Quantum Physics question (Heisenburg uncertainty principle)

[umagongdi]

Homework Statement

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.

Homework Equations

Heisenberg uncertainty principle is
ΔxΔp = h'/2

h' = Planck's constant over 2π = h/2π

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

 

[Dahaka14]

This is one of those questions where if you don't see what to do, you're kind of stuck. Look over the energy equation that you have found. Is there anything in that equation that can tell you the SIZE of the hydrogen atom?

Hint: Look at each variable and convince yourself what each means. Eventually you should see it. Post again if you are still having trouble!

 

[quZz]

E=-e²/(4πε0a) + h'/(8ma²) ?

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

Look - as a -> 0 energy E ~ const/a^2 and as a -> infinity E ~ -const/a (both constants are positive). Hence somewhere in between E(a) has to be minimal... thus you can estimate the size of the hydrogen atom in ground state.