# Schrodinger Equation and 1D Box

## Homework Statement

Trying to construct Shrodinger Equation given:
* mass: m

* Boundary Conditions: (potential)
V(x)=-Vo exp(-x/L) for 0<x≤L
V(x)=∞ for x≤0

## The Attempt at a Solution

(-h^2 / 2m ) (d^2 ψ / dx^2) + V(x)ψ = E * psi

Not sure how to incorporate step-wise V(x) into above eq.

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DrClaude
Mentor
Hi Litmus, welcome to PF!

* Boundary Conditions: (potential)
V(x)=-Vo exp(-x/L) for 0<x≤L
V(x)=∞ for x≤0

And what can you say about the wave function for ##x < 0##?

vela
Staff Emeritus
Homework Helper
You need to solve the Schrodinger equation in each region and then match the solutions at the boundaries. Your book should have examples of how to do this.

And what can you say about the wave function for x<0?
I don't understand. I've given conditions x<0, and x>L it's not specified. Why do we care?

You need to solve the Schrodinger equation in each region and then match the solutions at the boundaries. Your book should have examples of how to do this.
Like this?
(-h^2 / 2m ) (d^2 ψ / dx^2) + [-Vo exp(-x/L) ] ψ = E * psi
(-h^2 / 2m ) (d^2 ψ / dx^2) + ∞ψ = E * psi

Boundary is 0 so:
(-h^2 / 2m ) (d^2 ψ / dx^2) + [-Vo] ψ = E * psi

... how do I proceed?

I don't understand. I've given conditions x<0, and x>L it's not specified. Why do we care?

Like this?
(-h^2 / 2m ) (d^2 ψ / dx^2) + [-Vo exp(-x/L) ] ψ = E * psi
(-h^2 / 2m ) (d^2 ψ / dx^2) + ∞ψ = E * psi

Boundary is 0 so:
(-h^2 / 2m ) (d^2 ψ / dx^2) + [-Vo] ψ = E * psi

... how do I proceed?
The problem can be tackled in the following steps:

1. Use TISE to get a general form of the wavefunction
2. Solve for the constants in the general wavefunction using boundary conditions