Quantum Mechanics: Rigid Rotator

Rigid Rotator in Quantum Mechanics

Definition

• A rigid rotator is a model used in quantum mechanics to describe the rotational motion of a molecule or a system of particles.
• It is assumed that the particles are rigidly attached to each other and rotate as a single unit.

Classical vs. Quantum Mechanical Rigid Rotator

• In classical mechanics, the energy of a rigid rotator is continuous and can take on any value.
• In quantum mechanics, the energy of a rigid rotator is quantized, meaning it can only take on specific discrete values.

Energy Levels

• The energy levels of a rigid rotator are given by the equation: E = ħ²J(J+1) / 2I
  • ħ: reduced Planck constant
  • J: rotational quantum number (integer values 0, 1, 2, ...)
  • I: moment of inertia of the rotator
• The energy levels are degenerate, meaning that states with the same J but different M (magnetic quantum number) have the same energy.

Wave Functions

• The wave functions of a rigid rotator are spherical harmonics, YJM(θ, φ)
  • YJM: spherical harmonic function
  • θ: polar angle
  • φ: azimuthal angle
  • M: magnetic quantum number (-J ≤ M ≤ J)

Selection Rules

• The selection rules for transitions between energy levels of a rigid rotator are:
  • ΔJ = ±1 (change in rotational quantum number)
  • ΔM = 0, ±1 (change in magnetic quantum number)

Applications

• The rigid rotator model is used to describe the rotational spectra of molecules, such as diatomic molecules and symmetric tops.
• It is also used to study the rotational motion of particles in solids and liquids.

Rigid Rotator in Quantum Mechanics

Definition

• Rigid rotator is a model used to describe the rotational motion of a molecule or a system of particles where particles are rigidly attached to each other and rotate as a single unit.

Classical vs. Quantum Mechanical Rigid Rotator

• In classical mechanics, the energy of a rigid rotator is continuous and can take on any value.
• In quantum mechanics, the energy of a rigid rotator is quantized, meaning it can only take on specific discrete values.

Energy Levels

• Energy levels are given by the equation: E = ħ²J(J+1) / 2I
• ħ is the reduced Planck constant.
• J is the rotational quantum number (integer values 0, 1, 2,...).
• I is the moment of inertia of the rotator.
• Energy levels are degenerate, meaning that states with the same J but different M (magnetic quantum number) have the same energy.

Wave Functions

• Wave functions are spherical harmonics, YJM(θ, φ).
• YJM is the spherical harmonic function.
• θ is the polar angle.
• φ is the azimuthal angle.
• M is the magnetic quantum number (-J ≤ M ≤ J).

Selection Rules

• Selection rules for transitions between energy levels are:
  • ΔJ = ±1 (change in rotational quantum number).
  • ΔM = 0, ±1 (change in magnetic quantum number).

Applications

• Rigid rotator model is used to describe the rotational spectra of molecules, such as diatomic molecules and symmetric tops.
• It is also used to study the rotational motion of particles in solids and liquids.