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:
      <br>−rach22m</p></li></ul><h4id="5987eba5−7610−4bf3−930a−ab401bef1d00"data−toc−id="5987eba5−7610−4bf3−930a−ab401bef1d00"collapsed="true"seolevelmigrated="true">−</h4><h4id="e8c096f7−e29d−444d−9b82−c3e6bc92a864"data−toc−id="e8c096f7−e29d−444d−9b82−c3e6bc92a864"collapsed="false"seolevelmigrated="true">−</h4><h4id="9fc037c7−193a−44b6−86ab−acdd9c13782a"data−toc−id="9fc037c7−193a−44b6−86ab−acdd9c13782a"collapsed="false"seolevelmigrated="true">−</h4><h4id="9a1a43e6−1090−4c6a−a1bd−65b85c82c95a"data−toc−id="9a1a43e6−1090−4c6a−a1bd−65b85c82c95a"collapsed="false"seolevelmigrated="true">−</h4><p>−</p><ul><li><p>kEx=V+<br><br>- rac{h^2}{2m}</p></li></ul><h4 id="5987eba5-7610-4bf3-930a-ab401bef1d00" data-toc-id="5987eba5-7610-4bf3-930a-ab401bef1d00" collapsed="true" seolevelmigrated="true">-</h4><h4 id="e8c096f7-e29d-444d-9b82-c3e6bc92a864" data-toc-id="e8c096f7-e29d-444d-9b82-c3e6bc92a864" collapsed="false" seolevelmigrated="true">-</h4><h4 id="9fc037c7-193a-44b6-86ab-acdd9c13782a" data-toc-id="9fc037c7-193a-44b6-86ab-acdd9c13782a" collapsed="false" seolevelmigrated="true">-</h4><h4 id="9a1a43e6-1090-4c6a-a1bd-65b85c82c95a" data-toc-id="9a1a43e6-1090-4c6a-a1bd-65b85c82c95a" collapsed="false" seolevelmigrated="true">-</h4><p>-</p><ul><li><p>kE x = V +<br>

    • Transformation to Spherical Coordinates:

    • Variables transformation:

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

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

      • z=rimesextcos(heta)z = r imes ext{cos}( heta)

      • r=extsqrt(x2+y2+z2)r = ext{sqrt}(x^2 + y^2 + z^2)

      • extcos(heta)=raczextsqrt(x2+y2+z2)ext{cos}( heta) = rac{z}{ ext{sqrt}(x^2 + y^2 + z^2)}

      • exttan(extφ)=racyxext{tan}( ext{φ}) = rac{y}{x}

SCHRÖDINGER EQUATION

  • Operator for Momentum:

    • extOperator=−iexth<br>ext{Operator} = -i ext{h}<br>

  • Normalization Condition:

    • The integral must hold:
      <br>extZ10extd3r=1<br><br>ext{Z}_{1}^{0} ext{d}^3r = 1<br>

  • Wave Function Separation:

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

VOLUME ELEMENT

  • The volume element in spherical coordinates is:

    • extdV=r2extsin(θ)extdθextdφextdrext{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:

    • racext∂ext∂rR(r)Y(θ,φ)=racext∂ext∂θext∂ext∂φR(r)Y(θ,φ)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:
      extEψ(r,θ,φ)=−racexth22m∇2extψ(r,θ,φ)−race24πε0rextψ(r,θ,φ)<br>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)<br>

SCHRÖDINGER EQUATION FOR HYDROGEN ATOM

  • The format is:

    • −rach22m(rac1rracext∂2(rextψ(r,θ,φ))ext∂r2+rac1r2Λ2(θ,φ)extψ(r,θ,φ))+race24πε0rextψ(r,θ,φ)=Eextψ(r,θ,φ)- rac{h^2}{2m} \bigg( 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)\bigg) + 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:

    • extR(r)Y(θ,φ)ext{R}(r) Y(θ, φ)

  • Del Operator in Spherical Coordinates:
    <br>∇2=rac1rracext∂2ext∂r2+rac1r2Λ2<br><br>∇^2 = rac{1}{r} rac{ ext{∂}^2}{ ext{∂}r^2} + rac{1}{r^2}Λ^2 <br>

ANGULAR EQUATION OF SCHRÖDINGER EQUATION FOR HYDROGEN ATOM

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

    • racΛ2(θ,φ)Y(θ,φ)=−l(l+1)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:

    • −rach22mrracextd2(rR(r))dr2+…- 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:

    • extΘ(θ)=1extsin2(θ)racextd2extdφ2Φ(φ)+rac1extsinθracextdextdθextsinθracextdextdθΘ(θ)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,…n = 1, 2, 3, …

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

  • Magnetic Quantum Number: ml=−l,−(l−1),…,0,…,(l−1),lm_l = -l, -(l - 1), …, 0, …, (l - 1), l

  • Electron Spin Quantum Number: ms=±rac12m_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:

    • riangleE=−hcR<em>H(rac1n</em>22−rac1n12)riangle E = -hcR<em>H \bigg( rac{1}{n</em>2^2} - rac{1}{n_1^2} \bigg)

IONIZATION ENERGY

  • Ionization energy for Hydrogen: 1,312extkJ/mol1,312 ext{kJ/mol}

  • Ionization energy for Helium: 2,372extkJ/mol2,372 ext{kJ/mol}

  • Effective nuclear charge:

    • Z<em>e</em>eff=racI<em>atomI</em>HZ<em>e</em>{eff} = rac{I<em>{atom}}{I</em>H}

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.