How is the Exponential Term Derived in Time-Dependent Perturbation Theory?

Thread starter Rayan
Start date Feb 18, 2024
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Perturbation Theory Time

In summary, the exponential term in time-dependent perturbation theory is derived from the solution of the time-dependent Schrödinger equation using perturbation methods. It arises when considering a system's state evolving under a time-dependent Hamiltonian, where the first-order approximation leads to a transition amplitude expressed as an integral over time. This integral typically includes a factor that resembles an exponential function, reflecting the influence of the perturbation over time and the oscillatory nature of quantum states. The derivation emphasizes the role of the interaction Hamiltonian and the time evolution operator in determining how quantum states change in response to external influences.

Feb 18, 2024

Rayan

Homework Statement: A free particle of spin 1 is at rest in a magnetic field B, so that
$$ H_0 = −κ B_z S_z $$
where $S_z$ is the projection of the spin operator in the z direction. A harmonic perturbation
with frequency $ ω = κ B_z $ is applied in the x direction for a short time (one period of
oscillation only), so that the weak perturbation is
$$ V (t) = −κ B_x S_x sin(ωt) \, \, \, , \, \, \, for 0 < t < T = (2π/ω) $$

Relevant Equations: Calculate, using time-dependent perturbation theory the probability of observing $S_z = ℏ$ to the lowest non-vanishing order.

So I have the solution here and trying to understand what happened at the beginning of the second row! How did we get the exponential $$e^{i(\omega_m - \omega_0 ) t' }$$ ?

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Feb 18, 2024

Orodruin

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The left exponential acted on the bra and the right exponential acted on the ket. Since ##\ket{m}## is an eigenstate of ##\hat H_0## this replaces it by its corresponding eigenvalue.

FAQ: How is the Exponential Term Derived in Time-Dependent Perturbation Theory?

What is the basic idea behind time-dependent perturbation theory?

Time-dependent perturbation theory is a quantum mechanical framework used to study the effect of a time-dependent perturbation on a system. It is particularly useful for understanding how quantum states evolve over time when the Hamiltonian of the system includes a small time-dependent term in addition to the main, time-independent Hamiltonian.

How is the exponential term introduced in the derivation?

The exponential term arises from the solution to the differential equations governing the time evolution of the quantum states. Specifically, when solving the Schrödinger equation for the perturbed system, the time evolution of the states is often expressed in terms of an exponential function involving the integral of the perturbation over time.

Why is the exponential term important in time-dependent perturbation theory?

The exponential term is crucial because it encapsulates the time evolution of the system due to the perturbation. It allows for the calculation of transition probabilities between different quantum states, providing insights into how the system responds to the time-dependent perturbation.

Can you explain the role of the interaction picture in deriving the exponential term?

In the interaction picture, the state vectors evolve due to the perturbation while the operators evolve according to the unperturbed Hamiltonian. This separation simplifies the derivation of the exponential term, as it isolates the effect of the perturbation, leading to an expression for the state vectors that includes an exponential function of the integral of the perturbation.

How does the exponential term relate to Fermi's Golden Rule?

Fermi's Golden Rule is derived from time-dependent perturbation theory and uses the exponential term to calculate transition rates between quantum states. The rule states that the transition rate is proportional to the square of the matrix element of the perturbation and the density of final states, with the exponential term playing a key role in determining these matrix elements over time.