Abstract
The oscillator Hamiltonian in the coordinate representation is:
This is not just another one-dimensional example. It is a fundamental piece of the general theory.
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- 1.
The recurrence formula (16.10) also gives us a transcendental solution, but this is not acceptable for a wave function. For \(j \rightarrow \infty \), (16.10) becomes \(a_{j+2} \sim {2 \over j}a_{j}\); this is solved by \(a_{j} \sim {C \over ({j \over 2})!}\), with some constant C. In the even case, \(h= \sum _{j=2k}{C \over ({j \over 2})!}q^{j}=C\sum _{k}{1 \over k!}q^{2k} = C e^{q^{2}}\); this asymptotic behavior at large q leads to \(\psi \rightarrow \infty \) for \(x\rightarrow \infty ,\) and such a solution cannot be normalized. In the odd case, \(h= \sum _{j=2k-1}{C \over ({j \over 2})!}q^{j}=C\sum _{k}{1 \over (k-{1 \over 2})!}q^{2k-1}\), and since \(h > C\sum _{k}{1 \over k!}q^{2k-1} \sim {C\over q} e^{q^{2}},\) even this solution cannot be normalized.
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Cini, M. (2018). The Quantum Harmonic Oscillator. In: Elements of Classical and Quantum Physics. UNITEXT for Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-71330-4_16
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DOI: https://doi.org/10.1007/978-3-319-71330-4_16
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