The hydrogen atom, with its single proton and electron, is the simplest atomic system. Its energy levels, determined by the Schrödinger equation, form a discrete spectrum characterized by the principal quantum number n. This fundamental model lays the groundwork for understanding more complex atoms.

Quantum mechanics provides a more comprehensive description of the hydrogen atom than the Bohr model. It introduces wavefunctions and probability distributions, explaining features like angular momentum states and spectral line intensities. This approach forms the basis for understanding atomic structure and chemical bonding.

Hydrogen Atom Energy Levels

Schrödinger Equation for Hydrogen

• Three-dimensional partial differential equation incorporates Coulomb potential between proton and electron
• Separation of variables technique transforms problem into three one-dimensional equations
  • Radial equation
  • Polar equation
  • Azimuthal equation
• Radial equation solved using series solution method yields radial wave functions and energy eigenvalues
• Angular part of solution involves spherical harmonics (eigenfunctions of angular momentum operators)
• Complete solution results in wavefunctions characterized by three quantum numbers
  • n (principal)
  • l (angular momentum)
  • m (magnetic)

Energy Levels and Degeneracy

• Energy levels given by formula $E_n = -13.6 \text{ eV} / n^2$
  • n represents principal quantum number
• Degeneracy arises due to spherical symmetry of Coulomb potential
  • Multiple states with same energy for given principal quantum number
• Energy levels form discrete spectrum (Lyman series, Balmer series)
• Increasing n leads to closer spacing between energy levels
• Ground state (n=1) has lowest energy of -13.6 eV
• Excited states (n>1) have higher energies and larger spatial extent

Principal Quantum Number Significance

Energy and Spatial Distribution

• Primary determinant of electron's energy in hydrogen atom
• Larger n corresponds to higher energy states and more spread-out wavefunction
• Average distance from nucleus increases with n
• Influences number of nodes in radial wavefunction
  • Number of radial nodes = n - l - 1
• Sets range of possible values for angular momentum quantum number l
  • l can take values from 0 to n-1

Spectroscopic and Atomic Structure

• Directly related to concept of electron shells in multi-electron atoms
• Plays crucial role in periodic table organization (principal energy levels)
• Spectroscopic transitions between energy levels with different n values give rise to observed spectral lines
  • Lyman series (ultraviolet)
  • Balmer series (visible)
  • Paschen series (infrared)
• Inverse square dependence of energy on n explains
  • Convergence of spectral lines to series limit
  • Existence of ionization energy (energy required to remove electron from atom)

Bohr Model vs Quantum Mechanics

Similarities and Differences

• Both models predict identical energy levels for hydrogen atom
• Bohr model describes electrons in circular orbits
• Quantum mechanics replaces orbits with probability distributions defined by wavefunctions
• Both retain concept of quantized energy levels
• Quantum mechanics provides more comprehensive explanation for energy level origin
• Bohr model's allowed orbital radii correspond to average radii of electron probability distributions in quantum mechanical description for states with l = n - 1
• Both predict ground state energy of hydrogen atom as -13.6 eV
  • Quantum mechanics provides more rigorous derivation

Advantages of Quantum Mechanical Approach

• Explains features Bohr model cannot, such as existence of states with different angular momenta for given energy level
• Can be extended to multi-electron atoms and molecules (with increasing complexity)
• Provides deeper understanding of atomic structure and behavior
• Accurately predicts spectral line intensities and selection rules
• Explains phenomena like quantum tunneling and zero-point energy
• Forms basis for understanding chemical bonding and molecular structure