Quantum Mechanics

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Homework Statement


In classical electromagnetism, an accelerated charge emits electromagnetic radiation. In non-relativistic
limit, where the velocity of the electron is smaller than c, the total power radiated is given by the
Larmor formula, to wit P=2/3*e2*a2/c3, where a denotes the acceleration of the electron. I am expected to use energy conservation, dE/dt=P, to show that in the adiabatic approximation in which the orbit remains nearly circular at all times, the radius of the electron evolves with time as:
r3(t)=r3(0)-4r02ct, where r(0) is the initial radius at t=0 and r0=e2/(mc2) is the classical radius of the electron.

2. Homework Equations



The Attempt at a Solution


The general expression for energy in circular motion is:
E=1/2*m*ω2r2-e/r2
When I differentiate that wrt time and equate the result to P, I obtain the following:
md2r/dr2*dr/dt+2e(dr/dt)/r3=2/3*e*(d2r/dr2)2/c3 but I am not sure how to proceed. Any advice?
 

Answers and Replies

  • #2
STEMucator
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I'm not entirely sure, but using the fact ##dE = P \space dt##, and ##dE = m\omega^2r + \frac{2e}{r^3} \space dr##, I think you should solve:

$$\int P \space dt = \int [m\omega^2 + \frac{2e}{r^3}] \space dr$$
 
  • #3
BvU
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The general expression for energy in circular motion is:
E=1/2*m*ω2r2-e/r2
Doesn't look good to me.

Also, in this derivation you are supposed to make good use of a classical F = ma to give you an expression for a.
 

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