Griffiths QM Problem 6.1 | Parity Operator in 3D | Hydrogenic Orbitals & Eigenvalues
Автор: Vasu V
Загружено: 2024-04-05
Просмотров: 786
In this video, we solve Problem 6.1 from a Quantum Mechanics textbook step by step.
We analyze the parity operator (P̂) in three dimensions and explore how it acts on wavefunctions in different coordinate systems and on hydrogenic orbitals.
✅ (a) Show that applying the parity operator
P̂ψ(r) = ψ(−r)
is equivalent to a mirror reflection followed by a rotation.
✅ (b) For ψ expressed in polar coordinates, demonstrate that the parity operator transforms it as
P̂ψ(r, θ, φ) = ψ(r, π − θ, φ + π)
✅ (c) Show that for hydrogenic orbitals:
P̂ψₙₗₘ(r, θ, φ) = (−1)ˡ ψₙₗₘ(r, θ, φ)
and confirm that ψₙₗₘ is an eigenstate of the parity operator with eigenvalue (−1)ˡ.
💡 You will learn how to:
Understand the mathematical action of the parity operator in 3D
Apply spherical coordinate transformations under inversion
Explore how orbital angular momentum quantum number (ℓ) determines parity
Recognize parity symmetry in central potentials such as the hydrogen atom
Perfect for students studying Quantum Mechanics, Angular Momentum Theory, and Symmetry Operations!
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