Time-Dependent Potentials

The Interaction Picture

Consider $H = H_{0} + V (t)$ with $H_{0}$ being time-independent.

We then have a state: $| α, t_{0}; t ⟩ = \exp (\frac{i}{ℏ} H_{0} (t - t_{0})) | α, t_{0}; t ⟩_{S}$ .

Operators are: $A_{I} = \exp (\frac{i}{ℏ} H_{0} t) A_{S} \exp (- \frac{i}{ℏ} H_{0} t)$ .

Equation of state: $\frac{d A_{I}}{d t} = \frac{1}{i ℏ} [A_{I}, H_{0}]$ .

Assuming $t_{0} = 0$ for simplicity, $i ℏ \frac{\partial}{\partial t} | α, t_{0}; t ⟩_{I} = i ℏ \frac{\partial}{\partial t} (\exp (\frac{i}{ℏ} H_{0} t) | α, t_{0}; t ⟩_{S}) = - H_{0} \exp (\frac{i}{ℏ} H_{0} t) | α, t_{0}; t ⟩_{S} + \exp (\frac{i}{ℏ} H_{0} t) i ℏ \frac{\partial}{\partial t} | α, t_{0}; t ⟩_{S} = - H_{0} \exp (\frac{i}{ℏ} H_{0} t) | α, t_{0}; t ⟩_{S} + \exp (\frac{i}{ℏ} H_{0} t) (H_{0, S} + V_{(S)} (t)) | α, t_{0}; t ⟩_{S} = \exp (\frac{i}{ℏ} H_{0} t) V_{(S)} (t) | α, t_{0}; t ⟩_{S} = V_{I} \exp (\frac{i}{ℏ} H_{0} t) | α, t_{0}; t ⟩_{S} = V_{I} | α, t_{0}; t ⟩_{I}$ .

Thus,

\begin{matrix} (1) & i ℏ \frac{\partial}{\partial t} | α, t_{0}; t ⟩_{I} = V_{I} | α, t_{0}; t ⟩_{I} . \end{matrix}

Therefore, the time dependence is solely due to the time dependence of the potential.

Note: $V_{(S)} (t) = \exp (- \frac{i}{ℏ} H_{0} t) V_{I} \exp (\frac{i}{ℏ} H_{0} t)$ .

High-Level Example

$H_{0} | n ⟩ = E_{n} | n ⟩$ . Assume we have an initial state $| i ⟩$ . Now we turn on the time-dependent potential, $H = H_{0} + V (t)$ . What is/are our final state(s), with what probabilities?

$| α, t_{0}; t ⟩ = \sum_{n} c_{n} (t) | n ⟩$ . $P (t) = | c_{n} (t) |^{2}$ .

Ansatz: $| α, t_{0}; t ⟩ = \sum_{n} c_{n} \exp (- \frac{i}{ℏ} V_{I} t) | n ⟩$ .

More rigorously, letting $| α (t) ⟩ = | α, t_{0}; t ⟩$ ,

\begin{aligned} i ℏ \frac{\partial}{\partial t} | α (t) ⟩_{I} & = V_{I} | α (t) ⟩_{I} \\ i ℏ \frac{\partial}{\partial t} ⟨ n | α (t) ⟩ & = ⟨ n | V_{I} | α (t) ⟩ \\ i ℏ \frac{\partial}{\partial t} c_{n} (t) & = \sum_{m} ⟨ n | V_{I} | m ⟩ ⟨ m | α (t) ⟩ \\ = \sum_{m} ⟨ n | V_{I} | m ⟩ c_{m} (t) \\ = \sum_{m} ⟨ n | \exp (\frac{i}{ℏ} H_{0} t) V (t) \exp (- \frac{i}{ℏ} H_{0} t) | m ⟩ c_{m} (t) \\ = \sum_{m} ⟨ n | \exp (\frac{i}{ℏ} E_{n} t) V (t) \exp (- \frac{i}{ℏ} E_{m} t) | m ⟩ c_{m} (t) \\ = \sum_{m} \exp (\frac{i}{ℏ} (E_{n} - E_{m}) t) ⟨ n | V (t) | m ⟩ c_{m} (t) \\ = \sum_{m} \exp (i ω_{n m} t) V_{n m} (t) c_{m} (t) \\ i ℏ \frac{d c_{n} (t)}{d t} & = \sum_{m} V_{n m} \exp (i ω_{n m}) c_{m} (t) . \\ i ℏ \frac{d}{d t} (\begin{array}{c} {\dot{c}}_{1} \\ {\dot{c}}_{2} \\ \dots \end{array}) & = (\begin{array}{c} V_{11} & V_{12} \exp (i ω_{12} t) & \dots \\ V_{21} \exp (i ω_{21} t) & V_{22} & \dots \\ \dots & \dots & \dots \end{array}) (\begin{array}{c} c_{1} \\ c_{2} \\ \dots \end{array}) . \end{aligned}

This is useful for discrete system, i.e. non-infinite. Consider two-level systems: spin-1/2, laser or maser (NMR) with energies $E_{1}, E_{2}$ .

Example Two-Level System

Let $E_{1}, E_{2}$ , $H_{0} = E_{1} | 1 ⟩ ⟨ 1 | + E_{2} | 2 ⟩ ⟨ 2 |$ . Apply a potential, $V (t) = γ \exp (i ω t) | 1 ⟩ ⟨ 2 | + γ \exp (- i ω t) | 2 ⟩ ⟨ 1 |$ .

$i ℏ {\dot{c}}_{1} = V_{11} c_{1} + c_{2} V_{12} \exp (i ω_{12} t) = c_{2} V_{12} \exp (i ω_{12} t)$ . $i ℏ {\dot{c}}_{2} = V_{22} c_{2} + c_{1} V_{21} \exp (i ω_{21} t) = c_{1} V_{21} \exp (- i ω_{12} t)$ .

$| c_{2} (t) |^{2} = \frac{\frac{γ^{2}}{ℏ^{2}}}{\frac{γ^{2}}{ℏ^{2}} + \frac{(ω + ω_{12})^{2}}{4}} \sin (\sqrt{\frac{γ^{2}}{ℏ^{2}} + \frac{(ω + ω_{12})^{2}}{4}} t) = A \sin Ω t$ . $Ω$ is called the Rabi frequency. At resonance, $ω = ω_{2} 1$ , the amplitude is maximized and the Rabi frequency is $\frac{γ}{ℏ}$ and $| c_{2} (t) |$ can reach 1. Whereas, off resonance, $c_{2} (t)$ has a maximum occupation probability and is never 1.

Schrödinger Picture

State evolves and operators are time-independent.

Heisenberg Picture

State is time-independent and operators evolve.

\begin{matrix} (2) & \frac{d A_{H}}{d t} = \frac{1}{i ℏ} [A_{H}, H] . \end{matrix}