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Chapter Summary:

1. Spherical Symmetry

In the first book of Visual Quantum Mechanics, we considered mainly one- and two-dimensional systems. Now we turn to the investigation of three-dimensional systems. This chapter is devoted to the very important special case of systems with spherical symmetry.

In the presence of spherical symmetry, the Schrödinger equation has solutions that can be separated into a product of a radial part and an angular part. In this chapter, all possible solutions of the equation for the angular part will be determined once and for all.

We start by discussing symmetry transformations in general. In quantum mechanics, all symmetry transformations may be realized by unitary or antiunitary operators. We define the unitary transformations corresponding to rotations of a particle in R³. Their self-adjoint generators are the components of the orbital angular momentum L. We describe the angular-momentum commutation relations and discuss their geometrical meaning.

A quantum system is called invariant under a given symmetry transformation if the Hamiltonian commutes with the corresponding unitary operator. A particle moving under the influence of a potential V(x) is a spherically symmetric system (invariant under rotations) if the potential function depends only on the distance r from the origin. Spherical symmetry implies the conservation of the angular momentum and determines the structure of the eigenvalue spectrum of the Hamiltonian (degeneracy). The square L² and any component L_k of the angular momentum can be diagonalized simultaneously with the Hamiltonian of a spherically symmetric system. The structure of the common system of eigenvectors can essentially be derived from the angular-momentum commutation relations. In general, the possible eigenvalues of the angular-momentum operators are characterized by integer and half-integer quantum numbers. It turns out, however, that only integer quantum numbers occur in case of the orbital angular momentum.

The eigenvalues and eigenfunctions (spherical harmonics) of the orbital angular momentum are then determined explicitly. The spherical harmonics are the energy eigenfunctions of a particle whose configuration space is a sphere (rigid rotator). The rigid rotator can serve as a simple model for a diatomic molecule in its vibrational ground state.

The restriction of the eigenvalue problem to an angular-momentum eigenspace reduces the Schrödinger equation to an ordinary differential equation. We conclude the chapter with a brief discussion of this so-called radial Schrödinger equation.

• 1.1. A Note on Symmetry Transformations 2
  • 1.1.1. Rotations as symmetry transformations 2
  • 1.1.2. Symmetry transformations in quantum mechanics 4
  • 1.1.3. Realizations of symmetry transformations 5
  • 1.1.4. Invariance of a physical system 6
• 1.2. Rotations in Quantum Mechanics 7
  • 1.2.1. Rotation of vectors in R³ 7
  • 1.2.2. Rotation of wave functions 10
• 1.3. Angular Momentum 12
  • 1.3.1. Angular momentum in classical mechanics 12
  • 1.3.2. Angular momentum in quantum mechanics 12
  • 1.3.3. Commutation relations of the angular-momentum operators 14
  • 1.3.4. The meaning of the angular-momentum commutation relations 15
• 1.4. Spherical Symmetry of a Quantum System 17
  • 1.4.1. Conservation of angular momentum 17
  • 1.4.2. Spherically symmetric potentials 18
  • 1.4.3. Symmetry and degeneracy 19
• 1.5. The Possible Eigenvalues of Angular-Momentum Operators 21
  • 1.6. Spherical Harmonics 26
  • 1.6.1. Spherical coordinates 26
  • 1.6.2. Angular momentum in spherical coordinates 28
  • 1.6.3. Special topic: Properties of the spherical harmonics 32
• 1.7. Particle on a Sphere 34
  • 1.7.1. Classical particle on a sphere 34
  • 1.7.2. The rigid rotator 35
  • 1.7.3. Transition to quantum mechanics 35
  • 1.7.4. Dynamics of the rigid rotator 36
• 1.8. Quantization on a Sphere 38
  • 1.8.1. Comparison of classical and quantum probability densities 38
  • 1.8.2. Special topic: Curvilinear coordinates 41
• 1.9. Free Schrödinger Equation in Spherical Coordinates 44
  • 1.9.1. Solutions of the radial equation 44
  • 1.9.2. Special Topic: Properties of the Riccati-Bessel functions 45
  • 1.9.3. Special Topic: Expanding the plane wave 46
  • 1.9.4. Special topic: Spherical harmonics and the Fourier transformation 48
• 1.10. Spherically Symmetric Potentials 50
  • 1.10.1. The structure of the eigenvalue spectrum 50
  • 1.10.2. The vibrating rotator: A model of a diatomic molecule 53