Hydrogen Study Guide

HYDROGEN AND SCHRÖDINGER EQUATION

General Information

• Course: BCH 463

• Semester: Spring 2026

SPHERICAL COORDINATES

• Schrödinger Equation in Spherical Coordinates:

  • The equation can be expressed as:
    
    - rac{h^2}{2m}

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  • kE x = V +
    

  • Transformation to Spherical Coordinates:

  • Variables transformation:

    • x = r imes ext{sin}( heta) imes ext{cos}( ext{φ})

    • y = r imes ext{sin}( heta) imes ext{sin}( ext{φ})

    • z = r imes ext{cos}( heta)

    • r = ext{sqrt}(x^2 + y^2 + z^2)

    • ext{cos}( heta) = rac{z}{ ext{sqrt}(x^2 + y^2 + z^2)}

    • ext{tan}( ext{φ}) = rac{y}{x}

SCHRÖDINGER EQUATION

• Operator for Momentum:

  • ext{Operator} = -i ext{h}
    

• Normalization Condition:

  • The integral must hold:
    
    ext{Z}_{1}^{0} ext{d}^3r = 1
    

• Wave Function Separation:

  • In spherically symmetric cases, use:
    
    ext{ψ}(r, θ, φ) = R(r)Y(θ, φ)
    

VOLUME ELEMENT

• The volume element in spherical coordinates is:

  • ext{d}V = r^2 ext{sin}(θ) ext{d}θ ext{d}φ ext{d}r

PARTIAL DIFFERENTIAL EQUATIONS

• A separable format can be utilized:

  • rac{ ext{∂}}{ ext{∂}r} R(r)Y(θ, φ) = rac{ ext{∂}}{ ext{∂}θ} ext{∂} ext{∂}φ R(r)Y(θ, φ)

  • Equation in 3D physicist notation is of use:
    ext{E ψ}(r, θ, φ) = - rac{ ext{h}^2}{2m} ∇^2 ext{ψ}(r, θ, φ) - rac{e^2}{4πε_0 r} ext{ψ}(r, θ, φ)
    

SCHRÖDINGER EQUATION FOR HYDROGEN ATOM

• The format is:

  • - rac{h^2}{2m} igg( rac{1}{r} rac{ ext{∂}^2(r ext{ψ}(r, θ, φ))}{ ext{∂}r^2} + rac{1}{r^2} Λ^2(θ, φ) ext{ψ}(r, θ, φ)igg) + rac{e^2}{4πε_0 r} ext{ψ}(r, θ, φ) = E ext{ψ}(r, θ, φ)

• Component separation leads to:

  • Radial and angular components separated giving:

  • ext{R}(r) Y(θ, φ)

• Del Operator in Spherical Coordinates:
  
  ∇^2 = rac{1}{r} rac{ ext{∂}^2}{ ext{∂}r^2} + rac{1}{r^2}Λ^2
  

ANGULAR EQUATION OF SCHRÖDINGER EQUATION FOR HYDROGEN ATOM

• Angular momentum can be described, with the following equations involved:

  • rac{Λ^2(θ, φ)}{Y(θ, φ)} = -l(l + 1)

• Provides separate equations that depend on angular elements.

RADIOAL COMPONENTS

• Radial equation can be simplified outputting:

  • - rac{h^2}{2m r} rac{ ext{d}^2(rR(r))}{dr^2} + …

  • Rearrange terms coherent to energy level equations.

• Both angular and radial equations exist:

  • ext{Θ}(θ) = 1 ext{sin}^2(θ) rac{ ext{d}^2}{ ext{d}φ^2} Φ(φ) + rac{1}{ ext{sin}θ} rac{ ext{d}}{ ext{d}θ} ext{sin}θ rac{ ext{d}}{ ext{d}θ} Θ(θ)

QUANTUM NUMBERS

• Principal Quantum Number: n = 1, 2, 3, …

• Angular Momentum Quantum Number: l = 0, 1, 2, …, (n - 1)

• Magnetic Quantum Number: m_l = -l, -(l - 1), …, 0, …, (l - 1), l

• Electron Spin Quantum Number: m_s = ± rac{1}{2}

ELECTRONS IN MULTI-ELECTRON ATOMS

• The Pauli Exclusion Principle applies:

  • No two electrons have the same set of quantum numbers in the same atom.

ENERGY TRANSITIONS

• Energy transitions formula:

  • riangle E = -hcRH igg( rac{1}{n2^2} - rac{1}{n_1^2} igg)

IONIZATION ENERGY

• Ionization energy for Hydrogen: 1,312 ext{kJ/mol}

• Ionization energy for Helium: 2,372 ext{kJ/mol}

• Effective nuclear charge:

  • Ze{eff} = rac{I{atom}}{IH}

SHIELDING EFFECT

• The shielding effects influence energies experienced by electrons significantly , providing:

  • Differences for electron potentials highly observed.

ELECTRON CONFIGURATION RULES

• Building-up Principle: Fill lower energy levels first.

• Hund's Rule: Maximize spin alignment in degenerate orbitals when possible.

PAULI EXCLUSION PRINCIPLE

• Understood via antisymmetric wavefunction considerations and indistinguishable particle behavior.