# Quantum potential problem -- Particle confined in 1 dimension

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1. Jan 22, 2016

### gabz220

1. The problem statement, all variables and given/known data
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, cant show that the energy is equal to the equation above.
2. Relevant equations
Schrodinger equation

3. 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

Last edited: Jan 22, 2016
2. Jan 22, 2016

### PeroK

(***) You might want to check your differentiation.

3. Jan 22, 2016

### gabz220

I have, i cant see the mistake; i even wolfram alpha'd it and it gave me the same awnser

4. Jan 22, 2016

### PeroK

There must be one term without a factor of x. $f(x) = xg(x) \ \Rightarrow \ f''(x) = 2g'(x) + xg''(x)$

5. Jan 22, 2016

### gabz220

Your right, sorry, i copied it down wrong. I have been working with the correct derivative though and i still cant do it.
question now corrected

6. Jan 22, 2016

### PeroK

Hint: when are two polynomials equal?

7. Jan 22, 2016

### PeroK

You may also have mistyped the answer. I get $E = \frac{-me^4}{2(4\pi \epsilon \hbar)^2}$

8. Jan 22, 2016

### gabz220

Yeah that was the answer.
thanks for the help

9. Jan 22, 2016

### BvU

Hello gabz,

Interesting exercise.
In the future, Could you use brackets in appropriate places when posting ? For example, I assume your potential is $$V(x) = -{e^2\over 4 \pi \epsilon_0 \; x} \ \rm ?$$

10. Jan 22, 2016

### gabz220

yeah, sorry about that. Made quite a few mistakes, ill be more careful next time