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## Homework Statement

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.

## Homework Equations

Uncertainty principle

## 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?