Find potential energy using time-independent Schrodinger's equation

[eloiseh]

Homework Statement

In a region of space, a particle with mass m and with zero energy has a time-independent wave function ψ(x) = Axe^(−x^2/L^2) where A and L are constants.
Determine the potential energy U(x) of the particle.

Relevant Equations

The time-independent Schrodinger's equation

I had found what U(x) was equal to already by plugging in the wave function and simplifying, which is (2h^2/mL^4)(x^2 - 3L^2/2) by the way.

But the solution key that I have goes an extra step. After stating the equation of U(x) that I got, it says that: "U(x) is a parabola centred at x = 0 with U(0) = −3h^2/mL^2"

Does that mean that I have to plug 0 in for x for the right answer? And how to determine that U(x) is a parabola centred at x=0?

 

[PeroK]

I had found what U(x) was equal to already by plugging in the wave function and simplifying, which is (2h^2/mL^4)(x^2 - 3L^2/2) by the way.

Does that mean that I have to plug 0 in for x for the right answer? And how to determine that U(x) is a parabola centred at x=0?

Your solution is a parabola centred at $x=0$.

 

[eloiseh]

Your solution is a parabola centred at $x=0$.

Thank you so much!