Breakdown & Explanation of Key Equations in Quantum Mechanics and Spectroscopy- CH 7-9

E = hν

Used for: Calculating the energy of a photon. Scenario: Given frequency, find energy (or vice versa). Question Example: 'What is the energy of light with a frequency of 6.0 × 10^14 Hz?'

ν = λ/c

Used for: Converting between wavelength and frequency. Scenario: You're given wavelength (in meters, nm, etc.) and asked to find frequency. Question Example: 'Find the frequency of light with a wavelength of 500 nm.'

ν~ = 1/λ

Used for: Calculating wavenumber (used in IR spectroscopy). Scenario: IR or Raman spectroscopy problems. Question Example: 'Convert a wavelength of 2.5 µm into wavenumber (in cm^-1).'

λ = h/mv

Used for: Calculating de Broglie wavelength of a particle. Scenario: Quantum mechanics; electron diffraction. Question Example: 'What is the wavelength of an electron moving at 2.0 × 10^6 m/s?'

E = khν - ϕ

Used for: Photoelectric effect (ejection of electrons from metal surfaces). Scenario: Light hits a metal, calculate the kinetic energy of ejected electrons. Question Example: 'Find the kinetic energy of an electron if light of 4.0 eV hits a surface with a 2.5 eV work function.'

ψ = H^Eψ

Used for: Solving quantum systems (Schrödinger equation). Scenario: Conceptual questions or when asked to describe the energy levels or wavefunction of a quantum system. Question Example: 'Write the time-independent Schrödinger equation for a particle in a box.'

ΔxΔp ≥ 2ℏ

Used for: Heisenberg Uncertainty Principle. Scenario: Finding uncertainty in position or momentum. Question Example: 'If uncertainty in position is 1 × 10^-10 m, what is the minimum uncertainty in momentum?'

E = n^2h^2/(8mL^2)

Used for: Energy levels in a 1D particle in a box. Scenario: Quantum particle confined in a box of length L; solve for energy. Question Example: 'Calculate the energy of a particle in the n=2 state in a 1 nm wide box.'

E = mℏ^2l(l+1)/(2I)

Used for: Rotational energy of molecules (Rigid Rotor model). Scenario: Diatomic molecules rotating in space. Question Example: 'Calculate the rotational energy of a molecule with moment of inertia I.'

E = (vHO)v + hν(1/2)

Used for: Vibrational energy of a harmonic oscillator (molecular vibration). Scenario: IR spectroscopy or vibrational quantization. Question Example: 'What is the energy of the first excited vibrational state of a diatomic molecule?'

ν = (1/2π)√(k/m)

Used for: Finding vibrational frequency of a bond. Scenario: Harmonic oscillator model; force constant of bond. Question Example: 'Given force constant and reduced mass, find the vibrational frequency.'

E = -hcR_H/n^2

Used for: Hydrogen atom energy levels. Scenario: Calculate energy of an electron at quantum level n in hydrogen. Question Example: 'What is the energy of the n=3 level of a hydrogen atom?'

L = l + 1, l + 2, ..., |l - 1|

Used for: Combining orbital angular momentum quantum numbers. Scenario: Determining possible values of total orbital angular momentum in multi-electron atoms. Question Example: 'What are the possible values of L for l=1 and l=2?'

S = s + 1, s + 2, ..., |s - 1|

Used for: Combining spin angular momentum. Scenario: Finding total spin of multi-electron systems. Question Example: 'What are the possible values of S if two electrons are paired/unpaired?'

J = L + S, L + S - 1, ..., |L - S|

Used for: Total angular momentum (vector combination of orbital and spin). Scenario: Used in term symbols for atomic spectroscopy. Question Example: 'If L=1 and S=1, what are the possible values of J?'

b = (N - 2)/N * (N)

Used for: Electron pairing in electron configuration (Hund's Rule context). Scenario: Determining how many electron pairs exist in a configuration. Question Example: 'For a d⁶ configuration, calculate the number of electron pairs.'