Quantum Perturbation Theory and the Correspondence Principle
| Main Author: | Francesco R. Ruggeri |
|---|---|
| Format: | info publication-preprint |
| Terbitan: |
, 2020
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| Subjects: | |
| Online Access: |
https://zenodo.org/record/4086043 |
Daftar Isi:
- In (1), standard quantum perturbation theory is developed using Vo(x)+bV(x), W(x)=Wo(x)+bn1(x)+b*bn2(x)+... and E=Eo+E1b+b*bE2+... where b is small. These values are inserted into the time-independent Schrodinger equation and terms with like order of b collected. Multiplying by Wo and integrating over x yields an expression for Ei. Multiplying by Wk(x) and integrating yields a projection of <ni | k> where Wk is the kth eigenfunction of -1/2m d/dxd/dx +Vo(x). The projection <ni|Wo> is determined using <W |W>=1 and retaining terms of b to the power i. Thus, many terms are dropped from <W|W> and the approach used in (1) is a little tedious as noted in that reference. In this note, we consider W(x)=Wo(x)W1(x) which yields: -1/2m d/dx d/dx W1 + d/dx W1 d/dx Wo / Wo + b V(x) W1(x) = (E-Eo) W1(x) when inserted into the Schrodinger equation. The same <ni|Wk> projection results, as expected. For <ni|Wo>, however, we argue that <WoW1 | WoW1>=1 leads to a more direct and less tedious equation for evaluating this projection. We interpret the results which are based on eigenfunctions of Ho i.e. the Hamiltonian with bV(x) dropped. Finally, we compare the high energy quantum perturbed density with the classical (correspondence) principle i.e density = C/v(x) where v(x) is velocity. In the high energy classical limit, matters are not so simple because an expansion of the spatial density in positive powers of b is only valid away from the classical turning points. Near the turning points, a different power series in b holds and technically there is a region for which there is no power series as E-Vo and b(E1-V) are very similar in value.