🧮Mathematical Methods in Classical and Quantum Mechanics Unit 14 – Quantum Dynamics: Unitary Evolution

Key Concepts and Foundations

• Quantum dynamics describes the time evolution of quantum systems and how their states change over time
• Governed by the Schrödinger equation, a fundamental equation in quantum mechanics that determines the wave function of a system
• Unitary evolution preserves the normalization and orthogonality of quantum states, ensuring probability conservation
• Observables are represented by Hermitian operators, which have real eigenvalues corresponding to measurable quantities
• Commutation relations between observables determine their compatibility and the uncertainty principle
• Superposition principle allows quantum systems to exist in multiple states simultaneously until measured
• Entanglement is a unique quantum phenomenon where the states of multiple particles are correlated and cannot be described independently

Mathematical Framework

• Hilbert spaces provide the mathematical foundation for quantum mechanics, representing the state space of a quantum system
  • Complex vector spaces equipped with an inner product, allowing for the calculation of probabilities and expectation values
  • Basis vectors span the Hilbert space and can be used to express quantum states
• Linear operators act on the Hilbert space and represent observables and transformations
  • Hermitian operators have real eigenvalues and orthogonal eigenvectors, ensuring measurable quantities
  • Unitary operators preserve inner products and represent symmetries and time evolution
• Dirac notation (bra-ket notation) is a convenient way to represent quantum states and operators
  • Kets $|ψ⟩$ represent state vectors, while bras $⟨ψ|$ represent their dual vectors
  • Inner products are written as $⟨ψ|ϕ⟩$ and outer products as $|ψ⟩⟨ϕ|$
• Tensor products allow for the description of composite quantum systems
  • The Hilbert space of a composite system is the tensor product of the Hilbert spaces of its constituent parts
  • Entangled states cannot be written as a tensor product of individual states

Schrödinger Equation and Time Evolution

• The Schrödinger equation is a linear partial differential equation that describes the time evolution of a quantum system's wave function $Ψ(x, t)$
  • Time-dependent Schrödinger equation: $iℏ\frac{∂}{∂t}Ψ(x, t) = \hat{H}Ψ(x, t)$, where $\hat{H}$ is the Hamiltonian operator
  • Time-independent Schrödinger equation: $\hat{H}ψ(x) = Eψ(x)$, used for stationary states with well-defined energy $E$
• The Hamiltonian operator $\hat{H}$ represents the total energy of the system, consisting of kinetic and potential energy terms
• Solutions to the Schrödinger equation are wave functions that describe the probability distribution of a particle's position and momentum
• Time evolution of a quantum state is given by the unitary operator $\hat{U}(t) = e^{-i\hat{H}t/ℏ}$, which acts on the initial state $|ψ(0)⟩$ to give the state at time $t$: $|ψ(t)⟩ = \hat{U}(t)|ψ(0)⟩$
• The expectation value of an observable $\hat{A}$ in a state $|ψ⟩$ is given by $⟨\hat{A}⟩ = ⟨ψ|\hat{A}|ψ⟩$, which evolves in time according to the Ehrenfest theorem

Unitary Operators and Their Properties

• Unitary operators are linear operators that preserve the inner product between states and ensure probability conservation
  • Defined by the condition $\hat{U}^†\hat{U} = \hat{U}\hat{U}^† = \hat{I}$, where $\hat{U}^†$ is the adjoint of $\hat{U}$ and $\hat{I}$ is the identity operator
  • The adjoint of a unitary operator is its inverse: $\hat{U}^† = \hat{U}^{-1}$
• Unitary operators represent symmetries and transformations in quantum mechanics
  • Examples include rotation operators, translation operators, and the time evolution operator $e^{-i\hat{H}t/ℏ}$
• The product of two unitary operators is also unitary, allowing for the composition of transformations
• Unitary operators have eigenvalues of the form $e^{iθ}$, where $θ$ is a real number called the phase
• The exponential map relates Hermitian operators (generators) to unitary operators through $\hat{U} = e^{i\hat{A}}$, where $\hat{A}$ is a Hermitian operator

Time-Dependent vs Time-Independent Systems

• Time-independent systems have Hamiltonians that do not explicitly depend on time
  • The time-independent Schrödinger equation $\hat{H}ψ(x) = Eψ(x)$ is used to find stationary states and energy eigenvalues
  • Stationary states have wave functions that are separable into spatial and temporal parts: $Ψ(x, t) = ψ(x)e^{-iEt/ℏ}$
• Time-dependent systems have Hamiltonians that explicitly depend on time, $\hat{H}(t)$
  • The time-dependent Schrödinger equation $iℏ\frac{∂}{∂t}Ψ(x, t) = \hat{H}(t)Ψ(x, t)$ is used to describe the evolution of the system
  • The time evolution operator for a time-dependent Hamiltonian is given by the time-ordered exponential $\hat{U}(t) = \mathcal{T}exp(-\frac{i}{ℏ}∫_{0}^{t}\hat{H}(t')dt')$
• Adiabatic approximation applies to systems with slowly varying Hamiltonians
  • If the Hamiltonian changes slowly compared to the system's response time, the system remains in an instantaneous eigenstate of the Hamiltonian
• Sudden approximation applies to systems with rapidly changing Hamiltonians
  • If the Hamiltonian changes rapidly compared to the system's response time, the state of the system remains unchanged

Applications in Quantum Systems

• Quantum harmonic oscillator is a fundamental model for vibrational motion in molecules and lattices
  • Its Hamiltonian is $\hat{H} = \frac{\hat{p}^2}{2m} + \frac{1}{2}mω^2\hat{x}^2$, with evenly spaced energy levels $E_n = ℏω(n + \frac{1}{2})$
  • Coherent states are minimum uncertainty states that behave classically, representing the motion of a classical oscillator
• Quantum two-level systems (qubits) are the building blocks of quantum computation and information
  • Described by a 2D Hilbert space with basis states $|0⟩$ and $|1⟩$, representing the two possible states of the system
  • Unitary operations (quantum gates) manipulate the state of the qubit, enabling quantum algorithms and protocols
• Spin systems and magnetic resonance involve the interaction of spins with magnetic fields
  • The Hamiltonian for a spin in a magnetic field is $\hat{H} = -γ\vec{B} \cdot \vec{S}$, where $γ$ is the gyromagnetic ratio and $\vec{S}$ is the spin operator
  • Larmor precession occurs when a spin precesses around the magnetic field direction with frequency $ω = γB$
• Quantum tunneling allows particles to pass through potential barriers that they classically could not
  • Described by the wave function penetrating the barrier, leading to a non-zero probability of transmission
  • Applications include scanning tunneling microscopy (STM), alpha decay, and Josephson junctions

Problem-Solving Techniques

• Separation of variables is used to solve time-independent Schrödinger equations
  • The wave function is assumed to be separable into spatial and temporal parts, $Ψ(x, t) = ψ(x)T(t)$
  • Leads to separate equations for the spatial and temporal components, which can be solved independently
• Perturbation theory is used when the Hamiltonian can be split into a solvable part $\hat{H}_0$ and a small perturbation $\hat{V}$
  • Time-independent perturbation theory finds approximate energy eigenvalues and eigenstates
    • Corrections to the energy and states are calculated order by order in the perturbation strength
  • Time-dependent perturbation theory describes the evolution of a system under a time-dependent perturbation
    • Transition probabilities between states can be calculated using Fermi's golden rule
• Variational method finds upper bounds on the ground state energy of a system
  • The ground state energy is minimized over a family of trial wave functions, which are parameterized guesses for the true ground state
  • The Rayleigh-Ritz variational principle states that the expectation value of the Hamiltonian in any state is greater than or equal to the true ground state energy
• Numerical methods are used when analytical solutions are not possible
  • Examples include the split-operator method, finite difference methods, and the Crank-Nicolson method
  • These methods discretize space and time, approximating the continuous Schrödinger equation with a discrete set of equations that can be solved numerically

Connections to Classical Mechanics

• The correspondence principle states that quantum mechanics must reduce to classical mechanics in the limit of large quantum numbers
  • As the quantum numbers (e.g., energy levels) become large, the behavior of the system approaches the classical description
  • Examples include the Bohr correspondence principle for the hydrogen atom and the classical limit of the quantum harmonic oscillator
• Ehrenfest's theorem relates the time evolution of quantum expectation values to classical equations of motion
  • The expectation values of position and momentum obey the classical equations of motion, with the quantum force given by the expectation value of the gradient of the potential
  • Provides a connection between the quantum and classical descriptions of a system
• The Wigner-Weyl transformation maps quantum operators to classical phase space functions
  • The Wigner function is a quasi-probability distribution in phase space that represents a quantum state
  • Marginal distributions of the Wigner function give the probability distributions for position and momentum
• The path integral formulation of quantum mechanics expresses the transition amplitude between states as a sum over all possible classical paths
  • The action of each path determines its contribution to the total amplitude
  • In the classical limit, the path of stationary action dominates, recovering the classical equations of motion