Abstract
Radially symmetric problems appear if the interaction between two particles depends only on their separation r. We will first see how the dynamical problem of the motion of the two particles can be separated in terms of center of mass motion and relative motion and then write the effective Hamiltonian for the relative motion of the two particles in spherical coordinates.
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Notes
- 1.
See Appendix F for the calculation of the logarithm of an invertible matrix.
- 2.
Inversion of three axes is equivalent to inversion of one axis combined with a rotation.
- 3.
We could do the following calculations in slightly more general form without using hermiticity, and then find hermiticity of the finite-dimensional representations along the way.
- 4.
Stated differently, we leave out a factor 1 = ∫ 0 ∞ drr 2 | r〉〈r | .
- 5.
…or we could use total angular momentum, i.e. quantum numbers \(K,k,j \in \{\vert L -\ell\vert,\ldots,L+\ell\},m_{j} = M + m,L,\ell\).
- 6.
E. Schrödinger, Annalen Phys. 384, 361 (1926).
- 7.
W. Gordon, Annalen Phys. 394, 1031 (1929); M. Stobbe, Annalen Phys. 399, 661 (1930), see also [3]. Gordon and Stobbe normalized in the k scale, i.e. to δ(k − k′) instead of \(\delta (k - k')/k^{2}\).
- 8.
See e.g. N. Mukunda, Amer. J. Phys. 46, 910 (1978).
- 9.
N.F. Mott, Proc. Roy. Soc. London A 118, 542 (1928); W. Gordon, Z. Phys. 48, 180 (1928).
- 10.
D. Bohm, Phys. Rev. 85, 166 & 180 (1952).
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Dick, R. (2016). Central Forces in Quantum Mechanics. In: Advanced Quantum Mechanics. Graduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-25675-7_7
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DOI: https://doi.org/10.1007/978-3-319-25675-7_7
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