Schrodinger's equation

• Assume a particle-in-a-box scenario. x-axis is the position, y-axis is energy (which we already know must be discrete, so it’s basically n), two infinitely tall walls on two sides (at x=0 and x=L). The particle cannot be at or beyond the walls.
• Manipulate the Hamiltonian (expanded & substituted into the equation) such that only the second derivative of phi (wave function) is on the left side.
• The differential equation says that the wave function, when differentiated twice, will have the same form but multiplied by a constant ($E$).
• The wave function can be $A\cdot \sin kx$.
• After twice-differentiating the wave function in sine form and rearranging the Schrodinger’s equation, we get a formula for $E$ in terms of $k$.
• Square of wave function = probability of appearance
• definite integral of wave function sqaured (e.g. from 0 to box size) = 1 (total probability of apperance)
• $E=\frac{n^2{h}^2}{8m{L}^2}$
• Energy lower than the ground state energy (n = 1, a.k.a. zero-point energy). Since energy is discrete, the only energy level possible that is lower than the ground state energy is 0, and a energy of zero is impossible because the wave function will be 0 at all values (since n = 0), which violates the boundary condition