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Question
Consider a neutron and an electron bound to each other due to gravitational force. Assuming Bohr's quantization rule for angular momentum to be valid in this case, derive an expression for the energy of the neutron-electron system.
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Solution
According to Bohr's quantization rule,
Angular momentum of electron, L = `(nh)/(2pi)`
`rArr mv_er = (nh)/(2pi)`
`rArr v_e = (nh)/(2pirm_e)`
...(1)
Here,
n = Quantum number
h = Planck's constant
m = Mass of the electron
r = Radius of the circular orbit
ve = Velocity of the electron
Let mn be the mass of neutron.
On equating the gravitational force between neutron and electron with the centripetal acceleration,
`(Gm_nm_e)/r^2 = m_ev^2`
`rArr (Gm)/r = v^2`
Squaring (1) and dividing it by (2), we have
`(m_e^2v^2r)/(v_2) = (n^2h^2r)/(4pi^2Gm_n)`
`rArr m_e^2r^2 = (n^2h^2r)/(4pi^2Gm_n)`
`rArr r = (n^2h^2)/(4piGm_nm_e^2)`
`v_e = (nh)/(2pirm_e`
`rArr v_e = (nh)/(2pim_e xx ((n^2h^2)/(4pi^2Gm_nm_e^2))`
`rArr v_e = (2piGm_nm_e)/(nh)`
Kinetic energy of the electrons, K = `1/2 m_ev_e^2`
=`1/2 m_e ((2piGm_nm_e)/(nh))^2`
`= (4pi^2G^2m_n^2m_e^3)/(2n^2h^2)`
Potential energy of the neutron,`p = (-Gm_nm_e)/r`
Substituting the value of r in the above expression,
`P = (-Gm_em_n4pi^2Gm_um_e^2 )/(n^2h^2)`
`P = (-4pi^2G^2m_n^2m_e^3)/(n^2h^2)`
Total energy = K + P =`-(2pi^2G^2m_n^2m_e^2)/(2n^2h^2)= -(pi^2G^2m_n^2m_e^2)/(n^2h^2)`
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