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=nℏ, and the Newton's second law gives us F=ma=mrω2. A little arithmetic trick gives
(rω)3=(r2ω)(rω2)=mr2ωmmrω2m=nℏmFm=nℏFm2(rω)=(nℏFm2)1/3
Substitute this back to the equation we get the answer
E=3m(rω)22=3m((nℏFm2)1/3)22=32m(nℏFm2)2/3=32(m3/2nℏFm2)2/3=32(nℏF√m)2/3=32n2/3(ℏF√m)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 (ℏF√m)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.
... 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=nℏ, and the Newton's second law gives us F=ma=mrω2. A little arithmetic trick gives
(rω)3=(r2ω)(rω2)=mr2ωmmrω2m=nℏmFm=nℏFm2(rω)=(nℏFm2)1/3
Substitute this back to the equation we get the answer
E=3m(rω)22=3m((nℏFm2)1/3)22=32m(nℏFm2)2/3=32(m3/2nℏFm2)2/3=32(nℏF√m)2/3=32n2/3(ℏF√m)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 (ℏF√m)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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