Summary of Quantum Mechanics: Motion and Spin

Quantum Mechanics: Motion and Spin Explained for Students

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Introduction

Rotation in quantum mechanics reveals how classical continuous motion becomes discrete when waves must satisfy boundary conditions. This material focuses on the free rotor (particle constrained to move on a ring and on a sphere surface) and how topology (the cyclic geometry) enforces quantisation of angular momentum and rotational energy. We avoid topics covered elsewhere such as spin or bound/oscillator rotational models.

Definition: A free rotor is a particle constrained to move on a fixed-radius curve or surface with no potential energy on that manifold, so its kinetic energy is purely rotational.

1. Particle on a Ring (2D free rotor)

Physical setup

A particle of mass $m$ moves on a circle of radius $r$ in the $xy$-plane.
Coordinate: the polar angle $\varphi$ with $0 \leq \varphi < 2\pi$.
Potential: $V(\varphi)=0$ everywhere on the ring.

Classical picture

Linear momentum tangential to the circle is $p$.
Angular momentum about the $z$-axis is $J_z = p r$ (with sign depending on direction).
Moment of inertia is $I = m r^2$.
Classical rotational kinetic energy: $$E = \frac{p^2}{2m} = \frac{J_z^2}{2I}$$

Quantum picture: wavefunction on a circle

The Schrödinger equation reduces to motion in the angular coordinate $\varphi$ (radial coordinate fixed).
General solutions are plane waves in angle: $\psi(\varphi) = A e^{i k \varphi}$ where $k$ is related to momentum around the ring.

Definition: Single-valuedness means the physical wavefunction must give the same complex value at $\varphi$ and $\varphi + 2\pi$, because those points represent the same physical location.

Topology enforces the boundary condition: $$\psi(\varphi+2\pi)=\psi(\varphi).$$
This requires the phase factor $e^{i k 2\pi}=1$, so $k$ must be integer-valued. Introduce the magnetic quantum number $m_l$ (commonly denoted $m$ or $m_l$): $$k = m_l, \quad m_l \in \mathbb{Z}=0,\pm 1,\pm 2,\ldots$$
Allowed angular wavelengths (around the circumference $2\pi r$): $$\lambda_\varphi = \frac{2\pi r}{|m_l|}.$$ When $m_l=0$ the wave is constant around the ring.

Quantised angular momentum and energy

Angular momentum eigenvalues (about $z$): $$J_z = m_l \hbar.$$
Energy eigenvalues: $$E_{m_l} = \frac{J_z^2}{2I} = \frac{(m_l\hbar)^2}{2I} = \frac{\hbar^2 m_l^2}{2I}.$$
Degeneracy: states with $m_l$ and $-m_l$ have the same energy (except $m_l=0$), so energies for $|m_l|>0$ are doubly degenerate.

Wavefunctions and probability

Normalized angular eigenfunctions (choose normalization on $0\leq\varphi<2\pi$): $$\psi_{m_l}(\varphi) = \frac{1}{\sqrt{2\pi}} e^{i m_l \varphi}.$$
Probability density is uniform: $$|\psi_{m_l}(\varphi)|^2 = \frac{1}{2\pi},$$ independent of $\varphi$. Thus knowing $J_z$ exactly makes the angular position completely indefinite.

💡 Věděli jste?Fun fact: Did you know wavefunctions that change by a nontrivial phase after one full circuit, $\psi(\varphi+2\pi)=e^{i\alpha}\psi(\varphi)$ with $\alpha\neq 0$, are allowed physically only when an additional structure (like magnetic flux through the ring) or different topology is present; for a simple ring without such extra structure the phase must be zero.

Visual intuition

Higher $|m_l|$ corresponds to more oscillations of the complex wavefunction around the ring (shorter angular wavelength), and hence larger angular momentum and energy.

2. Particle on a Sphere (3D free rotor)

Physical setup

A particle of mass $m$ is constrained to the surface of a sphere of radius $r$.
Coordinates: polar angle $\theta$ (colatitude) with $0\leq\theta\leq\pi$ and azimuthal angle $\varphi$ with $0\leq\varphi<2\pi$.
Potential: $V(\theta,\varphi)=0$ on the surface.

Definition: The surface rotor (rigid rotor) is a model for molecules rotating freely in space when vibrational and electronic excitations are neglected.

Topology and boundary conditions

The wavefunction must be single-valued on the sphere: it must reproduce itse

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Free Rotor Quantisation

Klíčové pojmy: Particle on a ring: $\psi(\varphi+2\pi)=\psi(\varphi)$ enforces integer $m_l$, Ring angular momentum: $J_z=m_l\hbar$, Ring energy: $E_{m_l}=\hbar^2 m_l^2/(2I)$, Ring eigenfunctions: $\psi_{m_l}=\frac{1}{\sqrt{2\pi}}e^{i m_l\varphi}$, Particle on sphere: quantum numbers $\ell,m_l$ with $\ell=0,1,2,\dots$, Sphere angular momentum: $L^2=\ell(\ell+1)\hbar^2$, $L_z=m_l\hbar$, Sphere energy: $E_{\ell}=\hbar^2\ell(\ell+1)/(2I)$, Topology (boundary conditions) causes quantisation, Degeneracy: ring $\pm m_l$, sphere $2\ell+1$, Exact $J_z$ implies uniform $|\psi|^2$ around angle

## Introduction Rotation in quantum mechanics reveals how classical continuous motion becomes discrete when waves must satisfy boundary conditions. This material focuses on the free rotor (particle constrained to move on a ring and on a sphere surface) and how topology (the cyclic geometry) enforces quantisation of angular momentum and rotational energy. We avoid topics covered elsewhere such as spin or bound/oscillator rotational models. > Definition: A free rotor is a particle constrained to move on a fixed-radius curve or surface with no potential energy on that manifold, so its kinetic energy is purely rotational. ## 1. Particle on a Ring (2D free rotor) ### Physical setup - A particle of mass $m$ moves on a circle of radius $r$ in the $xy$-plane. - Coordinate: the polar angle $\varphi$ with $0 \leq \varphi < 2\pi$. - Potential: $V(\varphi)=0$ everywhere on the ring. ### Classical picture - Linear momentum tangential to the circle is $p$. - Angular momentum about the $z$-axis is $J_z = p r$ (with sign depending on direction). - Moment of inertia is $I = m r^2$. - Classical rotational kinetic energy: $$E = \frac{p^2}{2m} = \frac{J_z^2}{2I}$$ ### Quantum picture: wavefunction on a circle - The Schrödinger equation reduces to motion in the angular coordinate $\varphi$ (radial coordinate fixed). - General solutions are plane waves in angle: $\psi(\varphi) = A e^{i k \varphi}$ where $k$ is related to momentum around the ring. > Definition: Single-valuedness means the physical wavefunction must give the same complex value at $\varphi$ and $\varphi + 2\pi$, because those points represent the same physical location. - Topology enforces the boundary condition: $$\psi(\varphi+2\pi)=\psi(\varphi).$$ - This requires the phase factor $e^{i k 2\pi}=1$, so $k$ must be integer-valued. Introduce the magnetic quantum number $m_l$ (commonly denoted $m$ or $m_l$): $$k = m_l, \quad m_l \in \mathbb{Z}=0,\pm 1,\pm 2,\ldots$$ - Allowed angular wavelengths (around the circumference $2\pi r$): $$\lambda_\varphi = \frac{2\pi r}{|m_l|}.$$ When $m_l=0$ the wave is constant around the ring. ### Quantised angular momentum and energy - Angular momentum eigenvalues (about $z$): $$J_z = m_l \hbar.$$ - Energy eigenvalues: $$E_{m_l} = \frac{J_z^2}{2I} = \frac{(m_l\hbar)^2}{2I} = \frac{\hbar^2 m_l^2}{2I}.$$ - Degeneracy: states with $m_l$ and $-m_l$ have the same energy (except $m_l=0$), so energies for $|m_l|>0$ are doubly degenerate. ### Wavefunctions and probability - Normalized angular eigenfunctions (choose normalization on $0\leq\varphi<2\pi$): $$\psi_{m_l}(\varphi) = \frac{1}{\sqrt{2\pi}} e^{i m_l \varphi}.$$ - Probability density is uniform: $$|\psi_{m_l}(\varphi)|^2 = \frac{1}{2\pi},$$ independent of $\varphi$. Thus knowing $J_z$ exactly makes the angular position completely indefinite. Fun fact: Did you know wavefunctions that change by a nontrivial phase after one full circuit, $\psi(\varphi+2\pi)=e^{i\alpha}\psi(\varphi)$ with $\alpha\neq 0$, are allowed physically only when an additional structure (like magnetic flux through the ring) or different topology is present; for a simple ring without such extra structure the phase must be zero. ### Visual intuition - Higher $|m_l|$ corresponds to more oscillations of the complex wavefunction around the ring (shorter angular wavelength), and hence larger angular momentum and energy. ## 2. Particle on a Sphere (3D free rotor) ### Physical setup - A particle of mass $m$ is constrained to the surface of a sphere of radius $r$. - Coordinates: polar angle $\theta$ (colatitude) with $0\leq\theta\leq\pi$ and azimuthal angle $\varphi$ with $0\leq\varphi<2\pi$. - Potential: $V(\theta,\varphi)=0$ on the surface. > Definition: The surface rotor (rigid rotor) is a model for molecules rotating freely in space when vibrational and electronic excitations are neglected. ### Topology and boundary conditions - The wavefunction must be single-valued on the sphere: it must reproduce itse