Hermitian operators play a crucial role in quantum mechanics due to several important reasons:
Observables and measurements: In quantum mechanics, physical observables, such as position, momentum, energy, and angular momentum, are represented by mathematical operators. The outcomes of measurements in quantum mechanics are eigenvalues of these operators. Hermitian operators have real eigenvalues, which correspond to physically measurable quantities. Therefore, Hermitian operators are directly associated with observable quantities in quantum mechanics.
Probability interpretation: Quantum mechanics uses probabilistic interpretations to describe the behavior of particles. The probability of obtaining a particular measurement outcome is given by the squared magnitude of the corresponding eigenstate's coefficient in the wave function. For an operator representing an observable, the sum of the probabilities of all possible outcomes must be equal to 1. Hermitian operators guarantee that the eigenvalues are real and the eigenstates are orthogonal, ensuring that the probabilities are well-defined and consistent with the principles of quantum mechanics.
Conservation laws: In quantum mechanics, symmetries and conservation laws are intimately linked. Noether's theorem connects differentiable symmetries of a system to conserved quantities. Hermitian operators associated with conserved quantities, such as total energy or total angular momentum, commute with the system's Hamiltonian. This commutation relationship ensures that these quantities are conserved over time.
Uncertainty principle: The Heisenberg uncertainty principle is a fundamental aspect of quantum mechanics that imposes limitations on the simultaneous precision with which certain pairs of observables can be known. The uncertainty principle is mathematically expressed through the commutation relationship between operators. Hermitian operators are necessary to define the commutation relations accurately, allowing us to determine the limits on the simultaneous measurement of different observables.
Self-adjointness and spectral theorem: Hermitian operators are self-adjoint, meaning their adjoint operator is equal to the operator itself. This property ensures that the eigenvalues are real, and the eigenvectors form an orthonormal basis. The spectral theorem guarantees that any Hermitian operator can be diagonalized, enabling a convenient representation of the operator in terms of its eigenvalues and eigenvectors.
Overall, Hermitian operators provide a solid mathematical framework for describing physical observables, ensuring the probabilistic interpretation of quantum mechanics, connecting with conservation laws, defining uncertainty relations, and facilitating the diagonalization of operators for practical calculations and theoretical analysis.