ANSWER:

The potential energy is 0 inside the box (V=0 for 0L). We assume the walls have infinite potential energy to ensure that the particle has zero probability of being at the walls or outside the box. Doing so significantly simplifies our later mathematical calculations as we employ these boundary conditions when solving the Schrödinger Equation.

Step 2: Solve the Schrödinger Equation

The time-independent Schrödinger equation for a particle of mass m moving in one direction with energy E is

−ℏ22md2ψ(x)dx2+V(x)ψ(x)=Eψ(x)(1)

with

ℏ  is the reduced Planck Constant where  ℏ=h2π  

m is the mass of the particle

ψ(x)  is the stationary time-independent wavefunction

V(x) is the potential energy as a function of position

E  is the energy, a real number

This equation can be modified for a particle of mass m free to move parallel to the x-axis with zero potential energy (V = 0 everywhere) resulting in the quantum mechanical description of free motion in one dimension:

−ℏ22md2ψ(x)dx2=Eψ(x)(2)

This equation has been well STUDIED and GIVES a general solution of:

ψ(x)=Asin(kx)+Bcos(kx)(3)

where A, B, and K are constants.

Explanation: