Uncertainty principle (stationary state)

In summary, the uncertainty principle, also known as Heisenberg's uncertainty principle, is a fundamental principle in quantum mechanics that states that there is a limit to the precision with which certain pairs of physical properties of a particle can be known simultaneously. This applies to stationary states, where the properties of a particle do not change over time. The uncertainty principle is mathematically expressed as ΔxΔp ≥ ħ/2 and cannot be violated. It fundamentally changes our understanding of the physical world and has led to the development of new technologies.
  • #1
nazmus sakib
5
0
I have to derive
11 15 2015 4 21 AM Office Lens.jpg

I did all the way but stuck with the ">=" to "=" sign. what is the logic behind it ? is it safe to write "approximate" while taking the square-root on both side ? or the energy term "V" has gone during the calculation so it has only momentum "p" ?
 
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  • #2
It should be a ##\geq## sign. Concerning the homework (you should post in the homework section!) think about what's the commutator between ##\hat{x}## and ##\hat{H}## and what it has to do with the general Heisenberg uncertainty relations for arbitrary pairs of observables!
 

1. What is the uncertainty principle?

The uncertainty principle, also known as Heisenberg's uncertainty principle, is a fundamental principle in quantum mechanics that states that the more precisely the position of a particle is known, the less precisely its momentum can be known, and vice versa. This means that there is a limit to the precision with which certain pairs of physical properties of a particle can be known simultaneously.

2. How does the uncertainty principle relate to stationary states?

In quantum mechanics, a stationary state is a state in which a particle's properties, such as position and momentum, do not change over time. The uncertainty principle applies to stationary states by limiting the precision with which these properties can be known simultaneously. This means that even in a stationary state, there is a limit to how precisely we can know a particle's position and momentum.

3. What is the mathematical expression of the uncertainty principle?

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

4. Can the uncertainty principle be violated?

No, the uncertainty principle is a fundamental principle of quantum mechanics and cannot be violated. It is a consequence of the wave-particle duality of quantum particles and is supported by experimental evidence.

5. How does the uncertainty principle impact our understanding of the physical world?

The uncertainty principle fundamentally changes our understanding of the physical world by introducing the idea that certain properties of particles cannot be known with absolute precision. It also highlights the limitations of classical physics in describing the behavior of particles at the quantum level. The uncertainty principle has also led to the development of new technologies, such as electron microscopes, that allow us to observe and manipulate particles at a scale where the uncertainty principle becomes significant.

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