Minimum uncertainty of the momentum of a small particle

In summary, the minimum uncertainty of the momentum of a small particle with mass m=1g, confined within a region of width a=1cm, can be calculated using the equation Delta(p)*Delta(x)>=hbar/2. After solving for Delta(p), it is found that the product of uncertainties increases with increasing n, reaching a minimum at n=1 with a value of approximately 0.568. This value may be interpreted as the minimum uncertainty in the velocity of the particle.
  • #1
AndrejN96
26
0

Homework Statement


Find the minimum uncertainty of the momentum of a small particle with mass m=1g, which is confined within a region of width a=1cm.

Homework Equations


Delta(p)*Delta(x)>=hbar/2

The Attempt at a Solution



Delta(p)*Delta(x)=hbar/2
Delta(p)*10^(-2)=hbar/2
Delta(p)=10^2*hbar/2

This looks pretty straightforward to me, but the given mass in this problem is what confuses me.
 
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  • #2
I think you have it right & putting in the mass is a red herring. Maybe they meant to ask for the minimum uncertainty in the velocity.
 
  • #3
You might want to scan this:

http://en.wikipedia.org/wiki/Particle_in_a_box

Pull quote:

"The uncertainties in position and momentum (
b56546a86ab832a9b2a5b15f96519319.png
and
7aa41487a1a40b0077afa0c3331ba111.png
) are defined as being equal to the square root of their respective variances, so that:

a83e98aee94cda7f0410e16698b54ebf.png

This product increases with increasing n, having a minimum value for n=1. The value of this product for n=1 is about equal to 0.568
9dfd055ef1683b053f1b5bf9ed6dbbb4.png
..."
 

1. What is the minimum uncertainty of the momentum of a small particle?

The minimum uncertainty of the momentum of a small particle is known as the Heisenberg uncertainty principle, which states that it is impossible to precisely determine both the position and momentum of a particle simultaneously. This means that there will always be a minimum level of uncertainty in measuring the momentum of a small particle.

2. How is the minimum uncertainty of the momentum of a small particle calculated?

The minimum uncertainty of the momentum of a small particle is calculated using the Heisenberg uncertainty principle formula, which states that the uncertainty in the position of a particle multiplied by the uncertainty in its momentum must be greater than or equal to half of the reduced Planck's constant (h/2π).

3. What factors affect the minimum uncertainty of the momentum of a small particle?

The minimum uncertainty of the momentum of a small particle is affected by the mass and velocity of the particle. The smaller the mass and slower the velocity of the particle, the greater the uncertainty in its momentum. Additionally, the level of measurement accuracy also plays a role in the minimum uncertainty.

4. Can the minimum uncertainty of the momentum of a small particle be reduced?

No, according to the Heisenberg uncertainty principle, the minimum uncertainty of the momentum of a small particle cannot be reduced. This is because the act of measuring one quantity (e.g. momentum) will necessarily disturb the other quantity (e.g. position), resulting in a minimum level of uncertainty.

5. How does the minimum uncertainty of the momentum of a small particle relate to quantum mechanics?

The Heisenberg uncertainty principle, which dictates the minimum uncertainty of the momentum of a small particle, is a fundamental principle of quantum mechanics. It helps to explain the probabilistic nature of subatomic particles and the limitations of measuring their properties with precision.

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