21 Central Potentials and the Hydrogen Atom

When the Hamiltonian function is invariant under rotations, we then expect eigenspaces of the corresponding Hamiltonian operator to carry representations of $SO(3)$ . These spaces of eigenfunctions of a given energy break up into irreducible representations of $SO(3)$ , and we have seen that these are labeled by an integer $l=0,1,2,...$ . and have dimension $2l+1$ . This can be used to find properties of the solutions of the Schr¨odinger equation whenever one has a rotation invariant potential energy. We will work out what happens for the case of the Coulomb potential describing the hydrogen atom. This specific case is exactly solvable because it has a second not-so-obvious $SO(3)$ symmetry, in addition to the one coming from rotations of ${\mathbf{R}}^3$

中文目录

Chapter contents

• 21.1 Quantum particle in a central potential
• 21.2 so(4) symmetry and the Coulomb potential
• 21.3 The hydrogen atom
• 21.4 For further reading

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来源版本：2025-10-20

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