(Solved): Problem 1: Schrodinger's Equation - "Particle in a box" The time independent Schrodinger Equation ...

Problem 1: Schrodinger's Equation - "Particle in a box" The time independent Schrodinger Equation has the form: \[ \frac{\partial^{2} \psi}{\partial x^{2}}=\frac{2 m}{\hbar^{2}}[U(x)-E] \psi \] a) Considering the boundary conditions for an infinite potential (length, \( L \) particle in a box) demonstrate the validity of this equation. b) Determine the Energy of the confined levels in this system. c) What is the Normalization constant for the \( n=2 \) state. d) What is i) the probability of finding a particle in the confined \( n=2 \) state? And ii) What is the most likely position to find the particle?