Quantum Question: Help Understanding GUP Proof in Document

  • Thread starter latentcorpse
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Additionally, \hat{X}^2=(\Delta\hat{A}_t)^2 is just a shorthand for writing the integral of (\hat{A}-\langle\hat{A}\rangle_t)^2 since we know they are equal.
  • #1
latentcorpse
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Just reading through my notes and found a step I can't follow:

If you look at p3 of the follwing document, one of the lines in the proof of the Generalised Uncertainty Principle has a (2) next to it. I can't get from the line before it to that line.

Can anyone help me out?

http://www.ph.ed.ac.uk/teaching/course-notes/documents/64/786-lecture5.pdf


thanks
 
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  • #2
Use, equation (1) on that page to show that [tex][\hat{X},\hat{Y}]=[\hat{A},\hat{B}][/tex], then use the definition of expectation value
 
  • #3
hey. thanks. i have another question about it though.

in the line before (2), how does the [itex](\Delta \hat{A}_t)^2[/itex] term work? Surely when we square [itex]\hat{X}[/itex], we get [itex](\Delta \hat{A}_t)^2[/itex] as well as other stuff arising from the cross terms?

also in the very last line, where does he get the [itex]i[/itex] in the RHS from - I'm assuming it's so that we end up with a [itex]\geq[/itex] not a [itex]\leq[/itex] but i don't follow it...

thanks
 
  • #4
By definition, [tex](\Delta \hat{A}_t)^2=(\hat{A}-\langle\hat{A}\rangle_t)^2=\hat{X}^2[/tex]

And the [itex]i[/itex] is just a way to account for the negative sign since [tex]\langle i [\hat{A},\hat{B}]\rangle_t^2=i^2\langle [\hat{A},\hat{B}]\rangle_t^2=-\langle[\hat{A},\hat{B}]\rangle_t^2[/tex]
 
  • #5
but [itex]\Delta \hat{A}_t = \sqrt{\langle \hat{A^2}_t \rangle - \langle \hat{A}_t \rangle^2}[/itex]

so why do you get what you've written?
 
  • #6
The two definitions are equivalent:

[tex](\hat{A}-\langle\hat{A}\rangle_t)^2=\hat{A}^2-2\hat{A}\langle\hat{A}\rangle_t+\langle\hat{A}\rangle_t^2[/tex]

[tex]\implies \int_{-\infty}^{\infty} \Psi^{*}(x,t)(\hat{A}-\langle\hat{A}\rangle_t)^2\Psi(x,t)dx=\int_{-\infty}^{\infty} \Psi^{*}(x,t)\hat{A}^2\Psi(x,t)dx-2\langle\hat{A}\rangle_t \int_{-\infty}^{\infty} \Psi^{*}(x,t)\hat{A}\Psi(x,t)dx+\langle\hat{A}\rangle_t^2\int_{-\infty}^{\infty} \Psi^{*}(x,t)\Psi(x,t)dx[/tex]

[tex]=\langle\hat{A}^2\rangle_t-2\langle\hat{A}\rangle_t^2+\langle\hat{A}\rangle_t^2=\langle\hat{A}^2\rangle_t-\langle\hat{A}\rangle_t^2=(\Delta\hat{A}_t)^2[/tex]

[tex]\implies (\Delta\hat{A}_t)^2=(\hat{A}-\langle\hat{A}\rangle_t)^2[/tex]

since they both integrate to the same thing
 

1. What is the GUP proof in quantum mechanics?

The GUP (Generalized Uncertainty Principle) proof is a mathematical model that modifies the Heisenberg's uncertainty principle in quantum mechanics. It takes into account the effects of gravity and other fundamental forces on the measurement of a particle's position and momentum.

2. How does the GUP proof affect our understanding of quantum mechanics?

The GUP proof suggests that our current understanding of quantum mechanics may need to be modified to incorporate the effects of gravity and other fundamental forces. It also implies that there is a minimum measurable length and a maximum measurable momentum, which could have significant implications for our understanding of the universe.

3. What evidence supports the GUP proof?

There have been several theoretical and experimental studies that support the GUP proof. Some of these include studies on black hole entropy, the behavior of particles in a strong gravitational field, and observations of the cosmic microwave background radiation. However, more research and experimentation is needed to fully validate the GUP proof.

4. How does the GUP proof relate to string theory?

The GUP proof is closely related to string theory, which is a theoretical framework that attempts to reconcile general relativity and quantum mechanics. In string theory, the strings that make up the fabric of the universe have a minimum length, which is consistent with the minimum measurable length proposed in the GUP proof.

5. What are the potential implications of the GUP proof?

If the GUP proof is validated, it could lead to significant changes in our understanding of the universe and how it operates. It could also have practical applications, such as improving the accuracy of measurements in quantum technology. Additionally, the GUP proof could provide insights into the nature of space and time, and help us better understand the relationship between gravity and quantum mechanics.

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