Answer to Question #191350 in Quantum Mechanics for Sam Pavan

Question

 (a) Write down the Schrodinger equation for the electron in a hydrogen atom.

(b) Considering the central nature of the potential of the electron in hydrogen atom show that the solution of the Schrodinger wave equation is of the form:

Ψ(𝑟, 𝜃,𝜙) = 𝑅(𝑟)Θ(𝜃)Φ(𝜙), where Φ(𝜙)Θ(𝜃)𝑅(𝑟) satisfies the following differential equations

Answer 1

This is the two-parcel system of the hydrogen atom composed of an electron and a proton and the outward movement around its middle mass is similar to the movement of a single, reduced-mass particle. This diminished particle is positioned at r, where r is the vector defining the electron's location relative to the proton's position. The r length is the distance from the proton to the electron and the r direction is provided by the vector orientation from the proton to the electron. As the proton is much larger than the electron, the diminished mass is equal to the electron mass and the proton is situated at the heart of the mass in this champion.

Like any two-part system's internal movement can be described by the movement of a single particle with a decreased mass, the hydrogen atom representation has something in common with that of a diatomic molecule previously discussed. The Schrödinger hydrogen equation for the

$H^(r,θ,φ)ψ(r,θ,φ)=Eψ(r,θ,φ)$

Uses the same spherical co-ordinate kinetic energy operator, T. In contrast to the diatomic molecule where the length of a bond is constant and a rigid rotor model was used for the hydrogen atom, the distance r between two particles differ. The Hamiltonian hydrogen atom also has a potential term V to characterize the pull from the proton to the electron. This concept is the potential energy of Coulomb,

$V^{r}=-\frac{e^2}{4πϵ_0r}$

Where r is the distance from the proton to the electron. The potential energy from Coulomb depends on and does not depend on, the distance between the electron and the core. An opportunity of this kind is called core

It is advisable to change from x,y,z to spherical coordinates for the radius r, as well as for angles β measurement from the positive x-axis of the XY plane which can be between 0 and 2β μ l, determined between the positive z-axis and the XY plane.

The Schrödinger time indepdent equation for an electron around a positively charged nucleus then exists in spherical coordinates

${-\frac{ℏ_2}{2μr_2}[\frac{∂}{∂r}(r^2\frac{∂}{∂r})+\frac{∂}{sinθ∂θ}(sinθ\frac{∂}{∂θ})+\frac{1}{sin^2θ}\frac{∂^2}{∂φ^2}]−\frac{e^2}{4πϵ_0r}}ψ(r,θ,φ)=Eψ(r,θ,φ)$