and the last step involving the use of relations in (4.20). The author explains that inequality (4.20) holds for any two hermitian operators, including C and D. They go on to use equations (4.22) and (4.23) to write the right-hand side in terms of ΔA and ΔB, and ultimately derive the inequality (4.24) using [A,B] = [C,D]. There is a typo in (4.23) and it is questioned if the author meant to use A and B instead of C and D in (4.24). The summary ends with
  • #1
patric44
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39
Homework Statement
i am trying to proof the generalized uncertainty principle , and i am stuck at some point
Relevant Equations
(ΔC)^2 = <φ|A^2|φ>
hi guys
i am trying to follow a proof of the generalized uncertainty principle and i am stuck at the last step :
uncertanity.png

i am not sure why he put these relations in (4.20) :
$$(\Delta\;C)^{2} = \bra{\psi}A^{2}\ket{\psi}$$
$$(\Delta\;D)^{2} = \bra{\psi}B^{2}\ket{\psi}$$
i tried to prove these using the difination of the expectation value and had no success! any help will be appreciated, thanks.
 
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  • #2
Inequality (4.20) holds for any two hermitian operators. In particular, it holds for C and D. So you have
$$\lvert \langle [C,D] \rangle \rvert^2 \le 4 \langle C^2 \rangle \langle D^2 \rangle.$$ Then you can use (4.22) and (4.23) to write the righthand side in terms of ##\Delta A## and ##\Delta B##. And since [A,B] = [C,D], you end up with
$$\lvert \langle [A,B] \rangle \rvert^2 \le 4 (\Delta A)^2 (\Delta B)^2 .$$ Divide by 4 and take the square root to finish the derivation.

I'm wondering if the author meant to use A and B in (4.24) instead of C and D. There's also a pretty obvious typo in (4.23).
 
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  • #3
vela said:
Inequality (4.20) holds for any two hermitian operators. In particular, it holds for C and D. So you have
$$\lvert \langle [C,D] \rangle \rvert^2 \le 4 \langle C^2 \rangle \langle D^2 \rangle.$$ Then you can use (4.22) and (4.23) to write the righthand side in terms of ##\Delta A## and ##\Delta B##. And since [A,B] = [C,D], you end up with
$$\lvert \langle [A,B] \rangle \rvert^2 \le 4 (\Delta A)^2 (\Delta B)^2 .$$ Divide by 4 and take the square root to finish the derivation.

I'm wondering if the author meant to use A and B in (4.24) instead of C and D. There's also a pretty obvious typo in (4.23).
thanks so much it becomes clear now :smile: , i guess the author meant ##A,B##
 
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Related to Proof of the generalized Uncertainty Principle?

1. What is the generalized Uncertainty Principle?

The generalized Uncertainty Principle (GUP) is a modification of Heisenberg's Uncertainty Principle, which states that the position and momentum of a particle cannot be known simultaneously with perfect accuracy. The GUP extends this principle to include other physical quantities, such as energy and time, and suggests that there is a minimum measurable uncertainty in these quantities.

2. What is the significance of the GUP?

The GUP has significant implications for our understanding of quantum mechanics and the fundamental nature of reality. It suggests that there are limits to our ability to measure certain physical quantities, and that the uncertainty in these measurements is not just due to limitations in our technology, but is inherent in the fabric of the universe.

3. How is the GUP related to string theory?

String theory is a theoretical framework that attempts to reconcile the principles of quantum mechanics and general relativity. The GUP arises naturally in string theory, as it predicts that there is a minimum length scale in the universe. This minimum length scale is related to the uncertainty in position measurements, as described by the GUP.

4. Has the GUP been experimentally verified?

While there is currently no direct experimental evidence for the GUP, there have been studies that provide support for its predictions. For example, observations of black holes and other astronomical objects have shown that there is a minimum uncertainty in their position and momentum, as predicted by the GUP.

5. How does the GUP affect our understanding of the Big Bang?

The GUP has been proposed as a possible solution to the singularity problem in the Big Bang theory. By introducing a minimum length scale, the GUP suggests that there was never a true singularity at the beginning of the universe, but rather a state of maximum density. This idea is still being explored and studied by scientists.

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