{Uncertainty Principal} Uncertainty in position and wave-vector

In summary, the uncertainty principle, also known as Heisenberg's uncertainty principle, states that it is impossible to know both the exact position and momentum of a particle at the same time. This principle affects measurements by creating a degree of uncertainty in any measurement of a particle's position and momentum. It cannot be violated and only applies to particles at the atomic and subatomic level. The uncertainty principle is also closely related to the wave-particle duality of quantum mechanics, explaining how particles can exhibit both wave-like and particle-like behavior and the limitations of knowing their position and momentum simultaneously.
  • #1
Liquidmoose
2
0
A Wave packet is described by

[tex]\Psi[/tex](x)= sin [tex]\frac{kx}{x}[/tex]

Make reasonable estimates of the uncertainty in position and wavevector and show that this function obeys the uncertainty priciple

[tex]\Delta[/tex]x[tex]\Delta[/tex]k>1

solutionish...

sin [tex]\frac{kx}{x}[/tex] = sin ([tex]\pm[/tex][tex]\Delta[/tex]x[tex]\Delta[/tex]k) = 0

hence [tex]\Delta[/tex]x must = some interger of [tex]\pi[/tex]...?

this is about where i get lost don't know if I'm going in the right direction!?
 
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  • #2
isn't [tex]\sin{\frac{kx}{x}}=\sin{k}[/tex]? do you mean [tex]{\frac{\sin{kx}}{x}}[/tex] instead?
 
  • #3
yeah ta!
 

1. What is the uncertainty principle?

The uncertainty principle, also known as Heisenberg's uncertainty principle, states that it is impossible to know both the exact position and momentum of a particle at the same time. This is due to the wave-like nature of particles and the limitations of measurement tools.

2. How does the uncertainty principle affect measurements?

The uncertainty principle means that there will always be a degree of uncertainty in any measurement of a particle's position and momentum. The more accurately one quantity is measured, the less accurately the other can be measured.

3. Can the uncertainty principle be violated?

No, the uncertainty principle is a fundamental principle of quantum mechanics and has been extensively tested and supported by experiments. It is a fundamental limitation on our ability to measure particles.

4. Does the uncertainty principle apply to macroscopic objects?

No, the uncertainty principle only applies to particles at the atomic and subatomic level. Macroscopic objects, such as everyday objects, have such large mass and momentum that their uncertainties are negligible.

5. How does the uncertainty principle relate to the wave-particle duality?

The uncertainty principle is a fundamental aspect of the wave-particle duality of quantum mechanics. It explains how particles can exhibit both wave-like and particle-like behavior and how their position and momentum cannot be known simultaneously.

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