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Quantum Mechanics

  1. Nov 7, 2014 #1
    1. The problem statement, all variables and given/known data
    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. Relevant equations



    3. 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?
     
  2. jcsd
  3. Nov 7, 2014 #2

    Zondrina

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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:

    $$\int P \space dt = \int [m\omega^2 + \frac{2e}{r^3}] \space dr$$
     
  4. Nov 8, 2014 #3

    BvU

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    Gold Member

    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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