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Hi all i am doing past exam paper questions and this question i am not sure about, i check notes and books but can't find relevant information

Q1i.) If we try to confine an electron in a small region of size a, then this electron has to have a non-sero average kinetic energy, K in order to satisy Heisenberg uncertainty principle. Find an expression for the minimal kinetic energy K(a) as function of a.

Q1ii.) In a hydrogen atom the attraction between the electon and the nucleus, effectively confine the electron in a region of size a. The total energy of this electron is the sum of its potential energy U(a)=-e²/(4πε0a) and its kinetic energy K(a), as computed in part (ii). Find an estimate of the size of the hydrogen atom.

2. Relevant equations

Heisenberg uncertainty principle is

ΔxΔp = h'/2

h' = planck's constant over 2π = h/2π

3. The attempt at a solution

I have tried the question is this right?

Using Heisenberg uncertainty principle rearranged, p=h'/(2Δx) and replaced in kinetic energy equation, K=p²/2m and i got

K=(h')²/(8m(Δx)²), and in terms of a

K(a)=(h')²/(8ma²)

For part ii, is the total energy

E=-e²/(4πε0a) + h'/(8ma²) ?

I don't understand how to estimate the size of the hydrogen atom

thanks for helping

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# Quantum Physics question (Heisenburg uncertainty principle)

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