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gabz220
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Homework Statement
A particle with mass m and electric charge e is confined to move in one dimension along the x -axis. It experiences the following potential: V(x) = infinity when x<0, V(x) = -e^2/4*pi*ε*x when x≥0
For the region x ≥ 0 , by substituting in the Schrödinger equation, show that the wave function
u(x) = C*x*exp(-αx) can be a satisfactory solution of the Schrodinger equation so long as the constant α is suitably chosen. Determine the unique expression for α in terms of m , e and other fundamental constants. Note that C is a normalisation constant.then Show that the energy of the particle represented by u (x) is given by
E = -m*e^4/ 2(4*pi*ε*ħ)^2
This has me really stumped, can't show that the energy is equal to the equation above.
Homework Equations
Schrodinger equation
The Attempt at a Solution
attempt:
d^2(u(x))/dx^2 = ((E - V(x))2m/ħ^2)u(x) rearranging the Schrodinger equation
and
d^2(u(x))/dx^2 = (α^2 - 2α/x)C*x*exp(-αx)
so
((E - V(x))2m/ħ^2) = (α^2 - 2α/x) equating coefficients
this is where i get stuck. Any help apreciated
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