How can the uncertainty relation be written as such

In summary, the uncertainty relation can be written as Δλ Δx >= λ^2 /4π, where λ is the wavelength, Δλ is the uncertainty in the wavelength, and Δx is the uncertainty in the position. This can also be expressed as ΔpΔx >= h/2π, where p is the momentum and h is Planck's constant. To derive this relation, one can substitute p = h/λ and then manipulate the equation to get the final form.
  • #1
Abdul.119
73
2

Homework Statement


Show that the uncertainty relation can be written as
Δλ Δx >= λ^2 /4π

Homework Equations

The Attempt at a Solution


Ok the uncertainty relation is ΔpΔx >= h/2π , also p = h/λ , so substituting that I have Δh/λ Δx >= h/2π , then divide both sides by h, and multiply both sides by λ^2, so I get Δλ Δx >= λ^2 /2π , which is still not the same as the one given, I don't understand how the 2π becomes 4π
 
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  • #2
The minimum uncertainty is ##\displaystyle \frac{\hbar}{2}##.
 
  • #3
Abdul.119 said:
Ok the uncertainty relation is ΔpΔx >= h/2π
Are you sure about that?

Abdul.119 said:
also p = h/λ , so substituting that I have Δh/λ Δx >= h/2π
I don't understand what you are doing here. How can you have Δh?
 
  • #4
Abdul.119 said:
Δh/λ Δx >= h/2π
I assume you mean Δ(h/λ) Δx, which is hΔ(1/λ) Δx
Abdul.119 said:
multiply both sides by λ^2
But that isn't what you did.
Δ(1/λ) is not the same as 1/Δλ. What does it turn into?
 
  • #5
Right I had a mistake in the uncertainty relation that's why I was confused. I took care of it now, thank you for the help.
 

FAQ: How can the uncertainty relation be written as such

1. What is the uncertainty relation?

The uncertainty relation, also known as Heisenberg's uncertainty principle, is a fundamental concept 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.

2. How can the uncertainty relation be written?

The uncertainty relation can be mathematically written as ΔxΔp≥h/4π, where Δx represents the uncertainty in position, Δp represents the uncertainty in momentum, and h is the Planck constant.

3. Why is the uncertainty relation important?

The uncertainty relation is important because it sets a limit on the precision with which certain pairs of physical properties of a particle can be measured. It also has implications for the behavior of quantum systems and plays a crucial role in our understanding of the microscopic world.

4. How is the uncertainty relation related to wave-particle duality?

The uncertainty relation is closely related to the concept of wave-particle duality, which suggests that particles can exhibit both wave-like and particle-like properties. The uncertainty relation shows that it is impossible to precisely know the position and momentum of a particle at the same time, reinforcing the idea of duality.

5. Can the uncertainty relation be violated?

No, the uncertainty relation is a fundamental principle in quantum mechanics and cannot be violated. It is a mathematical consequence of the wave-like nature of particles at the quantum level and has been extensively tested and confirmed through experiments.

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