Prove Uncertainity Relation for Particle in a Box w/ Length L

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In summary, the conversation discusses a question about proving a particle in a box follows the Heisenberg Uncertainty Principle. The person asking the question believes it is illogical to use the principle to prove itself. The conversation suggests calculating the expectation values of x and x^2 separately and multiplying them, rather than using the Heisenberg Uncertainty Principle.
  • #1
string_theory
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Hi dear virtual friends,

I am now going to ask you something. I would be pleased if you would answer me.

In an exam I was asked to show that for a particle in a box of width length L , delta (x)delta(p)>=h/(4*pi) holds.
I think this is not a logical question. Because I think it is like asking something like this: Prove A using A.

I would be really grateful if you would write something regarding this.
Thank you.

P.S. My problem is not envolving you in doing my homework. I just wanted to know if what I think is right or wrong.
 
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  • #2
i think you were supposed to calculate Δx and Δp separately, then multiply the two together, and compare it to h/4pi.

you know, find the expectation value of x, and then the expectation value of x2 and all that jazz.

of course you are not allowed to use the Heisenberg Uncertainty principle to show that the Heisenberg Uncertainty principle is true. that is circular.
 
  • #3
Thank you very much for youranswer.
 

1. What is the Uncertainty Relation for a Particle in a Box with length L?

The Uncertainty Relation, also known as the Heisenberg Uncertainty Principle, states that the product of the uncertainties in the position and momentum of a particle must be greater than or equal to half of the reduced Planck's constant (h-bar).

2. How is the Uncertainty Relation derived for a Particle in a Box with length L?

The Uncertainty Relation for a Particle in a Box with length L is derived using the Schrodinger Equation and the boundary conditions for a particle confined to a one-dimensional box. This derivation shows that the minimum uncertainty in position is equal to half of the box length, while the minimum uncertainty in momentum is equal to half of the box's wave number.

3. What are the implications of the Uncertainty Relation for a Particle in a Box with length L?

The Uncertainty Relation has significant implications for the behavior of particles at the quantum level. It means that the more precisely we know the position of a particle, the less certain we can be about its momentum, and vice versa. This uncertainty is inherent in the nature of particles and cannot be eliminated.

4. Can the Uncertainty Relation be violated for a Particle in a Box with length L?

No, the Uncertainty Relation is a fundamental principle in quantum mechanics and cannot be violated. This means that the relationship between the uncertainties in position and momentum for a particle in a box with length L will always hold true, regardless of the specific values of these uncertainties.

5. How is the Uncertainty Relation for a Particle in a Box with length L related to the concept of wave-particle duality?

The Uncertainty Relation is closely connected to the wave-particle duality concept in quantum mechanics. It shows that particles can exhibit both wave-like and particle-like behavior, and that their position and momentum cannot be simultaneously known with certainty. This duality is a fundamental aspect of quantum mechanics and is essential for understanding the behavior of particles at the atomic and subatomic level.

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