Quantum Dynamics: Pictures and Transformations

The Three Pictures of Quantum Dynamics

As the video outlined, the different pictures are merely different ways of bookkeeping. The physically measurable quantities, namely the expectation values of operators, must remain the same regardless of the picture we use. This invariance is the key to transforming between them. An expectation value is a matrix element of the form $⟨A⟩=⟨\psi (t)|A|\psi (t)⟩$. By strategically inserting the identity operator $1=U^{†}U$, where $U$ is a unitary operator, we can shift the time dependence from the state vector $|\psi (t)⟩$ to the operator $A$, or distribute it between them.

The following text from a set of course notes by Prof. Curt Wittig provides a rigorous yet clear derivation of each picture. We will work through it step by step.

1. The Schrödinger Picture

This is the default framework we have used so far. Let's formalize its definition.

• State Vectors: The state vectors, denoted $|{\psi }_S(t)⟩$, contain the full time evolution of the system. They evolve according to the time-dependent Schrödinger equation:

  $$i\hslash \frac{d}{dt}|{\psi }_S(t)⟩=H_S|{\psi }_S(t)⟩$$

  The solution is, of course, $|{\psi }_S(t)⟩=U(t,t_0)|{\psi }_S(t_0)⟩$, where $U(t,t_0)$ is the time-evolution operator generated by the full Hamiltonian, $H_S$.

• Operators: Operators, denoted $A_S$, are typically time-independent unless they contain explicit time-dependence (e.g., an externally applied, time-varying field).

The expectation value is calculated as $⟨A⟩=⟨{\psi }_S(t)|A_S|{\psi }_S(t)⟩$.

Please consult the reading to solidify this definition.

2. The Heisenberg Picture

The Heisenberg picture offers a perspective that is often more aligned with classical intuition, where dynamical variables like position and momentum evolve in time. Here, the entire time evolution is transferred to the operators, while the state vectors remain fixed.

Let's derive this from the invariance of the expectation value:

Using the fact that $U$ is unitary ($U^{†}U=1$), we can rewrite this as:

This expression motivates the following definitions for the Heisenberg picture:

• State Vector: The Heisenberg state vector is constant in time, fixed at its initial value: $$|{\psi }_H⟩\equiv |{\psi }_S(t_0)⟩$$
• Operator: The Heisenberg operator absorbs all the time evolution: $$A_H(t)\equiv U^{†}(t,t_0)A_{S}U(t,t_0)$$

Notice that at $t=t_0$, $U(t_0,t_0)=1$, so the Schrödinger and Heisenberg pictures coincide.

The Heisenberg Equation of Motion

Since operators now evolve, they must obey an equation of motion. By differentiating $A_H(t)$ with respect to time and using the Schrödinger equation for $U$, we arrive at the Heisenberg equation of motion:

where $H_H=U^{†}{H}_{S}U$ and the last term represents the transformation of any explicit time dependence in the original Schrödinger operator. For a time-independent Hamiltonian where $A_S$ has no explicit time dependence, this simplifies to $\frac{d{A}_H}{dt}=\frac{1}{i\hslash }[A_H,H_S]$. This equation is the direct quantum analogue of the classical equation of motion involving Poisson brackets.

Now, let's turn back to the reading for a formal derivation.

3. The Interaction (Dirac) Picture

The Interaction picture is a powerful hybrid approach, essential for time-dependent perturbation theory, scattering theory, and QFT. It is used when the Hamiltonian can be split into two parts:

Here, $H_0$ is a time-independent, solvable part (e.g., the free-particle Hamiltonian), and $V(t)$ is a "perturbation" or "interaction" that is difficult to handle.

The core idea is to let the operators evolve according to the "easy" Hamiltonian $H_0$, while the state vectors evolve under the influence of the "difficult" interaction $V(t)$. This isolates the effect of the perturbation. The following video provides excellent physical motivation for this strategy.

As Prof. Zwiebach derived, we define the Interaction picture entities by transforming with the evolution operator associated with $H_0$, which we'll call $U_0(t,t_0)=e^{-i{H}_0(t-t_0)/\hslash }$.

• State Vector: The state vector evolves only due to the interaction. $$|{\psi }_I(t)⟩\equiv U_0^{†}(t,t_0)|{\psi }_S(t)⟩$$
• Operator: The operator evolves under the unperturbed Hamiltonian $H_0$. $$A_I(t)\equiv U_0^{†}(t,t_0)A_S{U}_0(t,t_0)$$

Equations of Motion in the Interaction Picture

By differentiating these definitions, we find two separate equations of motion:

1. The state vector evolves according to a new Schrödinger-like equation, but governed only by the interaction part of the Hamiltonian:

   $$i\hslash \frac{d}{dt}|{\psi }_I(t)⟩=V_I(t)|{\psi }_I(t)⟩$$

   where $V_I(t)=U_0^{†}(t,t_0)V(t)U_0(t,t_0)$ is the interaction Hamiltonian in the interaction picture.

2. The operators evolve according to a Heisenberg-like equation, but governed by the unperturbed Hamiltonian $H_0$:

   $$\frac{d{A}_I(t)}{dt}=\frac{1}{i\hslash }[A_I(t),H_0]+{\left(\frac{\partial A_S}{\partial t}\right)}_I$$

This separation is immensely useful. The rapid oscillations associated with $H_0$ are absorbed into the operators, leaving a hopefully "slower" evolution for the states, which can then be solved perturbatively using the Dyson series for $V_I(t)$.

Let's complete our study of the written notes for the formal derivation.

The table you just reviewed is a crucial summary. It encapsulates the definitions of states and operators, their governing equations, and the time-evolution operators in each picture. It's a valuable reference for navigating between these different but equivalent formulations of quantum dynamics.

Conclusion

In this lesson, we have moved beyond a single view of quantum dynamics to appreciate three distinct but equivalent frameworks. Each picture offers a different strategic advantage depending on the physical problem.

Key Takeaways:

• Quantum dynamics can be formulated in the Schrödinger, Heisenberg, or Interaction pictures, which differ in how they assign time evolution to state vectors and operators.
• In the Schrödinger picture, states evolve via the full Hamiltonian $H$, while operators are static. This is often the most intuitive picture for wave mechanics.
• In the Heisenberg picture, states are static, while operators evolve via the full Hamiltonian $H$. This picture's equation of motion closely resembles classical mechanics.
• In the Interaction picture, the Hamiltonian is split $H=H_0+V$. States evolve via the interaction part $V_I$, while operators evolve via the unperturbed part $H_0$. This is the cornerstone of time-dependent perturbation theory.
• The choice of picture is a matter of convenience; all three yield identical results for physical observables.

Understanding these different perspectives on time evolution is crucial. It not only provides practical tools for solving complex problems but also deepens our understanding of the role of time in quantum mechanics. In our next lesson, we will explore the time-energy uncertainty relation, whose interpretation is subtler than its position-momentum counterpart, partly because time is a parameter, not an operator, in the formalisms we've studied today.