Flashcards on Quantum Mechanics: Motion and Spin
Quantum Mechanics: Motion and Spin Explained for Students
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Quantum rotation: Free-rotor & topology
13 cards
Card 1
Question: For a particle of mass m constrained to a ring of radius r with V=0, how is the classical kinetic energy E expressed in terms of linear momentum p and
Answer: E = p^2/(2m) and, using angular momentum Jz = ±pr and I = mr^2, E = Jz^2/(2I).
Card 2
Question: What boundary condition must the wavefunction ψ(φ) on a ring satisfy and why?
Answer: ψ(φ) must be single-valued and reproduce itself after φ increases by 2π (ψ(φ+2π)=ψ(φ)), because points φ and φ+2π are identical on the ring. Non–singl
Card 3
Question: What allowed wavelengths (in terms of the circumference) arise from the single-valued condition on a ring, and what quantum number labels them?
Answer: Allowed wavelengths satisfy λ = 2πr/ m_l where m_l = 0, ±1, ±2, … (an integer). The integer m_l is the angular quantum number labeling allowed wavelen
Card 4
Question: How is the quantised angular momentum Jz of a particle on a ring expressed in terms of m_l and ħ?
Answer: Jz = m_l ħ, where m_l = 0, ±1, ±2, … (positive/negative values correspond to opposite rotation directions).
Card 5
Question: Write the quantised rotational energy levels for the particle on a ring in terms of m_l and the moment of inertia I.
Answer: E_{m_l} = (m_l^2 ħ^2)/(2I), where I = mr^2 and m_l = 0, ±1, ±2, … . Energy depends on m_l^2, so states ±m_l are degenerate.
Card 6
Question: What is the normalized ground-state wavefunction (m_l = 0) for a particle on a ring?
Answer: ψ_0(φ) = 1/√(2π), a constant around the circle (same value at all φ).
Card 7
Question: Give the general form of the normalized eigenfunctions for a particle on a ring with quantum number m_l.
Answer: ψ_{m_l}(φ) = (1/√(2π)) e^{i m_l φ}, with m_l = 0, ±1, ±2, … .
Card 8
Question: Why are only some angular momenta acceptable for the ring even though classically a continuum is allowed?
Answer: Because quantum wavefunctions must be single-valued after φ→φ+2π, only wavefunctions with integer m_l reproduce themselves on each circuit; other angu
Card 9
Question: How does increasing |m_l| affect the wavefunction, wavelength, and angular momentum on the ring?
Answer: As |m_l| increases the wavelength around the ring decreases, the number of nodes increases, and the magnitude of angular momentum |Jz| increases in st
Card 10
Question: What is the probability density |ψ(φ)|^2 for eigenstates on the ring and what does that imply about the particle's location?
Answer: |ψ_{m_l}(φ)|^2 = 1/(2π), independent of φ. Thus the particle’s angular position is completely indefinite for definite angular-momentum eigenstates.