🪐Principles of Physics IV Unit 3 – Quantum Operators and Observables

Key Concepts and Definitions

• Quantum operators mathematical entities represent physical quantities (position, momentum, energy) in quantum mechanics
• Observables measurable quantities in a quantum system correspond to Hermitian operators
• Eigenvalues possible outcomes of a measurement on a quantum system
• Eigenstates specific quantum states associated with a particular eigenvalue
• Commutator $[A, B] = AB - BA$ measures the degree of non-commutativity between two operators $A$ and $B$
  • Commuting operators $[A, B] = 0$ can be simultaneously measured with arbitrary precision
  • Non-commuting operators $[A, B] \neq 0$ cannot be simultaneously measured with arbitrary precision
• Uncertainty principle fundamental limit on the precision of simultaneous measurements of non-commuting observables (position and momentum)

Mathematical Foundations

• Hilbert space complex vector space with an inner product serves as the mathematical framework for quantum mechanics
  • Vectors in Hilbert space represent quantum states
  • Inner product $\langle \psi | \phi \rangle$ defines the overlap between two states $|\psi\rangle$ and $|\phi\rangle$
• Dirac notation compact way to represent quantum states and operators
  • Ket $|\psi\rangle$ represents a quantum state
  • Bra $\langle\psi|$ represents the dual of a ket
  • Operator $A$ acts on a ket from the left $A|\psi\rangle$
• Linear operators transform vectors in Hilbert space
  • Adjoint operator $A^\dagger$ satisfies $\langle A^\dagger \psi | \phi \rangle = \langle \psi | A \phi \rangle$
  • Hermitian operators $A = A^\dagger$ correspond to observables in quantum mechanics
• Spectral theorem states that any Hermitian operator can be decomposed into a sum of projection operators onto its eigenstates

Quantum Operators: Basics and Properties

• Position operator $\hat{x}$ represents the position of a particle in quantum mechanics
  • Eigenvalues of $\hat{x}$ are the possible outcomes of a position measurement
  • Eigenstates of $\hat{x}$ are the position eigenstates $|x\rangle$
• Momentum operator $\hat{p} = -i\hbar \frac{\partial}{\partial x}$ represents the momentum of a particle
  • Eigenvalues of $\hat{p}$ are the possible outcomes of a momentum measurement
  • Eigenstates of $\hat{p}$ are the momentum eigenstates $|p\rangle$
• Hamiltonian operator $\hat{H}$ represents the total energy of a quantum system
  • Eigenvalues of $\hat{H}$ are the possible energy levels of the system
  • Eigenstates of $\hat{H}$ are the energy eigenstates $|E\rangle$
• Commutation relations between operators determine their compatibility for simultaneous measurement
  • Position and momentum operators satisfy $[\hat{x}, \hat{p}] = i\hbar$, implying they cannot be simultaneously measured with arbitrary precision

Observables and Measurement

• Observables physical quantities that can be measured in a quantum system
  • Represented by Hermitian operators in the Hilbert space
  • Eigenvalues of an observable are the possible outcomes of a measurement
• Measurement process in quantum mechanics probabilistic and leads to the collapse of the wavefunction
  • Probability of measuring an eigenvalue $\lambda_i$ given by $|\langle \psi | \lambda_i \rangle|^2$, where $|\psi\rangle$ is the state of the system before measurement
  • After measurement, the system collapses into the eigenstate corresponding to the measured eigenvalue
• Expectation value average value of an observable $A$ in a given state $|\psi\rangle$ calculated as $\langle A \rangle = \langle \psi | A | \psi \rangle$
• Projection operators $P_i = |\lambda_i\rangle\langle\lambda_i|$ project a state onto the eigenstate corresponding to the eigenvalue $\lambda_i$

Eigenvalues and Eigenstates

• Eigenvalue equation $A|\psi\rangle = \lambda|\psi\rangle$ defines the eigenvalues and eigenstates of an operator $A$
  • Eigenvalue $\lambda$ is a scalar value associated with the eigenstate $|\psi\rangle$
  • Eigenstate $|\psi\rangle$ is a vector in Hilbert space that remains unchanged (up to a scalar factor) when acted upon by the operator $A$
• Degenerate eigenvalues occur when multiple linearly independent eigenstates correspond to the same eigenvalue
• Orthonormality eigenstates corresponding to different eigenvalues are orthogonal $\langle \psi_i | \psi_j \rangle = \delta_{ij}$ and normalized $\langle \psi_i | \psi_i \rangle = 1$
• Completeness relation $\sum_i |\psi_i\rangle\langle\psi_i| = I$ states that the eigenstates of an observable form a complete basis for the Hilbert space

Uncertainty Principle and Complementarity

• Heisenberg uncertainty principle $\Delta x \Delta p \geq \frac{\hbar}{2}$ sets a fundamental limit on the precision of simultaneous measurements of position and momentum
  • $\Delta x$ and $\Delta p$ are the standard deviations of position and momentum measurements, respectively
  • Similar uncertainty relations hold for other pairs of non-commuting observables (energy and time)
• Complementarity principle states that certain properties of a quantum system (wave-like and particle-like behavior) are mutually exclusive and cannot be observed simultaneously
• Bohr's interpretation of complementarity emphasizes the role of the measurement apparatus in determining the observed properties of a quantum system
• Entanglement non-classical correlation between quantum systems leads to stronger uncertainty relations (EPR paradox, Bell's inequality)

Applications in Quantum Systems

• Harmonic oscillator potential $V(x) = \frac{1}{2}m\omega^2x^2$ leads to evenly spaced energy levels $E_n = \hbar\omega(n + \frac{1}{2})$
  • Creation $a^\dagger$ and annihilation $a$ operators raise and lower the energy eigenstates, respectively
  • Ground state $|0\rangle$ is the lowest energy eigenstate of the harmonic oscillator
• Angular momentum operators $\hat{L}_x, \hat{L}_y, \hat{L}_z$ represent the components of angular momentum in quantum mechanics
  • Satisfy the commutation relations $[\hat{L}_i, \hat{L}_j] = i\hbar\epsilon_{ijk}\hat{L}_k$, where $\epsilon_{ijk}$ is the Levi-Civita symbol
  • Eigenvalues of $\hat{L}^2$ and $\hat{L}_z$ are quantized $l(l+1)\hbar^2$ and $m\hbar$, respectively, where $l$ is the angular momentum quantum number and $m$ is the magnetic quantum number
• Spin angular momentum intrinsic angular momentum of particles (electrons, protons) not associated with orbital motion
  • Described by the Pauli matrices $\sigma_x, \sigma_y, \sigma_z$ satisfying the commutation relations $[\sigma_i, \sigma_j] = 2i\epsilon_{ijk}\sigma_k$
  • Eigenvalues of $\sigma_z$ are $\pm 1$, corresponding to the spin-up $|\uparrow\rangle$ and spin-down $|\downarrow\rangle$ states

Problem-Solving Techniques

• Diagonalization finding the eigenvalues and eigenstates of an operator by solving the eigenvalue equation
  • Eigenvalues are the roots of the characteristic polynomial $\det(A - \lambda I) = 0$
  • Eigenstates are the non-zero solutions of the linear system $(A - \lambda I)|\psi\rangle = 0$
• Matrix representation expressing operators and states as matrices and vectors in a chosen basis
  • Matrix elements of an operator $A$ in the basis $\{|i\rangle\}$ given by $A_{ij} = \langle i|A|j\rangle$
  • Change of basis achieved through unitary transformations $U^\dagger A U$, where $U$ is a unitary matrix satisfying $U^\dagger U = I$
• Perturbation theory approximating the eigenvalues and eigenstates of a perturbed system $H = H_0 + \lambda V$, where $H_0$ is the unperturbed Hamiltonian, $V$ is the perturbation, and $\lambda$ is a small parameter
  • First-order correction to the energy eigenvalues $E_n^{(1)} = \langle n|V|n\rangle$, where $|n\rangle$ is an eigenstate of $H_0$
  • First-order correction to the eigenstates $|n^{(1)}\rangle = \sum_{m \neq n} \frac{\langle m|V|n\rangle}{E_n - E_m}|m\rangle$
• Variational method approximating the ground state energy and wavefunction of a quantum system by minimizing the expectation value of the Hamiltonian over a trial wavefunction $|\psi_\text{trial}\rangle$
  • Upper bound on the ground state energy $E_0 \leq \frac{\langle \psi_\text{trial}|H|\psi_\text{trial}\rangle}{\langle \psi_\text{trial}|\psi_\text{trial}\rangle}$
  • Optimal trial wavefunction obtained by varying the parameters of $|\psi_\text{trial}\rangle$ to minimize the expectation value of $H$