Quantum Dynamics, Time Evolution & Alternative Pictures

1. Postulate of Unitary Time Evolution

In quantum mechanics, if a system is prepared in state $|\psi(t_0)\rangle$ at initial time $t_0$, its subsequent state $|\psi(t)\rangle$ at any time $t > t_0$ is generated by a linear time-evolution operator $\hat{U}(t, t_0)$:

$$|\psi(t)\rangle = \hat{U}(t, t_0)|\psi(t_0)\rangle$$

Conservation of total probability requires that the norm of the state vector be invariant under temporal evolution:

$$\langle \psi(t) | \psi(t) \rangle = \langle \psi(t_0) | \hat{U}^\dagger(t, t_0) \hat{U}(t, t_0) | \psi(t_0) \rangle = \langle \psi(t_0) | \psi(t_0) \rangle = 1$$

This holds for all initial states if and only if $\hat{U}(t, t_0)$ is strictly unitary:

$$\hat{U}^\dagger(t, t_0) \hat{U}(t, t_0) = \hat{U}(t, t_0) \hat{U}^\dagger(t, t_0) = \hat{I}$$

Additionally, the evolution operator satisfies the group composition property:

$$\hat{U}(t, t_1)\hat{U}(t_1, t_0) = \hat{U}(t, t_0), \quad \hat{U}(t_0, t_0) = \hat{I}, \quad \hat{U}^{-1}(t, t_0) = \hat{U}(t_0, t)$$

2. Infinitesimal Time Evolution and the Hamiltonian Generator

Consider an infinitesimal time step $dt$. Since $\hat{U}(t_0, t_0) = \hat{I}$, $\hat{U}(t_0 + dt, t_0)$ can be expanded to first order in $dt$:

$$\hat{U}(t_0 + dt, t_0) = \hat{I} - \frac{i}{\hbar}\hat{\Omega} \, dt$$

The unitarity condition $(\hat{I} + \frac{i}{\hbar}\hat{\Omega}^\dagger dt)(\hat{I} - \frac{i}{\hbar}\hat{\Omega} dt) = \hat{I} + \frac{i}{\hbar}(\hat{\Omega}^\dagger - \hat{\Omega})dt + \mathcal{O}(dt^2) = \hat{I}$ demands that $\hat{\Omega}^\dagger = \hat{\Omega}$. By Planck's quantum hypothesis, the Hermitian generator of time translations is the energy observable, the Hamiltonian $\hat{H}$:

$$\hat{\Omega} = \hat{H}$$

Forming the difference quotient:

$$\frac{\partial}{\partial t}\hat{U}(t, t_0) = \lim_{dt \to 0} \frac{\hat{U}(t+dt, t_0) - \hat{U}(t, t_0)}{dt} = \lim_{dt \to 0} \frac{(\hat{I} - \frac{i}{\hbar}\hat{H}(t)dt - \hat{I})\hat{U}(t, t_0)}{dt}$$ $$i\hbar \frac{\partial}{\partial t}\hat{U}(t, t_0) = \hat{H}(t)\hat{U}(t, t_0)$$

3. Integration Cases: From Conservative Systems to the Dyson Series

Three cases govern the explicit mathematical form of $\hat{U}(t, t_0)$:

1. Time-Independent Hamiltonian ($\hat{H} \ne f(t)$): $$\hat{U}(t, t_0) = \exp\left(-\frac{i}{\hbar}\hat{H}(t - t_0)\right) = \sum_{n=0}^\infty \frac{1}{n!} \left(-\frac{i}{\hbar}\hat{H}(t - t_0)\right)^n$$
2. Commuting Time-Dependent Hamiltonian ($[\hat{H}(t_1), \hat{H}(t_2)] = 0$): $$\hat{U}(t, t_0) = \exp\left(-\frac{i}{\hbar}\int_{t_0}^t \hat{H}(t')\,dt'\right)$$
3. General Non-Commuting Time-Dependent Hamiltonian: Direct integration leads to the integral equation: $$\hat{U}(t, t_0) = \hat{I} - \frac{i}{\hbar}\int_{t_0}^t \hat{H}(t_1)\hat{U}(t_1, t_0)\,dt_1$$ Iterative substitution generates the famous Dyson series: $$\hat{U}(t, t_0) = \hat{I} + \sum_{n=1}^\infty \left(-\frac{i}{\hbar}\right)^n \int_{t_0}^t dt_1 \int_{t_0}^{t_1} dt_2 \cdots \int_{t_0}^{t_{n-1}} dt_n \hat{H}(t_1)\hat{H}(t_2)\cdots \hat{H}(t_n)$$ Using the time-ordering meta-operator $\mathcal{T}$, which rearranges operators in descending chronological order: $$\hat{U}(t, t_0) = \mathcal{T}\exp\left(-\frac{i}{\hbar}\int_{t_0}^t \hat{H}(t')\,dt'\right)$$

1. The Schrödinger Equation of Motion

In the Schrödinger picture, all time dependence resides within the state vectors $|\psi_S(t)\rangle$, while observable operators $\hat{A}_S$ remain constant in time (unless they possess explicit parametric time dependence, such as a time-varying external electric field).

Applying the time evolution operator equation to $|\psi_S(t)\rangle = \hat{U}(t, 0)|\psi_S(0)\rangle$:

$$i\hbar \frac{\partial}{\partial t}|\psi_S(t)\rangle = i\hbar \frac{\partial \hat{U}(t, 0)}{\partial t}|\psi_S(0)\rangle = \hat{H}\hat{U}(t, 0)|\psi_S(0)\rangle$$ $$i\hbar \frac{\partial}{\partial t}|\psi_S(t)\rangle = \hat{H}|\psi_S(t)\rangle$$

Projecting this abstract ket equation onto continuous coordinate eigenstates $\langle x|$ yields the familiar partial differential Schrödinger wave equation:

$$i\hbar \frac{\partial \psi(x, t)}{\partial t} = \left(-\frac{\hbar^2}{2m}\nabla^2 + V(x)\right)\psi(x, t)$$

2. Stationary States and Constant Expectation Values

If the Hamiltonian is time-independent, its energy eigenstates satisfy $\hat{H}|E_n\rangle = E_n |E_n\rangle$. Expanding the initial state in the energy basis:

$$|\psi_S(0)\rangle = \sum_n c_n |E_n\rangle \implies |\psi_S(t)\rangle = e^{-i\hat{H}t/\hbar}\sum_n c_n |E_n\rangle = \sum_n c_n e^{-i E_n t/\hbar} |E_n\rangle$$

If the system is prepared in a pure energy eigenstate $|\psi_S(0)\rangle = |E_k\rangle$, its state vector evolves solely by an overall global phase factor:

$$|\psi_S(t)\rangle = e^{-i E_k t/\hbar}|E_k\rangle$$

The expectation value of any time-independent observable $\hat{A}_S$ in this state is strictly constant:

$$\langle \hat{A}_S \rangle(t) = \langle \psi_S(t) | \hat{A}_S | \psi_S(t) \rangle = e^{iE_k t/\hbar} \langle E_k | \hat{A}_S | E_k \rangle e^{-iE_k t/\hbar} = \langle E_k | \hat{A}_S | E_k \rangle$$

Hence, energy eigenstates are termed stationary states: neither probability densities nor physical expectation values depend on time.

1. Formulation of the Heisenberg Picture

Werner Heisenberg formulated quantum dynamics by holding state vectors fixed at their initial values while transferring the entire temporal evolution onto the operators. By definition, expectation values must be identical in both pictures:

$$\langle \hat{A} \rangle(t) = \langle \psi_S(t) | \hat{A}_S | \psi_S(t) \rangle = \langle \psi_S(0) | \hat{U}^\dagger(t, 0) \hat{A}_S \hat{U}(t, 0) | \psi_S(0) \rangle \equiv \langle \psi_H | \hat{A}_H(t) | \psi_H \rangle$$

This establishes the unitary transformation between Schrödinger and Heisenberg pictures:

$$|\psi_H\rangle \equiv |\psi_S(0)\rangle, \quad \frac{\partial}{\partial t}|\psi_H\rangle = 0$$ $$\hat{A}_H(t) \equiv \hat{U}^\dagger(t, 0) \hat{A}_S \hat{U}(t, 0)$$

2. The Heisenberg Equation of Motion

Differentiating the Heisenberg operator $\hat{A}_H(t)$ with respect to time:

$$\frac{d\hat{A}_H}{dt} = \frac{\partial \hat{U}^\dagger}{\partial t} \hat{A}_S \hat{U} + \hat{U}^\dagger \hat{A}_S \frac{\partial \hat{U}}{\partial t} + \hat{U}^\dagger \left(\frac{\partial \hat{A}_S}{\partial t}\right) \hat{U}$$

Since $i\hbar \frac{\partial \hat{U}}{\partial t} = \hat{H}\hat{U} \implies \frac{\partial \hat{U}}{\partial t} = -\frac{i}{\hbar}\hat{H}\hat{U}$, and taking the Hermitian adjoint yields $\frac{\partial \hat{U}^\dagger}{\partial t} = \frac{i}{\hbar}\hat{U}^\dagger \hat{H}$:

$$\frac{d\hat{A}_H}{dt} = \frac{i}{\hbar}\hat{U}^\dagger \hat{H} \hat{A}_S \hat{U} - \frac{i}{\hbar}\hat{U}^\dagger \hat{A}_S \hat{H} \hat{U} + \left(\frac{\partial \hat{A}_S}{\partial t}\right)_H$$

Noting that $\hat{U}^\dagger \hat{H} \hat{U} = \hat{H}_H$, we arrive at the celebrated Heisenberg Equation of Motion:

$$\frac{d\hat{A}_H}{dt} = \frac{1}{i\hbar}[\hat{A}_H, \hat{H}_H] + \left(\frac{\partial \hat{A}}{\partial t}\right)_H$$

3. Connection to Classical Mechanics: Poisson Brackets & Ehrenfest's Theorem

The Heisenberg equation of motion displays a profound isomorphism with the classical Hamilton-Jacobi equation:

$$\frac{df}{dt} = \{f, H\}_{\text{classical}} + \frac{\partial f}{\partial t}$$

Dirac established the canonical quantization postulate by mapping classical Poisson brackets to quantum commutators:

$$\{\cdot, \cdot\}_{\text{classical}} \longleftrightarrow \frac{1}{i\hbar}[\cdot, \cdot]$$

Taking the expectation value of the Heisenberg equation yields Ehrenfest's Theorem:

$$\frac{d}{dt}\langle \hat{A} \rangle = \frac{1}{i\hbar}\langle [\hat{A}, \hat{H}] \rangle + \left\langle \frac{\partial \hat{A}}{\partial t} \right\rangle$$

Specifically, for a particle in a potential $V(\hat{x})$:

$$\frac{d}{dt}\langle \hat{x} \rangle = \frac{\langle \hat{p} \rangle}{m}, \quad \frac{d}{dt}\langle \hat{p} \rangle = -\left\langle \frac{\partial V(\hat{x})}{\partial \hat{x}} \right\rangle$$

Quantum expectation values follow classical Newtonian equations of motion, provided the spatial spread of the wavepacket is small compared to the scale of spatial variation of the potential.

1. Motivation and Partitioning of the Hamiltonian

In many advanced quantum systems, the total Hamiltonian can be split into an unperturbed exactly solvable part $\hat{H}_0$ and a small or time-dependent perturbation $\hat{V}(t)$:

$$\hat{H}(t) = \hat{H}_0 + \hat{V}(t)$$

The Dirac (interaction) picture provides an intermediate representation where the fast trivial evolution governed by $\hat{H}_0$ is factored into the operators, while the slow non-trivial transition dynamics governed by $\hat{V}(t)$ is transferred to the state vectors.

2. State Vectors and Operators in the Interaction Picture

The state vector in the interaction picture is defined by stripping off the unperturbed evolution:

$$|\psi_I(t)\rangle \equiv e^{i\hat{H}_0 t/\hbar}|\psi_S(t)\rangle$$

For observables to yield identical expectation values, $\langle \psi_S(t) | \hat{A}_S | \psi_S(t) \rangle = \langle \psi_I(t) | \hat{A}_I(t) | \psi_I(t) \rangle$:

$$\hat{A}_I(t) \equiv e^{i\hat{H}_0 t/\hbar}\hat{A}_S e^{-i\hat{H}_0 t/\hbar}$$

3. Dual Equations of Motion

Differentiating the interaction state ket $|\psi_I(t)\rangle$:

$$i\hbar \frac{\partial}{\partial t}|\psi_I(t)\rangle = i\hbar\left(\frac{i\hat{H}_0}{\hbar}e^{i\hat{H}_0 t/\hbar}|\psi_S(t)\rangle + e^{i\hat{H}_0 t/\hbar}\frac{\partial |\psi_S(t)\rangle}{\partial t}\right)$$ $$= -\hat{H}_0|\psi_I(t)\rangle + e^{i\hat{H}_0 t/\hbar}(\hat{H}_0 + \hat{V}(t))|\psi_S(t)\rangle$$ $$= -\hat{H}_0|\psi_I(t)\rangle + \hat{H}_0|\psi_I(t)\rangle + e^{i\hat{H}_0 t/\hbar}\hat{V}(t)e^{-i\hat{H}_0 t/\hbar}|\psi_I(t)\rangle$$ $$i\hbar \frac{\partial}{\partial t}|\psi_I(t)\rangle = \hat{V}_I(t)|\psi_I(t)\rangle$$

State vectors evolve solely under the interaction potential $\hat{V}_I(t)$. Conversely, differentiating the operator $\hat{A}_I(t)$ yields:

$$\frac{d\hat{A}_I}{dt} = \frac{1}{i\hbar}[\hat{A}_I, \hat{H}_0] + \left(\frac{\partial \hat{A}}{\partial t}\right)_I$$

Operators evolve strictly under the unperturbed Hamiltonian $\hat{H}_0$. The interaction picture is the foundational framework for time-dependent perturbation theory, S-matrix scattering, and relativistic quantum field theory.

1. Heisenberg Equations of Motion for Coordinate and Momentum

Consider the one-dimensional harmonic oscillator with time-independent Hamiltonian:

$$\hat{H} = \frac{\hat{p}^2}{2m} + \frac{1}{2}m\omega^2 \hat{x}^2$$

Applying the Heisenberg equation of motion to the position operator $\hat{x}_H(t)$:

$$\frac{d\hat{x}_H}{dt} = \frac{1}{i\hbar}[\hat{x}_H, \hat{H}] = \frac{1}{i\hbar}\left[\hat{x}_H, \frac{\hat{p}_H^2}{2m}\right] = \frac{1}{i\hbar}\frac{1}{2m}(2i\hbar\hat{p}_H) = \frac{\hat{p}_H(t)}{m}$$

Applying it to the momentum operator $\hat{p}_H(t)$:

$$\frac{d\hat{p}_H}{dt} = \frac{1}{i\hbar}[\hat{p}_H, \hat{H}] = \frac{1}{i\hbar}\left[\hat{p}_H, \frac{1}{2}m\omega^2\hat{x}_H^2\right] = \frac{1}{i\hbar}\frac{1}{2}m\omega^2(-2i\hbar\hat{x}_H) = -m\omega^2\hat{x}_H(t)$$

2. Exact Operator Solutions

Differentiating $\frac{d\hat{x}_H}{dt}$ once more with respect to time:

$$\frac{d^2\hat{x}_H}{dt^2} = \frac{1}{m}\frac{d\hat{p}_H}{dt} = -\omega^2 \hat{x}_H(t)$$

This is an operator second-order linear differential equation, with the exact operator solution:

$$\hat{x}_H(t) = \hat{x}_H(0)\cos(\omega t) + \frac{\hat{p}_H(0)}{m\omega}\sin(\omega t)$$ $$\hat{p}_H(t) = m\frac{d\hat{x}_H}{dt} = \hat{p}_H(0)\cos(\omega t) - m\omega \hat{x}_H(0)\sin(\omega t)$$

Notice that the equal-time commutator remains invariant for all time $t$:

$$[\hat{x}_H(t), \hat{p}_H(t)] = [\hat{x}(0)\cos\omega t + \frac{\hat{p}(0)}{m\omega}\sin\omega t, \hat{p}(0)\cos\omega t - m\omega\hat{x}(0)\sin\omega t]$$ $$= \cos^2(\omega t)[\hat{x}(0), \hat{p}(0)] - \sin^2(\omega t)[\frac{\hat{p}(0)}{m\omega}, m\omega\hat{x}(0)] = (\cos^2\omega t + \sin^2\omega t)[\hat{x}(0), \hat{p}(0)] = i\hbar\hat{I}$$

3. Evolution of Ladder Operators

In terms of the creation and annihilation operators:

$$\frac{d\hat{a}_H}{dt} = \frac{1}{i\hbar}[\hat{a}_H, \hbar\omega(\hat{a}_H^\dagger\hat{a}_H + 1/2)] = \frac{\hbar\omega}{i\hbar}[\hat{a}_H, \hat{a}_H^\dagger\hat{a}_H] = -i\omega\hat{a}_H(t)$$ $$\implies \hat{a}_H(t) = \hat{a}_H(0)e^{-i\omega t}, \quad \hat{a}_H^\dagger(t) = \hat{a}_H^\dagger(0)e^{+i\omega t}$$

This illustrates that in the Heisenberg picture, the ladder operators rotate in the complex phase plane at the classical frequency $\omega$.

1. Two-Level State Space Hamiltonian

Consider a quantum two-level system with unperturbed basis states $\{|1\rangle, |2\rangle\}$, energies $E_1 = \hbar\omega_1$ and $E_2 = \hbar\omega_2$, coupled by an oscillating classical driving field of angular frequency $\omega$:

$$\hat{H}(t) = \begin{pmatrix} E_1 & V_{12} e^{i\omega t} \\ V_{21} e^{-i\omega t} & E_2 \end{pmatrix}, \quad V_{21} = V_{12}^* \equiv \frac{\hbar\Omega_R}{2}$$

where $\Omega_R = \frac{2|V_{12}|}{\hbar}$ is the on-resonance Rabi frequency. Expanding the general state vector $|\psi(t)\rangle = c_1(t)|1\rangle + c_2(t)|2\rangle$, the time-dependent Schrödinger equation $i\hbar \frac{d}{dt}\begin{pmatrix} c_1 \\ c_2 \end{pmatrix} = \hat{H}\begin{pmatrix} c_1 \\ c_2 \end{pmatrix}$ yields the coupled system:

$$i\dot{c}_1 = \omega_1 c_1 + \frac{\Omega_R}{2} e^{i\omega t} c_2$$ $$i\dot{c}_2 = \omega_2 c_2 + \frac{\Omega_R}{2} e^{-i\omega t} c_1$$

2. Rotating Wave Transformation & Exact Rabi Solution

Defining the transition frequency $\omega_0 \equiv \omega_2 - \omega_1$ and the detuning $\Delta \equiv \omega - \omega_0$, we transform amplitudes to remove high-frequency oscillations:

$$c_1(t) = a_1(t) e^{-i\omega_1 t}, \quad c_2(t) = a_2(t) e^{-i(\omega_2 - \Delta)t}$$

Assuming the system is initially prepared in state $|1\rangle$, so $a_1(0) = 1$ and $a_2(0) = 0$, solving the coupled differential equations yields the probability of finding the system in excited state $|2\rangle$ at time $t$:

$$\mathcal{P}_{1\to 2}(t) = |c_2(t)|^2 = \frac{\Omega_R^2}{\Omega_{\text{eff}}^2} \sin^2\left(\frac{\Omega_{\text{eff}} t}{2}\right)$$

where the generalized effective Rabi frequency $\Omega_{\text{eff}}$ is defined by:

$$\Omega_{\text{eff}} \equiv \sqrt{\Omega_R^2 + \Delta^2}$$

3. Physical Interpretation: Resonance, Power Broadening & Flopping

Two key physical consequences emerge:

1. Complete Inversion at Resonance ($\Delta = 0$): When the driving frequency exactly matches the Bohr transition frequency ($\omega = \omega_0$), the transition probability simplifies to: $$\mathcal{P}_{1\to 2}(t) = \sin^2\left(\frac{\Omega_R t}{2}\right)$$ At time $t = \pi / \Omega_R$ (termed a $\pi$-pulse), the population is completely inverted into state $|2\rangle$ with 100% probability. A pulse of duration $t = \pi / (2\Omega_R)$ (a $\pi/2$-pulse) creates an equal coherent superposition $\frac{1}{\sqrt{2}}(|1\rangle - i|2\rangle)$.
2. Detuning Suppression: As the detuning $|\Delta|$ increases, the maximum possible transition probability drops to $\left(\frac{\Omega_R}{\Omega_{\text{eff}}}\right)^2 = \frac{\Omega_R^2}{\Omega_R^2 + \Delta^2} < 1$, while the oscillation frequency increases to $\Omega_{\text{eff}} > \Omega_R$.