Estimate the momentum of an electron confined to an atomic nucleus of radius a, and deduce a formula for its speed. By comparing the speed to the escape velocity explain why electrons cannot remain in the nucleus.
The Attempt at a Solution
Uncertainty Principle: ∆x∆p≈h (1)
If the electron were confined to the nucleus, the uncertainty in its position ∆x would be the radius of the nucleus. This means that the uncertainty in its momentum is given by rearranging (1) to:
∆p= ∆(mv)≈h/∆x => ∆v≈h/(m ∆x) (2)
The uncertainty in the velocity ∆v is all we know about the velocity so this is the velocity that the electron may have. (Is this explained right? How may I word this better – I get confused between uncertainty and actual velocity.)
= 7.3×10^11 (this is faster than speed of light so must be wrong?)
Potential Energy of an electron in the nucleus of a Bohr atom is PE= e^2/(4πε_0 a) where the radius of the nucleus is a~10^(-15) metres.
Putting in values => PE=2.3×10^(-13) Joules
Escape velocity can be calculated from KE = PE.
=> v = √(2PE/m_e ) =√((2 ×2.3×10^(-13))/(9.11×10^(-31) ))=707 106 781
=> v ≈ 7×10^8 ms^(-1)
So Escape Velocity ≈7×10^8 ms^(-1)
This is faster than the speed of light so I must have done something wrong as this would imply that the electron cannot escape unless its velocity is greater than the speed of light, therefore the electron would be bound???