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We normally observe classical behaviour due to the time dependent schrodinger equation in simple quantum systems when we introduce 'Gaussian wavepackets' which have bell shaped uncertainty in energy, momentum and position. A wavefunction which is often constructed by tapering the energy eigenstates. This can be seen in the case of a particle in a step function, harmonic oscillator, or under gravity (exhibits bouncing). This is especially true when $dL>>\hbar$ and the mean of quantum number $n$ is large.

Can this be modelled for a hydrogen atom? For example let us construct an atom with a small Rydberg constant, and have our initial state as $G(n,l,m)\psi_{nlm}(r)$ summed over all n,l,m where G is a tapered bell curve in all 3 variables with a large mean, i.e n = 50, l = 25 and m = 0. Then we plot this |psi|^2 as probability cloud and allow it to evolve over time. What does the shape of this probability cloud look like? Does it approximate classical behaviour such as orbital motion?

EDIT: My question asks in particular for the evolution of simple Gaussian coefficients in n,l,m. I'm not sure what the other question appears to model, or whether they have used the same approach.

Just to be clear I'm not asking for a necessarily Gaussian wavepacket. The distribution in n,l,m should be bell shaped. This is much simpler

  • It's very technical, but in phase-space quantum mechanics this correspondence is achievable. J Dahl and M Springborg, Mol Phys 47 (1982) 1001-1019; Phys Rev A36 (1988) 1050- 1062; Phys Rev A59 (1999) 4099-4100; J Chem Phys 88 (1988) 4535-4547 ... – Cosmas Zachos Mar 23 '21 at 13:33
  • Adding Cosmas's links explicitly, in the hope that Altmetric will once day work like it promises to: https://doi.org/10.1080/00268978200100752 https://doi.org/10.1103/PhysRevA.36.1050 https://doi.org/10.1103/PhysRevA.59.4099 https://doi.org/10.1063/1.453761 – Emilio Pisanty Mar 23 '21 at 14:26
  • can someone inform me about why it's still closed –  Mar 23 '21 at 15:14
  • Reopening is not immediate -- your edit has put it in a review queue, and it can take a few hours / couple of days to get through it. – Emilio Pisanty Mar 23 '21 at 16:45
  • That said, your edit is rather self-contradictory. If you're specifically asking about a gaussian distribution over $n,l,m$, then that's a very weirdly specific thing to ask about but sure, it's a legitimate question and it's distinct from the linked duplicate. However, your final paragraph opens the field to arbitrary bell-shaped distributions, and this includes the case of the linked duplicate. Thus, as currently phrased, the closure is correct -- the linked duplicate does answer this question as it's presently worded. – Emilio Pisanty Mar 23 '21 at 16:49
  • Regarding the narrower reading of what the evolution of the specific wavefunction you detail in your second paragraph would be, it's very close to this site's understanding of homework-and-exercises questions. This is quite different to the policy on the maths site, and it holds regardless of the origin of the question. Please take a minute to read the guidelines for homework and exercise questions as well as check-my-work questions to understand those differences. – Emilio Pisanty Mar 23 '21 at 16:55

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Yes, you can add many contributions $$\psi(r,\theta,\phi,t) = \sum_{nlm}G_{nlm}\ \psi_{nlm}(r,\theta,\phi)\ e^{-iE_nt/\hbar}$$ where the $G_{nlm}$ are large only around some averages $n_{\text{avg}},l_{\text{avg}},m_{\text{avg}}$. Then you can get a wave packet which will propagate approximately along a classical path (i.e. a Kepler orbit). This follows from the classical limit saying that the expected position and momentum will approximately follow the classical trajectories.

  • If $n_{\text{avg}} \approx l_{\text{avg}}$, then the packet will move along a classical circular path.
  • If $n _{\text{avg}}>l_{\text{avg}}$, then the packet will move along a classical elliptical path.
  • If $m_{\text{avg}}\approx\pm l_{\text{avg}}$, then the packet will move in the equatorial plane (counterclockwise or clockwise, respectively).
  • For other $m_{\text{avg}}$ in between the packet will move in a tilted orbit.

See for example the animation at "A Tiny Solar System After All".
enter image description here

Thomas Fritsch
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