(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Solving the following differential equation with the given boundary conditions:

[tex]\hbar^2 \frac{d^2}{dx^2}\psi (x) = 2mE\psi (x), \ \ \ \ \ \forall \ \hbar^2,\ m,\ E > 0 [/tex]

[tex]\psi(a) = \psi(-a) = 0 [/tex]

2. Relevant equations

3. The attempt at a solution

[tex]\hbar^2 \frac{d^2}{dx^2}\psi (x) = 2mE\psi (x) [/tex]

[tex]\frac{d^2}{dx^2}\psi (x) = \frac{2mE}{\hbar^2}\psi (x) [/tex]

[tex]\frac{d^2}{dx^2}\psi (x) - \frac{2mE}{\hbar^2}\psi (x) = 0 [/tex]

Let

[tex]\frac{2mE}{\hbar^2} = \rho[/tex]

Then

[tex]\frac{d^2}{dx^2}\psi (x) - \rho \psi (x) = 0 [/tex]

The characteristic polynomial for this ODE shall be

[tex]r^2 - \rho = 0 [/tex]

[tex]r^2 = \rho ,\ r_1 = -\sqrt{\rho},\ r_2 = \sqrt{\rho} [/tex]

Therefore

[tex]\psi (x) = c_1e^{r_1 x} + c_2e^{r_2 x} [/tex]

[tex]\psi (x) = c_1^{-\sqrt{\rho} x} + c_2e^{\sqrt{\rho} x} [/tex]

By using the given conditions the only possible solution I can get is

[tex]c_1 = c_2 = 0,\ \psi (x) = 0 [/tex]

As I get to something like

[tex]e^{4 \sqrt{\rho} a} = 1 [/tex]

And since

[tex]\rho,\ a > 0[/tex]

So

[tex]\psi(x) = 0 [/tex]

seems to be the only solution.

Is this it?

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# First Order Homogeneous ODE

**Physics Forums | Science Articles, Homework Help, Discussion**