Quantum Mechanics & Spectroscopy Review

A comprehensive set of Q&A flashcards covering foundational quantum mechanics, spectroscopy, atomic structure, and statistical thermodynamics concepts drawn from the lecture notes.

What is blackbody radiation?

Electromagnetic energy emitted by an ideal object that absorbs all radiation and emits energy solely as a function of its temperature.

Which classical crisis did blackbody radiation create and why?

The Ultraviolet Catastrophe, because classical physics predicted an infinite energy density at very short wavelengths.

How did Planck resolve the Ultraviolet Catastrophe?

By proposing that the energy of each oscillator is quantized, occurring only in discrete packets (E = nhν).

State de Broglie’s wavelength formula.

λ = h / (mv), where h is Planck’s constant, m is particle mass, and v is velocity.

Why do macroscopic objects show negligible wave behavior?

Their large mass makes the de Broglie wavelength extremely small compared to observable scales.

What key result is illustrated by the double-slit experiment with electrons?

Particles exhibit wave-particle duality, producing an interference pattern when not observed but behaving like particles when detected.

Calculate the velocity of an electron with λ = 10 nm (given in notes).

v ≈ 7.27 × 10⁴ m s⁻¹.

What symbol denotes the wavefunction in quantum mechanics?

Ψ (psi).

Is the wavefunction a real physical wave?

No; it is a mathematical function whose modulus squared gives probability.

List the four basic acceptability criteria for a wavefunction.

Continuous, single-valued & finite, square-integrable, and normalizable.

Define a Hermitian operator.

An operator whose eigenvalues are real and represent measurable physical quantities.

What is implied if two operators commute (AB = BA)?

They share a common set of eigenfunctions and their observables can be simultaneously measured with exact values.

Write the eigenvalue equation for an operator  acting on Ψ.

ÂΨ = aΨ, where a is the eigenvalue.

How is the expectation value ⟨Â⟩ of an observable calculated?

⟨Â⟩ = ∫Ψ* Â Ψ dτ.

Provide the time-dependent Schrödinger equation.

ĤΨ(x,t) = iħ ∂Ψ/∂t.

What does the Pauli Exclusion Principle state for fermions?

No two identical fermions can occupy the same quantum state simultaneously; the total wavefunction must be antisymmetric.

Give the Born probability interpretation.

|Ψ|² = Ψ*Ψ gives the probability density of finding a particle at a point.

What integral condition normalizes a 1-D wavefunction between a and b?

∫ₐᵇ Ψ*Ψ dx = 1.

Define orthogonality for two wavefunctions Ψi and Ψj.

∫Ψi* Ψj dτ = 0.

State the Heisenberg Uncertainty Principle in its common form.

Δx Δp ≥ ħ/2.

Describe a free particle in quantum mechanics.

Has zero potential energy; energy is purely kinetic and not quantized.

Why are energies in the 1-D particle-in-a-box quantized?

Boundary conditions (Ψ = 0 at walls) allow only standing-wave solutions with discrete wavelengths and energies.

What physical model does the rigid rotor represent?

Rotation of a molecule with fixed bond length and constant moment of inertia.

Why are harmonic oscillator energy levels equally spaced?

The quadratic potential leads to solutions with energy Eₙ = (n + ½)ħω.

Write the 1-D time-independent Schrödinger equation.

−(ħ²/2m)(d²Ψ/dx²) + V(x)Ψ = EΨ.

For a particle in a box of length L, give the allowed energies.

Eₙ = n²h² / (8mL²), n = 1,2,3…

What is the mean position ⟨x⟩ for a symmetric particle-in-a-box state?

⟨x⟩ = L/2.

Which motions of molecules are usually NOT observed in spectroscopy?

Translational motions.

Which spectroscopy probes rotational transitions?

Microwave spectroscopy.

What molecular property must change for IR absorption to occur?

Dipole moment.

What property must change for Raman activity?

Polarizability.

Give one example of a molecule IR active but Raman inactive in CO₂.

The asymmetric stretching mode is IR active but Raman inactive.

Why are all vibrational modes of H₂O IR active?

Because water has polar O–H bonds whose dipole moment changes during vibration.

Who discovered Raman spectroscopy and won a Nobel Prize in 1930?

C. V. Raman.

What is a hydrogenic atom?

An atom or ion with only one electron (e.g., He⁺, Li²⁺).

State the Rydberg equation for hydrogen spectral lines.

1/λ = R_H (1/n₁² − 1/n₂²).

What is the Bohr radius (a₀) expression?

a₀ = 4πɛ₀ħ² / (me e²).

Summarize the Born–Oppenheimer approximation.

Electron motion is solved assuming nuclei are fixed because nuclei are much heavier and move more slowly.

What extra term appears in the Hamiltonian of many-electron atoms but not one-electron atoms?

Electron–electron repulsion term e²/(4πɛ₀ r_ij).

Define a macrostate in statistical thermodynamics.

A set of macroscopic properties (e.g., P, V, T) describing the system.

What is a microstate?

A specific configuration giving position and momentum of every particle in the system.

Write the expression for the canonical partition function q.

q = Σi e^(−βEi), where β = 1/(k_B T).

Give the Boltzmann distribution for probability P_i.

Pi = e^(−βEi) / q.

At 300 K, which energy level (0 eV, 0.1 eV, 0.2 eV) is most probable for the sample three-level system?

E₀ = 0 eV, because lower energy states have higher Boltzmann factors.

What is the computed partition function for the three-level particle at 300 K (given)?

q ≈ 1.0212.

What is the probability of finding that particle in E₁ (0.1 eV) at 300 K?

P₁ ≈ 0.0204 (≈ 2.0 %).

Why are energies of a free particle continuous, unlike a particle in a box?

Because no boundary conditions restrict the wavelength.

Give the energy formula for a rigid rotor.

E_J = (ħ² / 2I) J(J + 1), J = 0,1,2…

What is the expectation value ⟨x⟩ for a harmonic oscillator?

Zero; the average position over time is at the equilibrium point.

What does square-integrable mean for Ψ?

∫|Ψ|² dτ is finite, ensuring a finite total probability.

How is an acceptable wavefunction affected by an infinite potential wall?

It must vanish (Ψ = 0) at the wall.

Explain commute relation and simultaneous observables with an example.

Position and momentum do not commute ([x, p] ≠ 0), so they cannot be simultaneously known precisely.

For the operator d²/dx², what is the eigenvalue of sin(kx)?

−k².

Is cos(3x + 5) an eigenfunction of d²/dx²? If so, eigenvalue?

Yes; eigenvalue −9.

Why does the free particle have equal probability everywhere?

Because |Ψ|² is constant for plane-wave solutions.

Give one application of the particle-in-a-box model.

Quantum wells or electrons confined in nanoscale structures.

State one application of the rigid rotor model.

Predicting rotational spectra of diatomic molecules.

What is the Helmholtz free energy relation to the partition function?

F = −k_BT ln q.

What are Stokes and anti-Stokes lines?

Frequency shifts in Raman spectra corresponding to energy lost or gained by photons during scattering.

Which molecules are Raman active but IR inactive among homonuclear diatomics?

H₂, N₂, O₂.

What distinguishes macrostates of ice versus water vapor?

Different aggregate structures and energy distributions, though each macrostate comprises many microstates.