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fluidistic
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Homework Statement
Estimate the zero point energy of an electron confined into a region of [tex]10^{-14}m[/tex] which is around the magnitude of the nucleus. Compare this energy with the potential energy between a proton and an electron and the gravitational potential energy with the same distance [tex]10^{-14}m[/tex]. Discuss on the possibility that an electron can be found inside a nucleus.
Homework Equations
[tex]E_p_1=-\frac{ke^2}{r}[/tex]
[tex]E_p_2=-\frac{Gm_em_p}{r}[/tex].
[tex]\int _{0}^{10^{-14}} |\Psi (x)|^2dx=1[/tex].
The Attempt at a Solution
I guess I must find that the zero point energy of the electron is greater than both the potential energy gravitational and Coulombian and so there's a possibility to find it inside a nucleus.
What I've done is to write the time independent Schrodinger's equation with a potential function V(x)=0. Because I assumed that there were walls of potential surrounding the electron and inside the region its potential would be 0.
The solution to Schrodinger's equation gave me [tex]\Psi (x)=c_1 e^{-\frac{2m_eEx}{\hbar}} + c_2[/tex]. Just to be sure, the equation I "solved" is [tex]-\frac{\hbar}{2m_e} \frac{d^2 \Psi}{dx^2}=E\Psi[/tex].
But I'm not given any initial conditions, maybe only contour conditions...
I need some help in order to find [tex]c_1[/tex] and [tex]c_2[/tex].
Then I guess I must see if [tex]|\Psi |^2[/tex] is normalized by evaluating [tex]\int _{0}^{10^{-14}} |\Psi (x)| ^2dx[/tex] and it must equal to 1?
If it's not, I just multiply [tex]\Psi[/tex] by a constant?