Quantum Mechanics: Energies and Dynamics of Microscopic Systems

Quantized energy levels

It means the particle can only have specific, discrete energy values—no in-between values.

Quantum number n in the particle-in-a-box model

Because that would imply zero energy and a zero wavefunction everywhere, which violates the uncertainty principle.

Energy quantization in the 1D box model

The boundary conditions that require the wavefunction to be zero at the walls of the box.

Energy spacing as n increases in the 1D box model

The spacing between energy levels increases as n2.

Equation for energy levels of a particle in a 1D box

En = 8nm2hL22

Particle's energy in the box

Due to the Heisenberg uncertainty principle—confined particles must have some kinetic energy.

Wave type in the 1D box model

A standing wave with nodes at the walls.

Significance of wavefunction ψ(x) in quantum mechanics

It describes the amplitude of the particle's probability distribution in space.

Physical property of ψ(x)²

The probability density of finding the particle at position x.

Number of nodes in the wavefunction as n increases

The number of nodes increases with n, indicating higher energy states.

de Broglie equation λ = hp

That particles like electrons have wave properties, including a wavelength.

Effect of increasing box length L on energy levels

It decreases the energy level values—larger L means smaller energy spacing.

Antinodes in n = 3 state wavefunction

There will be 3 half-waves, so 2 internal nodes and 3 antinodes.

Mass effect on energy levels in a 1D box

Larger mass m results in lower energy levels, since En.

Physical situation simulated by the 1D box model

A particle confined between two perfectly rigid, impenetrable walls (e.g., electron in a quantum dot).

Principle preventing a particle in a box from being completely still

The uncertainty principle.

Energy levels in a 1D box model spacing

Because energy is proportional to n2, so the spacing grows with n.

Particle existence outside box boundaries in the 1D box model

No, the particle's probability of being outside the box is zero due to infinite potential walls.

Effect of increasing n on λ (wavelength)

λ decreases; higher energy states correspond to shorter wavelengths.

Zero-point energy implication about quantum particles

They always retain some motion/energy even in the lowest energy state—never at rest.

Operator in quantum mechanics

A mathematical entity that acts on a wavefunction to extract information about a physical observable (e.g., position, energy, momentum).

Eigenfunction

A wavefunction that, when operated on by a quantum mechanical operator, returns a constant (eigenvalue) times the original function.

Eigenvalue

The measurable value obtained when an observable is measured and the system is in an eigenfunction of the corresponding operator.

Equation Ôψ = aψ

The operator Ô acting on the wavefunction ψ yields the eigenvalue a times ψ, indicating ψ is an eigenfunction of Ô.

Physical interpretation of an eigenfunction

A system in an eigenfunction of an operator will always yield the same measurement (the eigenvalue) for that observable.

Expectation value of an observable

The average value obtained from many identical measurements of a quantum system in the same state.

Mathematical definition of expectation value

It is given by ⟨Ô⟩ = ∫ψ(x) Ô ψ(x) dx, where ψ is the complex conjugate of ψ.

|ψ(x)|²

The probability density of finding a particle at position x.

Normalization condition ∫|ψ(x)|² dx = 1

It ensures the total probability of finding the particle somewhere in space is 100%.

Hamiltonian operator

The total energy operator of the system, consisting of kinetic and potential energy components.

Time-independent Schrödinger equation

Ĥψ = Eψ, where Ĥ is the Hamiltonian operator, ψ is the wavefunction, and E is the total energy.

Kinetic energy operator in 1D

As T̂ = −(ħ² / 2m) d²/dx², where ħ is the reduced Planck's constant and m is the mass.

Momentum operator in one dimension

p̂ = −iħ (d/dx), showing momentum is related to the wavefunction's rate of change.

Hermitian operator

An operator whose eigenvalues are always real and whose eigenfunctions are orthogonal, used to represent observables.

Observables and Hermitian operators

Because they ensure that measurement results (eigenvalues) are real and physically meaningful.

Superposition principle in quantum mechanics

A general wavefunction can be written as a sum of eigenfunctions of an operator, each with a complex coefficient.

Quantum measurement

The wavefunction collapses into one of the eigenfunctions of the measured observable's operator.

Probability of obtaining a specific eigenvalue

The square of the magnitude of the coefficient (|c|²) associated with that eigenfunction.

Commutator

The commutator [A, B] = AB − BA measures whether two operators are compatible; if non-zero, they can't be simultaneously measured precisely.

Measurement in quantum mechanics

Because unless the system is in an eigenstate of the observable, measurements yield a distribution of possible outcomes.

Energy levels in a particle-in-a-box

Boundary conditions force wavefunctions to fit as standing waves, allowing only specific energy levels.

Lowest energy level in a box

The wavefunction must have at least one half-wavelength to satisfy boundary conditions, resulting in zero-point energy.

Zero-energy state in quantum mechanics

A zero-energy state would require a flat wavefunction, which violates the boundary condition ψ = 0 at the walls.

Spacing between energy levels as n increases

The spacing increases because energy is proportional to n², not linear.

Increasing box length L

A larger box reduces the curvature of wavefunctions, lowering their kinetic energy.

Number of nodes in a 1D box wavefunction

The quantum number n; there are n−1 nodes for a given state.

Probability of finding a particle at a node

The wavefunction ψ(x) equals zero at nodes, so ψ(x)² = 0 at those positions.

Higher energy states and nodes

Higher quantum numbers result in wavefunctions with more oscillations to fit boundary conditions.

Degeneracy in a 3D box

Different combinations of quantum numbers can produce the same total energy.

Degeneracy in symmetrical boxes

Equal box lengths allow for multiple permutations of quantum numbers yielding identical energy sums.

Mass of the particle and energy in the box

A higher mass results in lower energy levels, since energy is inversely proportional to mass.

Wavefunction ψ(x) in a box model

It describes the amplitude of the particle's position; its square gives the probability density.

Nodal planes in a 3D particle-in-a-box

Surfaces within the box where the probability of finding the particle is zero.

ψ(x)²

It's the square of the wavefunction, which represents probability and must be non-negative.

Lower quantum states wavefunctions

Lower energy means fewer oscillations, resulting in broader, simpler wavefunctions.

Increasing quantum number n

It adds more nodes and oscillations, making the wavefunction more complex.

Energy in smaller dimensions

Energy increases because confinement raises the curvature of the wavefunction.

Same energy in different states

Because total energy depends on the sum of squared quantum numbers, not their order.

Zero-point energy principle

The Heisenberg Uncertainty Principle — confining a particle increases its minimum kinetic energy.

Rigid rotator in quantum mechanics

A rigid rotator is a model for a molecule where the bond length between atoms is fixed, and the molecule can rotate freely without stretching or vibrating.

Rotational energy levels equation

El = B l(l + 1), where B = 2I and l = 0,1,2,...

Quantum number l

It represents the rotational energy level or the total angular momentum quantum number.

Moment of inertia (I)

It depends on the reduced mass (μ) of the molecule and the square of the bond length (r): I = μr².

Energy spacing between rotational levels

It increases linearly, since ΔE = 2B(l + 1).

Selection rule for rotational transitions

Δl = ±1.

Energy levels in rigid rotator model

No, they get farther apart as l increases.

Spectral lines in rotational spectra

Because the energy differences between levels increase linearly with l.

Degeneracy in rotational energy levels

It's the number of distinct states (orientations) that have the same energy: 2l + 1.

Zero-point rotational energy of a molecule

It is zero. For l = 0, the rotational energy E0 = 0.

Zero-point energy for rotation

Because the lowest allowed rotational state (l = 0) corresponds to no rotation, and this has exactly zero energy.

Rotational constant B and bond length

B decreases, because moment of inertia increases with r², and B ∝ 1/I.

Isotopic substitution effect on rotational spectrum

It changes the reduced mass μ, which alters the moment of inertia and shifts the positions of spectral lines.

Dimensional model for molecular rotation

A 3D rigid rotor model using quantum numbers l and ml.

Spherical harmonics Ylm(θ,ϕ)

They represent the angular part of the rotational wavefunction, describing how the molecule's orientation is distributed in space.

Variables for spherical harmonics

θ (polar angle) and ϕ (azimuthal angle).

Quantum number for angular motion in 2D ring model

The quantum number m, where m = 0,±1,±2,...

Electromagnetic spectrum for rotational transitions

The microwave region.

Rotational spectroscopy and bond lengths

By measuring the rotational constant B, which is related to the moment of inertia and hence to bond length r.

Pattern distinguishing rotational from vibrational energy levels

Rotational energy levels increase quadratically with l, and there is no zero-point energy, unlike vibrational levels which start above zero.

Harmonic oscillator

A physical model where two atoms are connected by a spring representing the chemical bond.

Hooke's Law

Describes a restoring force proportional to displacement, expressed as F = −kx.

Force constant (k)

Measures bond stiffness; higher k means a stronger or stiffer bond.

Vibrational quantum number (v)

Designates the vibrational energy level, where v = 0 is the ground state, v = 1 is the first excited state, etc.

Zero-point energy (ZPE)

The energy that prevents the vibrational energy of a molecule from being zero, even at the ground state.

Reduced mass (μ)

Defined as μ = (m1 * m2) / (m1 + m2), it simplifies the two-body system into a single effective particle for vibrational analysis.

Vibrational frequency (ν)

Determined by the force constant (k) and reduced mass (μ), where stronger bonds vibrate faster and lighter atoms vibrate faster.

Vibrational energy levels

In a harmonic oscillator, they are equally spaced by hν, unlike electronic or rotational levels.

Vibrational energy equation

For a harmonic oscillator, E = (v + 1/2)hν.

Selection rule for vibrational transitions

Δv = ±1, meaning only transitions between adjacent levels are allowed.

IR active condition

A vibrational mode must cause a change in the dipole moment of the molecule during vibration.

O=O symmetric stretch IR inactivity

O=O is a nonpolar, symmetric molecule; its vibration doesn't change the dipole moment.

Anharmonicity effect

Causes energy levels to become closer together as v increases and can lead to dissociation at high energy.

Overtones in IR spectroscopy

Weaker transitions allowed due to anharmonicity, expressed as Δv = ±2, ±3, ...

Morse potential

A potential energy curve that models real vibrational motion more accurately than the parabolic curve.

Dissociation energy (De)

The energy needed to break the bond and separate atoms completely.

Vibrational modes in nonlinear molecules

A nonlinear molecule with N atoms has 3N - 6 vibrational modes.

Types of vibrational motions

The two main types in polyatomic molecules are stretching (changes in bond length) and bending (changes in bond angle).

Vibrational modes in linear molecules

A linear molecule has 3N - 5 vibrational modes.

IR spectroscopy in molecular analysis

It detects specific vibrational frequencies (fingerprints) to identify functional groups or compounds.