1. The problem statement, all variables and given/known data 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. 2. Relevant equations Uncertainty principle 3. 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.) ∆v≈(6.626×10^(-34))/(9.11×10^(-31)×10^(-15) )=(6.626×10^12)/9.11 = 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. KE=(m_e v^2)/2=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???