Examples of eigenfunctions in the following topics:
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- The functions $Y_{lm}$ are eigenfunctions of the angular momentum operator.
- This means quantum-mechanically that the Hamiltonian commutes with the angular momentum operator, and that the wavefunctions that satisfy the Hamiltonian also are eigenfunction of the angular momentum operator (${\bf L}={\bf r}\times {\bf p}$).We have
- The angular eigenfunctions take this form regardless of the form of the central potential.
- They are simply the eigenfunctions of the angular momentum operator.
- Because the radial eigenfunctions for different values of $l$ satisfy different equations, there is no orthogonality relation for the radial wavefunctions with different $l$ values.
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- You can find the eigenfunctions of the Morse potential on Wikipedia.
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- We can imagine the operator $H$ as a matrix that multiplies the state vector $\psi$, so this equation is an eigenvalue equation with $E$ as the eigenvalue and $\psi$ as an eigenvector (or eigenfunction) of the matrix (or operator) $H$.The Hamiltonian classically is the sum of the kinetic energy and the potential energy of the particles.
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- Although the resulting energy eigenfunctions (the orbitals) are not necessarily isotropic themselves, their dependence on the angular coordinates follows completely, generally from this isotropy of the underlying potential.
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- Although the resulting energy eigenfunctions (the orbitals) are not necessarily isotropic themselves, their dependence on the angular coordinates follows generally from this isotropy of the underlying potential.
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- You can find the eigenfunctions of the Morse potential on Wikipedia.