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Friday, May 1, 2015

Exploring Quantum Physics - Week 5 Question 7, 8, 9

Question:

... snip ...
Compute the energies of a particle under a field of constant force towards the center using Bohr's model.

Solution:

The particle is going in circular motion, so it is running on a particular plane, choose coordinate system so that the plane is the xy plane (i.e. z-coordinate is always zero), the trajectory of the particle can then be simplified to (rcosωt,rsinωt). The velocity is the first derivative (rωsinωt,rωcosωt), and the acceleration would be the second derivative (rω2cosωt,rω2sinωt)

Now let's work on the magnitudes, |v|=rω, |a|=rω2.

The total energy is given by

E=p22m+Fr=(mv)22m+mar=mv22+mar=m(rω)22+m(rω2)r=m(rω)22+m(rω)2=3m(rω)22

So we need to find rω. The Bohr's model give us mr2ω=m(rω)r=mvr=n, and the Newton's second law gives us F=ma=mrω2. A little arithmetic trick gives

(rω)3=(r2ω)(rω2)=mr2ωmmrω2m=nmFm=nFm2(rω)=(nFm2)1/3

Substitute this back to the equation we get the answer

E=3m(rω)22=3m((nFm2)1/3)22=32m(nFm2)2/3=32(m3/2nFm2)2/3=32(nFm)2/3=32n2/3(Fm)2/3

This formula solved question 7, 8, 9 as follow:

Question 7 requires the quantity that is a factor that is independent of n, that would be (Fm)2/3.

Question 8 ask for the number part of the ground state, that would be 3212/3=1.5.

Question 9 ask for the exponent, so it is 23.

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