Exam preparation separation of variables and the Schrodinger equation
Channel: CHEMISTRY MATHEMATICS
Date: 2021-08-26

The Schr¨odinger equation in a central potential is examined. The
separation of variables procedure is used to turn this partial differential
equation into a set of ordinary differential equations. The angular
equations are solved, first for a particle confined to a ring and then for
a particle confined to a sphere, thereby building up, one dimension at
a time, toward the eigenstates of the hydrogen atom. Special attention
is paid to linear combinations of states and time-dependent states.
The properties of the spherical harmonics are explored, including
a brief introduction to angular-momentum raising and lowering operators. The relationship of spherical harmonics to spin 1 systems is

discussed.


Consider the spherical coordinates, (r, ψ, θ), for the center of a particle that may be a point particle or a spatially distributed one. The variable r is the distance from the center of a force field, ψ is the azimuthal (latitude) angle and θ is the orbital (longitude) angle. Let S denote the column vector of these three coordinates.

Suppose some physical quantity is a function of the spherical coordinates of the center of a particle, say P(r, ψ, θ). In quantum analysis P could be the probability density for the particle. The separation of variables assumption is that the function P(r, ψ, θ) is the product of three functions, each depending on only one of the variables; i.e.,

P(r, ψ, θ) = R(r)Ψ(ψ)Θ(θ)
When the separation-of-variables assumption is applied to the partial differential equation arising from Schrödinger's time-independent equation it leads to three ordinary differential equation that are more easily solved than the partial differential one. The problem is that the essential nature of a particle is that it has a trajectory that gives its coordinates as a function of time, say (r(t), ψ(t), θ(t)). If the trajectory is periodic with a period T then (r(t+T),ψ(t+T), θ(t+T))=(r(t), ψ(t), θ(t)). The time variable may be eliminated by solving for t as a function of θ and making the other coordinates functions of θ.

The Quantum Analysis
Schrödinger's wave mechanics is and has been the preferred formulation of quantum mechanics almost from its creation in the late 1920's. It generates a partial differential equation whose squared magnitude is the probability density function (PDF) for the particles of a system. The analysis starts with the Hamiltonian function for the system. For an electron in the field of a proton the Hamiltonian function is:

H = ½p²/m − α/r
where p is the momentum of the electron, m is its mass and r is the distance from the proton. The parameter α is usually represented as 1/(4πε0).
The Hamiltonian function H of a system is converted into a Hamiltonian operator H^ by replacing the momentum squared, p², with −h²∇², where h is Planck's constant divided by 2π and ∇² is the Laplacian for the coordinate system. The time-independent Schrödinger's equation is then

H^Ω = EΩ
where Ω, called the wave function, which Schrödinger left unspecified but now is interpreted as such that |Ω|² is the probability density function (PDF) for the system. E is the energy of the system.




this channel coversQuantum Mechanics – II
Schrodinger Wave Equation for a Particle in a Three Dimensional Box
#The #Concept #of #Degeneracy #Among #Energy #Levels #for #a #Particle #in #Three #Dimensional #Box
#Schrodinger #Wave #Equation #for a #Linear Harmonic Oscillator & Its Solution by Polynomial Method
Zero Point Energy of a Particle Possessing Harmonic Motion and its Consequence
#Schrodinger #Wave #Equation #for #Three #Dimensional #Rigid #Rotator
#Energy #of #Rigid #Rotator
#Space #Quantization
#Schrodinger #Wave #Equation #for #Hydrogen #Atom, #Separation #of #Variable #in #Polar #Spherical #Coordinates #and #its #Solution
#Principal, #Azimuthal #and #Magnetic #Quantum #Numbers #and #the #Magnitude #of #Their #Values
#Probability #Distribution #Function
#Radial #Distribution #Function
#Shape #of #Atomic #Orbitals (s,p & d)#Thermodynamics – II
#Clausius-Clapeyron #Equation
#Law #of #Mass #Action #and #Its #Thermodynamic #Derivation
#Third #Law #of #Thermodynamics (#Nernest #Heat #Theorem, #Determination of #Absolute #Entropy, #Unattainability #of #Absolute #Zero)
 #Electrochemistry – II: Ion Transport in Solutions
#Ionic #Movement Under the Influence of an Electric Field
#Mobility #of #Ions
#Ionic #Drift #Velocity #and #Its #Relation #with #Current #Density
#Einstein #Relation #Between #the #Absolute #Mobility #and #Diffusion #Coefficient
#The #Stokes-Einstein #Relation
#The #Nernst-Einstein #Equation
#Walden’s Rule
#The #Rate-Process #Approach #to #Ionic #Migration
#The #Rate #Process #Equation #for #Equivalent #Conductivity
#Force #Nernst-Planck #Flux #Equation
#Ionic #Drift #and #Diffusion #Potential
#The #Onsager #Phenomenological #Equations
#The #Basic #Equation #for #the #Diffusion
#Planck-Henderson #Equation #for #the #Diffusion #Potential