Mind Network - Samuel Solomon

Schrödinger Equation

Spherical Symmetric Potential
          It is best to begin our discussion of solving quantum mechanical problems in spherical coordinates with (one of) the easiest potentials out there: spherically symmetric potentials. Spherically symmetric potentials are potentials that have no angular variations: V(r, theta, phi) => V(r).
          Before we begin our analysis, let us quickly remind ourselves of our spherical coordinates definition (some books vary). We define our spherical coordinates as follows:

Spherical Coordinates
          It is important to note that even among scientists, people tend to define these coordinates differently. But as long as we are internally consistent, it is okay. To be clear, my coordinates are defined as follows:
       Theta: The angle the point makes with the z axis
       Phi: The angle the point makes with the x-axis in the x-y plane
       r: the distance between the point and the origin (Some call it 'Rho')

We are now ready to work with our Schrödinger equation (in spherical coordinates) below (see Schrödinger Equation (Spherical Coordinates) page for a recap):

Hamiltonian: Spherical Coordinates
          Altogether, this equation looks daunting, but we can still apply our standing wave approximation to the spherical form of the Schrödinger equation. Since we are working with a spherically symmetric potential, we will only separate out the radial solution from the angular component. We can write our ansatz out explicitly below:
Seperation of Variables Ansatz
          Please note that I am using the notation 'R(r)' and 'Y(theta, phi)' for the radial and angular solution respectively. This is just to match the most common notation used for spherically symmetric potentials online.
          We can now plug in our ansatz back into the Schrödinger equation:

Schrödinger Equation (Spherical Coordinates): Separation of Variables
In order to make sure everyone is following, let us review some key steps below:
       4: Schrödinger equation with the spherical Laplacian (purple), spherically symmetric potential, and ansatz (equation 3)
       4 to 5: Move V(r) and the constant in front of the Laplacian to the LHS. Distribute 'R' and 'Y' throughout the Laplacian
       5 to 6: Multiply both sides by 'r^2' and divide both sides by 'R(r) * Y(theta, phi)'
       6 to 7: Bring the black term with the 'V(r) - E' to the RHS, so that the equation equals zero (a constant)

          We can immediately notice in equation 7 that the dark red terms only depends on the variable 'r' and the pink terms only depend on the angular components 'theta' and 'phi.' This means that if I move the radius of the particle, but not the angle, the whole dark red term CANNOT change (it is a constant). If I move the angle of the particle, but not the radius, the whole pink term CANNOT change (the dark red term will not change and zero is just a constant). Hence, both the pink and the red term are CONSTANTS. To solve any spherically symmetric potential, all you have to do is solve the dark red term (the radial component) and the pink term (the angular component). We can explicitly right this below:

Schrödinger Equation: Radial and Angular Component
          Equation 9 may look a bit familiar. In fact, it has a hidden L^2 operator inside. We can pull it out below:
Schrödinger Equation: Angular Component
          This leads us to our full radial and angular equations below:
Schrödinger Equation: Radial and Angular Component
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  • Quantum Mechanics I
    • Introduction to Waves (The Wave Equation)
    • Introduction to Waves (The Wave Function)
    • Motivation for Quantum Mechanics (Photoelectric effect)
    • Motivation for Quantum Mechanics (Compton Scattering)
    • Motivation for Quantum Mechanics (Black Body Radiation)
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    • Introduction to Commutation
    • Expectation Values of Operators
    • Quantum Uncertainty (Defining Uncertainty)
    • Quantum Uncertainty (Heisenberg's Uncertainty Principle)
    • The Schrödinger Equation (The "Derivation")
    • The Schrödinger Equation (How to use it)
    • No Degeneracy in 1-Dimension
    • Parity Operator
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    • Free Particle (In a Vacuum)
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    • Quantum Tunneling (Alpha Decay Example)
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    • Quantum Harmonic Oscillator (Brute Force Solution)
    • Quantum Harmonic Oscillator (Ladder Operators)
    • Quantum Harmonic Oscillator (Expectation Values)
    • Bringing Quantum to 3D (Cartesian Coordinates)
    • Free Particle
    • Infinite Cubic Well (3D Particle in a Box)
    • Quantum Harmonic Oscillator
    • Schrödinger Equation (Spherical Coordinates)
    • Angular Momentum (Experiments)
    • Angular Momentum (Operators)
    • Angular Momentum (Ladder Operators)
    • Schrödinger Equation (Spherical Symmetric Potential)
    • Infinite Spherical Well (Radial Solution)
    • One Electron Atom (Radial Solution for S-orbital)
    • Hydrogen Atom (Angular Solution; Spherically Symmetric)
    • Hydrogen Atom (Radial Solution; Any Orbital)
    • Hydrogen atom (Recap)
  • Quantum Mechanics II
  • Nuclear Fusion
    • Introduction to Fission (Energy Extraction)
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    • Choosing Fusion Reactants
  • Special Relativity
    • Terminology and Notation
    • Galilean Transformation
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  • Home
  • Quantum Mechanics I
    • Introduction to Waves (The Wave Equation)
    • Introduction to Waves (The Wave Function)
    • Motivation for Quantum Mechanics (Photoelectric effect)
    • Motivation for Quantum Mechanics (Compton Scattering)
    • Motivation for Quantum Mechanics (Black Body Radiation)
    • Bohr Model of the Atom
    • Wave-Particle Duality (The Wave Function Motivation)
    • Problems with the Wave Function
    • Introduction to Quantum Operators (The Formalism)
    • Introduction to Dirac Notation
    • Introduction to Quantum Operators (The Hermitian and the Adjoint)
    • Introduction to Commutation
    • Expectation Values of Operators
    • Quantum Uncertainty (Defining Uncertainty)
    • Quantum Uncertainty (Heisenberg's Uncertainty Principle)
    • The Schrödinger Equation (The "Derivation")
    • The Schrödinger Equation (How to use it)
    • No Degeneracy in 1-Dimension
    • Parity Operator
    • Quantum Mechanics' Core Postulates
    • Free Particle (In a Vacuum)
    • Particle in a Box (Infinite Square Well)
    • Bound States (The Mathematical Setup)
    • Bound States (The Shooting Method)
    • Bound States (Patching Solutions Together)
    • Patching Solutions (Finite, Infinite, and Delta Function Potentials)
    • Delta Function Potential Well
    • Scatter States (A Touch on Dispersion)
    • Scatter States (Reflection, Transmission, Probability Current)
    • Scatter States (Worked Example)
    • Scatter States (Elastic Collision)
    • Quantum Tunneling (Constant Potential)
    • Quantum Tunneling (Changing Potential)
    • Quantum Tunneling (Alpha Decay Example)
    • Quantum Harmonic Oscillator (Classical Mechanics Analogue)
    • Quantum Harmonic Oscillator (Brute Force Solution)
    • Quantum Harmonic Oscillator (Ladder Operators)
    • Quantum Harmonic Oscillator (Expectation Values)
    • Bringing Quantum to 3D (Cartesian Coordinates)
    • Free Particle
    • Infinite Cubic Well (3D Particle in a Box)
    • Quantum Harmonic Oscillator
    • Schrödinger Equation (Spherical Coordinates)
    • Angular Momentum (Experiments)
    • Angular Momentum (Operators)
    • Angular Momentum (Ladder Operators)
    • Schrödinger Equation (Spherical Symmetric Potential)
    • Infinite Spherical Well (Radial Solution)
    • One Electron Atom (Radial Solution for S-orbital)
    • Hydrogen Atom (Angular Solution; Spherically Symmetric)
    • Hydrogen Atom (Radial Solution; Any Orbital)
    • Hydrogen atom (Recap)
  • Quantum Mechanics II
  • Nuclear Fusion
    • Introduction to Fission (Energy Extraction)
    • Introduction to Fusion (Applications and Challenges)
    • Choosing Fusion Reactants
  • Special Relativity
    • Terminology and Notation
    • Galilean Transformation
  • Statistical Thermodynamics
  • Chemical Thermodynamics
  • Ionization Radiation
  • Multivariable Calculus
    • Vectors
    • Dot Product
    • Cross Product
    • Rotating Vectors
    • Level Curves
    • Gradients
    • Directional Derivatives
  • Differential Equations
  • Contact