Question

I have the answer key but I do not fully understand it. Please help! I beleive that d is the distance between the electron and the proton.

10. Calculate the speed of the proton in the ground state of Bohr's hydrogen atom Solution: As outlined in the statement of Problem 32, the effect of the proton's motion is to replace me μ in the results of the Bohr theory. Therefore, the separation distance between the proton and the electron is given by (24) with n1, and the common angular velocity of the orbit is uh (27) By working in a choice of coordinates where the center of mass lies at the origin (done so that rpl and rel are the radii of each particle's orbit around h cer of mass). we have that +mere (28) +m The radii of the proton's and the electron's orbits are related by mpp-mere and p+re-d (29) where the left side of the above is necessary for (28) to be satisfied, and the right side of the above follows from the fact that the proton and the electron lie on the same line through the center of mass. The two equations in (29) can then be solved to obtain 12 meke(30) =-=-a0 = mp mp Thus, the (linear) velocity of the proton in the ground state is 1190 3.97 x 10-6e (31) There is a more direct route involving the vanishing center of mass velocity. By virtue (32) of mp + me we have that the speeds of the proton and the electron are related by mptpmee and pe (33) here m is the reduced mass velocity (this second equation is due to the fact that v(rp ) Vp - ve). Solving (33) for ve and p, we obtain (34) me mp (mp+melao yielding the same result.

Answer 1

Suppose initially people have gone through this problem by assuming proton as stationary and an electron is orbiting around the proton. But if we see center of mass frame of reference then, we need to make some modification in every expression that is mass of electron have to replace by mass of reduced mass of electron and proton mu where 1/mu = 1/m_e + 1/m_p mu = m_e m_p/m_e + m_p and distance between proton and electron d_j = r_p + r_e This is case, where electron and proton both are moving around the center of mass position. \r\n and electron and proton are always at 180 degrees phase apart. So that r_p notequalto r_e as v_p notequalto v_e (velocity) and they move with common angular velocity, given by omega_I = angular momentum/moment of Inertia omega_I = L_I/mu d_j^2 = mu h/m_e^2 a_0^2 [you need to change m_e rightarrow mu/m_e in every expression] And if we consider center

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