Vanessa V. Destura, MSU-IIT Starting with an eigenvalue equation, where Also, we set So, substituting the given values, we find Now, by separation of variables, the equation yields to Let . So, Note that So, Thus, the associated Legendre differential equation is Since the solutions are symmetric in , assume . So, Using Frobenius method […]

Derive . Solution: The radial equation is given by But for a hydrogen atom, Substituting this into the equation above and introducing dimensionless variables or and or , where is the ionization energy, we have Recall from Bohr’s model and . Substituting this to the above equation, we get . Multiplying both sides by , […]

Problems: 1.) Show that 2.) Show that   Solutions: (1) Given that , . But note that , so . Now, recall the radial equation . But . Substitution and rearranging gives: . To find , treat the electron charges as continuous variable Multiplying both sides by and integrating over r, Note that: and So, […]

Problems: 1.) Show that 2.) Show that Solutions: To show that and , we can use Kramer’s relation which is given as where (1) For : But note that: and . So, Therefore, . (2) For : since Therefore, .

Problem: Derive the radial function of hydrogen atom, . Solution: Given the equation , For and : Now, the recursion relation is given as Hence, and  Thus, where Solving for the normalization constant, C: Therefore,