Derivation of Uncertainty Princple

In summary, the conversation discussed the Generalized Uncertainty Principle in Modern Quantum Mechanics by J.J. Sakurai. The principle states that the square of the standard deviation of operator A multiplied by the square of the standard deviation of operator B must be greater than or equal to a certain value. The conversation also touched on the omission of the second term in the inequality, which Sakurai explained is because it only makes the relation stronger and is always positive. This makes the uncertainty principle look "nicer" and easier to understand.
  • #1
ChrisLM
9
0
Referrence to Modern Quantum Mechanics, J.J.Sakurai,
I found the "Generalized Uncertainty Principle" is that
<( s.d. of operator A)^2> <( s.d. of operator B)^2> >= 1/4 |<[A,B]>|^2 + 1/4 |<{s.d of A, s.d of B}>|^2

I hope it is not difficult to read, as I don't know how to type it formally.
I would like to ask why the inequality finally omitted the second (the anticommutator) term ?
Sakurai said it is because the second term can only make the inequality relation stronger.
I can't understand. Besides, I am a UG Year 1 students, but familiar to Dirac Notation as I am doing UG research on Quantum Information Theory. I hope anyone can explain it in a easy way ^^ Thank You
 
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  • #2
The second term is always positive. Therefore if LHS >= First + Second, LHS >= First will always be true. And omitting the second term makes the uncertainty principle look "nicer".
 
  • #3
Hi ChrisLM! :smile:

oooh, nicksauce :smile: has beat me to it!

anyway, have a geq: ≥, and try using the X2 tag just above the Reply box :wink:)
 

1. What is the uncertainty principle?

The uncertainty principle, also known as Heisenberg's uncertainty principle, is a fundamental concept in quantum mechanics that states that it is impossible to simultaneously know the exact position and momentum of a subatomic particle.

2. Who first proposed the uncertainty principle?

The uncertainty principle was first proposed by German physicist Werner Heisenberg in 1927 as part of his uncertainty relations.

3. How is the uncertainty principle mathematically expressed?

The uncertainty principle is mathematically expressed as ΔxΔp ≥ ħ/2, where Δx is the uncertainty in position, Δp is the uncertainty in momentum, and ħ is the reduced Planck's constant.

4. What are the implications of the uncertainty principle?

The uncertainty principle has several implications, including the limitations of our ability to measure and predict the behavior of subatomic particles, the wave-particle duality of matter, and the interconnectedness of the universe.

5. Can the uncertainty principle be violated?

No, the uncertainty principle is a fundamental principle in quantum mechanics and has been supported by numerous experiments. It is a key component of our understanding of the behavior of particles on a microscopic level.

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