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I have two contradictory result about the amplitude of single mode field when it is quantized

  • In C. Gerry and P. Knight's Introductory quantum optics, the amplitude is given by $$\sqrt{\frac{2\omega^2}{V\epsilon_0}},$$ which puts the hamiltonian of radiation in the cavity at $H=\frac{1}{2}(p^2 + \omega^2q^2)$ and the electric field as $E=\sqrt{\frac{2\omega^2}{V\epsilon_0}} q(t)\sin{kz} \ \ \hat{x}$

  • There is also another suggestion in Alain Aspect's lecture Quantum Optics - Canonical quantization of a single mode on youtube, where Aspect suggests that amplitude is $$\sqrt{\frac{\hbar\omega}{V\epsilon_0}},$$ which puts the hamiltonian at $H=\hbar\omega|\alpha|^2$ and the electric field as $E = i \sqrt{\frac{\hbar\omega}{V\epsilon_0}} \alpha(t)\sin{kz}\ \ \textbf{e}$

Which one is correct? or is there what make me confused?

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There isn't a "correct" or an "incorrect" choice. The two are just two different, equivalent ways to handle the book-keeping of the different constants involved in the formalism, and different conventions in terms of working with real-valued quadratures (like Gerry and Knight's $q$ and $p$) or complex-valued ones (like Aspect's $\alpha$).

Neither choice has any impact on the content of the physics.

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  • $\begingroup$ Thank you very much $\endgroup$ – min Fe Sep 25 at 4:25

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