1- If the functions 1 and 2 are solutions of the Schrödinger equation for a particle, then show that a,4,+az2, where a, and az are arbitrary constants, is also a solution of the same equation.

A particle in a 3-dimensional quadratic box with box length L has an energy given by h² E = (n+n+n). The degeneracies of the first, second, and 8mL² third level are a. e. 1, 2, 3 1, 3, 3 b. 1, 3, 1 c. 3, 3, 3 d. 1, 2, 2

2. A wave function is a linear combination of 1s, 2s, and 3s orbitals: y(r) = N(0.25 y,s + 0.50W2, +0.30W). Find the normalization constant N, knowing that 1s, 2s, and 3s orbitals are normalized.

Review schrodinger equations not dependent on 3D time in ball coordinates -V² + V(f) ) µ(F) = E Þ(*) 2m F = (r,0,9) and E is the energy system. Assume the potential is only radial function V (f) = V(r). With %3D 'To solve the Schrodinger equation above apply the method ψη, θ φ) -R(r)P (θ) Q (φ) variable separation problem : By defining the separation constant in the angular function 0 as (1 + 1) Also show that P(0) = P"(cos 0) angular function solution P(0) can be written as Polynomial Associated Legendre The combined angular functions of the sphere are known as the "harmonic function of the sphere" (spherical harmonics) Y(0, p) x Pi" (cos 0)e±imp