Why Can't the Probability Exceed 1 in Time-Dependent Perturbation Theory?

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SUMMARY

The discussion centers on the limitations of time-dependent perturbation theory in quantum mechanics, specifically regarding the transition probability from the state \(\psi_{100}\) to \(\psi_{210}\) in a hydrogen atom under a uniform magnetic field with magnitude \(\mathcal{E}\). The derived expression for the transition probability is \(\frac{131072}{59049} \frac{e^2 \mathcal{E}^2 a_o^2}{(E_2 -E_1)^2} \sin^2(\frac{E_2-E_1}{2 \hbar}t)\). It is established that increasing \(\mathcal{E}\) beyond a certain threshold invalidates the perturbative approximation, leading to probabilities exceeding 1, which is not physically meaningful.

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  • Understanding of quantum mechanics principles, specifically perturbation theory.
  • Familiarity with the hydrogen atom model and its energy states.
  • Knowledge of the mathematical formulation of transition probabilities in quantum systems.
  • Basic grasp of the implications of magnetic fields on quantum states.
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  • Study the limitations of perturbation theory in quantum mechanics.
  • Explore the derivation of transition probabilities in quantum systems.
  • Investigate the effects of strong magnetic fields on atomic states.
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Students and researchers in quantum mechanics, particularly those focusing on perturbation theory and atomic transitions in external fields.

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[SOLVED] time-dependent perturbation theory

Homework Statement


My book uses time-dependent perturbation theory to derive the following expression for the transition of \psi_{100} to \psi_{210} in the hydrogen atom in a uniform magnetic field with magnitude \mathcal{E}

\frac{131072}{59049} \frac{e^2 \mathcal{E}^2 a_o^2}{(E_2 -E_1)^2} \sin^2(\frac{E_2-E_1}{2 \hbar}t)

What I don't understand is why you cannot just increase \mathcal{E} until the probability goes above 1?

Homework Equations


The Attempt at a Solution

 
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Because it's 'perturbation theory'. If the field becomes large, then whole approximation that the field can be treated as a 'perturbation' goes out the window and the formula is invalid.
 

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