A Jaynes-Cummings Hamiltonian: Where did the time dependence go?

yucheng
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Semiclassically, the electric field varies harmonically in time, but sometimes, in the JC Hamiltonian, the time dependence disappears. What???
Consider the interaction of a two level atom and an electric field (semiclassically, we treat the field as 'external' i.e. not influenced by the atom; the full quantum treats the change in the field as well)

Electric field in semiclassical Hamiltonian: plane wave

##H_{int,~semiclassical}=-\mu \cdot E=-\mu \cdot E_{0}\cos \nu t##

Electric field in Jaynes-Cummings Hamiltonian, single mode i.e. plane wave
(Schrodinger picture)

##H_{int}=\hbar g(\sigma _{+}a+\sigma _{-}a^{\dagger })##

\bigskip

We realize ##H_{int}## is time independent! So where did the time-harmonic
dependence go? How does this compare to the classical case?

Also, how are we supposed to go to the interaction picture, with a constant
hamiltonian?

Furthermore, ##H_{int,~semiclassical}## is time-dependent, but isn't this the
Schrodinger picture Hamiltonian? Shouldn't it be time-independent?

Possibly related: Sakurai and Napolitano, Modern Quantum Mechanics: constant
perturbation turned on at t=0!?

Also see:
https://www.physicsforums.com/threa...hen-the-hamiltonian-is-time-dependent.971007/
 
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It is also possible to add a time-dependent drive term to the J-C Hamiltonian.
However, in the simples case you have a situation where the energy is continuously moving between the cavity and the two-level systems; i.e. it is a closed system and there is no real time-dependence.

Note that the J-C Hamiltonian in the "strong driving" regime gets really complicated.
 
Demystifier said:
In the JC Hamiltonian, the field is quantized. It's not a semiclassical approximation.
Yes I understand that it's quantized, but... why must find dependence disappear if it's quantized?

f95toli said:
it is a closed system and there is no real time-dependence.
Perhaps this is a good hint. Closed system=energy conservation=no time dependence, but:
How do we know if the system is closed?
Why no time dependence=energy conservation?....
 
yucheng said:
Yes I understand that it's quantized, but... why must find dependence disappear if it's quantized?
Because the time dependence of operators usually disappears in the Schrodinger picture.
 
If we release an electron around a positively charged sphere, the initial state of electron is a linear combination of Hydrogen-like states. According to quantum mechanics, evolution of time would not change this initial state because the potential is time independent. However, classically we expect the electron to collide with the sphere. So, it seems that the quantum and classics predict different behaviours!
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. Towards the end of the first lecture for the Qiskit Global Summer School 2025, Foundations of Quantum Mechanics, Olivia Lanes (Global Lead, Content and Education IBM) stated... Source: https://www.physicsforums.com/insights/quantum-entanglement-is-a-kinematic-fact-not-a-dynamical-effect/ by @RUTA

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