Time-Dependent Perturbation Theory and Transition Probabilities

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Hello! Welcome to the first lesson in our module on Approximation Methods for Quantum Dynamics.

In the previous module, we delved into the complex but static world of atomic structure, learning how to classify the energy eigenstates of multi-electron atoms using coupling schemes and term symbols. We now pivot from "what are the states?" to a more dynamic question: "how does a system transition between these states?" This is the realm of time-dependent quantum mechanics.

This lesson introduces one of the most powerful and widely used tools in quantum physics: time-dependent perturbation theory. We'll develop the formalism to calculate how a quantum system, initially in a specific energy eigenstate, responds to a small, time-varying external influence. This is the fundamental theory behind processes like the absorption and emission of light by atoms, a topic of immense practical importance.

By the end of this lesson, you will be able to derive the transition probability between two quantum states to first order in the perturbation.

1. The Challenge of Time-Dependence

Let's consider a system described by a Hamiltonian $H_{0}$ whose eigenstates $∣ n ⟩$ and eigenvalues $E_{n}$ we know.
$H_{0} ∣ n ⟩ = E_{n} ∣ n ⟩$
Now, we apply a time-dependent perturbation, $δ H (t)$ , so the full Hamiltonian is:
$H (t) = H_{0} + δ H (t)$
A crucial consequence of this time dependence is that the concept of "energy eigenstates" for the full Hamiltonian $H (t)$ is no longer well-defined. The states of the system are no longer stationary. Our task is to solve the full Time-Dependent Schrödinger Equation (TDSE):
$i ℏ \frac{\partial}{\partial t} ∣Ψ (t)⟩ = H (t) ∣Ψ (t)⟩$
Solving this directly is usually impossible. We need an approximation scheme, and that requires a new way of looking at time evolution.

The following video introduces the standard approach. It explains why our usual methods fail and motivates the use of the interaction picture, a clever hybrid of the Schrödinger and Heisenberg pictures that is perfectly suited for this problem.

L9.1 The interaction picture and time evolution

This video from MIT OpenCourseWare, 'L9.1 The interaction picture and time evolution', sets up the problem of time-dependent perturbations and masterfully derives the Schrödinger equation in the interaction picture.

Please watch from the beginning to 26:35. Pay close attention to: The physical setup of a perturbation that is 'on' for a finite time (4:36). The motivation for the interaction picture: factoring out the 'easy' part of the time evolution due to H₀ (10:07). The final result: a new Schrödinger equation where the evolution is driven only by the perturbation in the interaction picture (15:39).

2. The Interaction Picture and the Perturbative Solution

Let's summarize the key results from the video.

The time evolution due to the full Hamiltonian is complicated. The core idea of the interaction picture is to separate the time evolution generated by $H_{0}$ from that generated by $δ H (t)$ .

We define an interaction picture state $∣ Ψ_{I} (t)⟩$ as:
$∣ Ψ_{I} (t)⟩ \equiv e^{i H_{0} t /ℏ} ∣ Ψ_{S} (t)⟩$
where $∣ Ψ_{S} (t)⟩$ is the usual Schrödinger picture state. This definition essentially "removes" the time evolution that the state would have under $H_{0}$ alone. If $δ H (t) = 0$ , then $∣ Ψ_{I} (t)⟩$ would be constant in time.

The time evolution of this new state is governed by a new Schrödinger equation:
$i ℏ \frac{d}{d t} ∣ Ψ_{I} (t)⟩ = H_{I} (t) ∣ Ψ_{I} (t)⟩$
where $H_{I} (t)$ is the perturbation Hamiltonian in the interaction picture:
$H_{I} (t) \equiv e^{i H_{0} t /ℏ} δ H (t) e^{- i H_{0} t /ℏ}$
This is a powerful result. The evolution of the interaction-picture state is driven solely by the perturbation. This makes it the ideal starting point for a perturbative solution.

To proceed, we expand the unknown state $∣ Ψ_{I} (t)⟩$ in the complete basis of $H_{0}$ eigenstates, $∣ n ⟩$ :
$∣ Ψ_{I} (t)⟩ = \sum_{n} c_{n} (t) ∣ n ⟩$
The coefficients $c_{n} (t)$ are the time-dependent amplitudes for finding the system in state $∣ n ⟩$ . The probability is $P_{n} (t) = ∣ c_{n} (t) ∣^{2}$ . Our goal is to find these coefficients.

The following reading derives the first-order approximation for these coefficients and the resulting transition probability.

Chapter 4 - Time Dependent Perturbation Theory

The document 'Chapter 4 - Time Dependent Perturbation Theory' by Prof. Zwiebach (the lecturer in the video) provides the corresponding written derivation. It's a clear and concise path from the interaction picture to the final formula for first-order transition probability.

Please read section 4.2, 'Perturbative solution', up to equation (4.2.22). Focus on how the equations are solved iteratively, assuming the system starts in a definite state |n⟩ at t=0. The main result you are looking for is the expression for the first-order transition probability, P^(1)_(m←n)(t).

3. First-Order Transition Probability

Let's consolidate the result from the reading. We assume the system starts at $t = 0$ in a specific initial eigenstate $∣ i ⟩$ . We want to find the probability of finding it in a different final state $∣ f ⟩$ (where $f \neq = i$ ) at a later time $t$ .

To first order in the perturbation $δ H$ , the probability amplitude for the transition is:
$c_{f}^{(1)} (t) = \frac{1}{i ℏ} \int_{0}^{t} ⟨ f ∣ H_{I} (t^{'}) ∣ i ⟩ d t^{'}$
Plugging in the definition of $H_{I} (t^{'})$ and using $H_{0} ∣ n ⟩ = E_{n} ∣ n ⟩$ , we get:

⟨ f ∣ H_{I} (t^{'}) ∣ i ⟩ = ⟨ f ∣ e^{i H_{0} t^{'} /ℏ} δ H (t^{'}) e^{- i H_{0} t^{'} /ℏ} ∣ i ⟩ = e^{i (E_{f} - E_{i}) t^{'} /ℏ} ⟨ f ∣ δ H (t^{'}) ∣ i ⟩

Let's define the Bohr frequency for the transition as $ω_{f i} \equiv (E_{f} - E_{i}) /ℏ$ . The amplitude becomes:
$c_{f}^{(1)} (t) = \frac{1}{i ℏ} \int_{0}^{t} e^{i ω_{f i} t^{'}} ⟨ f ∣ δ H (t^{'}) ∣ i ⟩ d t^{'}$
The probability of transition is the modulus squared of this amplitude. This gives us the central formula of first-order time-dependent perturbation theory:

P_{i \to f}^{(1)} (t) = ∣ c_{f}^{(1)} (t) ∣^{2} = \frac{1}{ℏ} \int_{0}^{t} e^{i ω_{f i} t^{'}} ⟨ f ∣ δ H (t^{'}) ∣ i ⟩ d t^{'}^{2}

This equation tells us that the transition probability depends on three things:

The matrix element $⟨ f ∣ δ H (t^{'}) ∣ i ⟩$ , which measures how strongly the perturbation connects the initial and final states. If this is zero, the transition is "forbidden" to first order.
The energy difference between the states, which appears in the oscillating term $e^{i ω_{f i} t^{'}}$ .
The duration over which the perturbation acts, from 0 to $t$ .

4. Application: A Constant Perturbation

To see how this works, let's analyze the simplest case: a perturbation that is constant in time, $δ H (t) = V$ , and is switched on at $t = 0$ . This is a good model for understanding sudden changes to a system.

The following video works through this calculation and, more importantly, analyzes the physical meaning of the result.

L10.2 Transitions with a constant perturbation

In 'L10.2 Transitions with a constant perturbation', Prof. Zwiebach applies the general formula we just derived to the specific case of a constant perturbation.

Please watch from the beginning to 15:19. The key parts are: The straightforward calculation of the integral for the constant perturbation (3:08). The analysis of the resulting probability when E_f ≠ E_i (the non-degenerate case), which leads to oscillations (7:42). The analysis of the special case when E_f = E_i (the degenerate case), which leads to quadratic growth in time (12:56).

Let's analyze the result derived in the video. For a constant perturbation $V$ , the integral is simple, and the transition probability is:

$P_{i \to f}^{(1)} (t) = \frac{∣ ⟨ f ∣ V ∣ i ⟩ ∣ ^{2}}{ℏ ^{2}} \frac{s i n ^{2} ( ω _{f i} t /2 )}{( ω _{f i} /2 ) ^{2}}$

This function has two distinct behaviors:

Case 1: Non-degenerate states ( $E_{f} \neq = E_{i}$ )
The probability oscillates in time between 0 and a maximum value of $\frac{4∣ ⟨ f ∣ V ∣ i ⟩ ∣ ^{2}}{( E _{f} - E _{i} ) ^{2}}$ . The transition is "virtual"—the system borrows energy $Δ E = E_{f} - E_{i}$ for a short time, but the probability to remain in the final state does not grow indefinitely. Notice that the probability is strongly suppressed for transitions to states with very different energies.
Case 2: Degenerate states ( $E_{f} = E_{i}$ )
Here, $ω_{f i} = 0$ . Using the limit $lim_{x \to 0} \frac{s i n ( a x )}{a x} = 1$ , the expression becomes:
$P_{i \to f}^{(1)} (t) = \frac{∣ ⟨ f ∣ V ∣ i ⟩ ∣ ^{2}}{ℏ ^{2}} t^{2}$
The probability grows quadratically with time! This indicates a real, secular transfer into the final state. However, this quadratic growth cannot last forever, as probability is capped at 1. This tells us that first-order perturbation theory breaks down for long times when making transitions between degenerate states.

Test your understanding!

Consider a spin-1/2 particle in a magnetic field $B_{0} z^$ , so $H_{0} = ω_{0} S_{z}$ where $ω_{0} = γ B_{0}$ . The eigenstates are spin-up $∣ ↑ ⟩$ and spin-down $∣ ↓ ⟩$ . At $t = 0$ , a small, constant magnetic field $B_{x} x^$ is applied, so the perturbation is $V = ω_{x} S_{x}$ where $ω_{x} = γ B_{x}$ .

If the particle starts in the spin-up state $∣ ↑ ⟩$ at $t = 0$ , what is the probability of finding it in the spin-down state $∣ ↓ ⟩$ at a later time $t$ , according to first-order perturbation theory? Do these states satisfy the degenerate or non-degenerate case?

Show answer

Identify states and energies:
The initial state is $∣ i ⟩ = ∣ ↑ ⟩$ with energy $E_{i} = + \frac{ℏ ω _{0}}{2}$ .
The final state is $∣ f ⟩ = ∣ ↓ ⟩$ with energy $E_{f} = - \frac{ℏ ω _{0}}{2}$ .
Since $E_{f} \neq = E_{i}$ , this is the non-degenerate case. The Bohr frequency is $ω_{f i} = (E_{f} - E_{i}) /ℏ = (- \frac{ℏ ω _{0}}{2} - \frac{ℏ ω _{0}}{2}) /ℏ = - ω_{0}$ .
Calculate the matrix element:
We need $⟨ ↓ ∣ V ∣ ↑ ⟩ = ω_{x} ⟨ ↓ ∣ S_{x} ∣ ↑ ⟩$ .
Using $S_{x} = \frac{ℏ}{2} (∣ ↑ ⟩ ⟨ ↓ ∣ + ∣ ↓ ⟩ ⟨ ↑ ∣)$ , we find:
$⟨ ↓ ∣ S_{x} ∣ ↑ ⟩ = \frac{ℏ}{2} ⟨ ↓ ∣ (∣ ↑ ⟩ ⟨ ↓ ∣ + ∣ ↓ ⟩ ⟨ ↑ ∣) ∣ ↑ ⟩ = \frac{ℏ}{2} ⟨ ↓ ∣ ↓ ⟩ = \frac{ℏ}{2}$ .
So, the matrix element is $⟨ ↓ ∣ V ∣ ↑ ⟩ = \frac{ℏ ω _{x}}{2}$ .
Apply the formula:
Using the formula for the non-degenerate case:
$P_{↑\to↓}^{(1)} (t) = \frac{∣ ⟨ ↓ ∣ V ∣ ↑ ⟩ ∣ ^{2}}{ℏ ^{2}} \frac{s i n ^{2} ( ω _{f i} t /2 )}{( ω _{f i} /2 ) ^{2}}$
$P_{↑\to↓}^{(1)} (t) = \frac{∣ℏ ω _{x} /2 ∣ ^{2}}{ℏ ^{2}} \frac{s i n ^{2} ( - ω _{0} t /2 )}{( - ω _{0} /2 ) ^{2}} = \frac{ℏ ^{2} ω _{x}^{2} /4}{ℏ ^{2}} \frac{s i n ^{2} ( ω _{0} t /2 )}{ω _{0}^{2} /4} = (\frac{ω _{x}}{ω _{0}})^{2} sin^{2} (\frac{ω _{0} t}{2})$
This is the probability of a spin flip. It oscillates with the Larmor frequency $ω_{0}$ , and its amplitude is proportional to $(B_{x} / B_{0})^{2}$ , which makes sense for a small perturbation.

Conclusion

Today we have built the essential machinery for calculating how quantum systems respond to external influences. This is a significant step from analyzing static systems to understanding quantum dynamics.

Key Takeaways:

For a time-dependent Hamiltonian $H = H_{0} + δ H (t)$ , the interaction picture is the ideal framework. In this picture, the state evolution is driven only by the perturbation $H_{I} (t) = e^{i H_{0} t /ℏ} δ H (t) e^{- i H_{0} t /ℏ}$ .
To first order, the probability of a transition from an initial state $∣ i ⟩$ to a final state $∣ f ⟩$ is given by:
$P_{i \to f}^{(1)} (t) = \frac{1}{ℏ} \int_{0}^{t} e^{i ω_{f i} t^{'}} ⟨ f ∣ δ H (t^{'}) ∣ i ⟩ d t^{'}^{2}$
For a constant perturbation turned on at $t = 0$ , this leads to an oscillating probability for $E_{f} \neq = E_{i}$ and a probability that grows as $t^{2}$ for $E_{f} = E_{i}$ .

Preview of the next lesson:
The $t^{2}$ growth for degenerate states and the oscillating behavior for non-degenerate states might seem a bit strange. In many real-world scenarios, like an atom absorbing light and becoming ionized, we observe a constant transition rate, meaning the probability grows linearly with time. How do we get from our current result to a linear rate? The answer lies in considering transitions not to a single final state, but to a continuum of final states. In our next lesson, we will do exactly this, leading us to one of the most celebrated results in quantum mechanics: Fermi's Golden Rule.