Heisenberg Uncertainty Relations - angular momentum and angular displacement

[SJT]

Homework Statement

Starting from one of the more familiar Heisenberg Uncertainty Relations, derive the Uncertainty Relation involving angular momentum and angular displacement and explain its significance.

Homework Equations

The relevant uncertainty relationship is that between uncertainty in position and momentum:

x .  p ≳ ℏ

The Attempt at a Solution

For the first part of the problem:

We know that in rotational mechanics that angular displacement is the length of an arc s which is calculated from

s = r . 

where r is the radius and  is the angle subtended. We also know that momentum p is the product of the particle’s mass m and velocity v:

p = m . v

Combining these where arc length s replaces x,  is the uncertainty in angle subtended and v is the uncertainty in velocity:

x . p ≳ ℏ

r .   . m . v ≳ ℏ

 . m . v . r ≳ ℏ

and angular momentum L = m . v . r where r and p are perpendicular, so

 . L ≳ ℏ

For the second part of the problem I am unsure. We have not covered atomic structures (Bohr) or Schrödinger yet. My attempt is below, but I am unsure because there is nothing we have covered in lectures yet which provide any context, and this (last) question in the assignment has not been covered in class.

Attempt:

This is a significant result because of its relevance to atomic structure. The result tells us that not only that as we measure angular displacement or momentum, as we improve the accuracy of one measurement we increase uncertainty of the other, but also that angular momentum is quantised.

In the early 20th century following Rutherford’s discovery of structure of the atom, subsequent investigation had difficulty explaining why the electrons orbiting the nucleus didn’t radiate electromagnetic radiation, and from that loss of energy, eventually spiral into the nucleus so that the atom collapsed. Niels Bohr proposed that there are stationery states in which the electrons can orbit without radiating electromagnetic radiation where the angular momentum of the electron is a positive integer multiple of ℏ.


Am I on track or have I missed the mark completely? Thanks in advance for any feedback/suggestions.

S.

 

[fireboy115]

The actual formula is ∆L∆Ø≥h, not h-bar. I am student too, so my explanation of this is going to be ad hoc and could be wrong, but the equation is from my professor, so I have faith in it. I believe that you failed to account for the fact that the arc length cannot just be plugged in for x. http://en.wikipedia.org/wiki/Sine#Relation_to_the_unit_circle"

If we are considering a wave that has a perfect sin curve, then one wavelength of the wave is 2π, therefore it becomes 2π*r*∆Ø*m*∆v≥h-bar --->r∆Ø∆vm≥h