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.

Answer 1

An expression for the energy of the neutron-electron system:

• As per the Bohr’s quantization rule,

• Electron’s angular momentum, L = nh2π

• ⇒mver = nh2πrme          …(1)

where,

• v is the electron velocity  
• h = Constant of Planck  
• m = Electron mass  
• n = Quantum number  
• r = Circular orbit’s radius

Let the neutron’s mass be mn.

• When the gravitational force is equated between electron and neutron with the centripetal acceleration,

• ⇒Gmnr = v2                   …(2)

• When (1) is squared and divided by (2), we have

• ⇒ve = 2πGmnmenh  

• Electron’s kinetic energy, K=12meve2

• Neutron’s potential energy,

• P = -Gmnmer

• When the value of r is substituted in the above expression,

• P = -4π2G2mn2me3n2h2

• Total energy = P + K

• =-π2G2mn2me2n2h2