Chapter 20: Operators in Quantum Mechanics

Operators in Quantum Mechanics introduces the central role of operators (A) as a specialized mathematical framework, distinct from standard algebra, used to define quantum-mechanical quantities and equations. Operators function by acting upon a state vector (psi), which represents a specific physical state, to generate a new resulting state (phi). This operation can be described numerically by a set of complex numbers or a matrix, where the matrix elements (A-i-j) provide the quantitative description of the operator acting between base states. Quantum mechanical operators corresponding to physically observable, measurable quantities are often Hermitian or self-adjoint, meaning the operator equals its adjoint (B-dagger equals B). A fundamental achievement detailed in the chapter is the derivation of the generalized formula for calculating the average value (or expectation value) of any physical quantity A for a given system state. This expectation value formula is then applied to specific observable quantities, illustrating that in the coordinate representation (using the wave function psi of x), the position operator (x) corresponds simply to multiplication by x, while the momentum operator (p-x) must be represented by a differential operator: h-bar over i times the partial derivative with respect to x. The concept of the Hamiltonian operator (H) is established as the operator for the total energy of the system, combining the kinetic energy (related to the squared momentum operator) and potential energy. The chapter also introduces the angular momentum operator (L-z) and discusses the critical concept of non-commuting operators, noting that in quantum mechanics, the order of operations matters (A times B does not equal B times A). This non-commutation is quantified by the fundamental relationship between position and momentum. Finally, the derivation for the rate of change of average values with time connects the quantum description back to classical physics, demonstrating that the equations governing the time rate of change of average position and average momentum mirror Newton's laws of motion, solidifying the idea that Schrödinger's differential equation approach and Heisenberg's matrix algebra are mathematically equivalent theories.