- #1
weguihapzi
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Homework Statement
a) For a region, the potential energy experienced by a particle is:[tex]\frac{-e^2}{4\pi \varepsilon _{0}x}[/tex]
where e is electric charge.For this region, by substituting into the Schrodinger equation, show the wave function:
[tex]u(x)=Cxe^{-\alpha x}[/tex]
can be a satisfactory solution. And determine the unique expression for α in terms of m, e, and other fundamental constants.b) Show the energy of the particle represented by u(x) is given by:
[tex]E=\frac{-me^4}{2(4\pi \varepsilon _{0}hbar)^2}[/tex]
Homework Equations
TISE for this potential:
[tex]\frac{-hbar^2}{2m}\frac{d^2\Psi }{dx^2}-\frac{e^2}{4\pi \varepsilon _{0}x}\Psi = E\Psi[/tex]
The Attempt at a Solution
It's been a long time since I've done any quantum exercises, and just trying to get myself back into it, so please bear with me if this seems like total nonsense.
For part a) so far, I've tried differentiating the given wave function twice, and plugging that in the TISE, along with the undifferentiated wavefunctions.
I've then tried rearranging this equation to get in terms of α, but end up E (not sure how to get rid of this) with other non-fundamental constants in the answer, so I was wondering if I'm on the right track, or whether I'm way off.As for part b), I'm really not sure. Is that simply a case of rearranging the equation to get E? Or would I need to use an operator to get this?
Again, I can only apologise for my (very) rusty knowledge, so any help is very much appreciated! Thank you.