Quantum Mechanical Operators
Click the keywords to read more about it
Operator: A rule that transforms a given function into another function. Operators are fundamental in quantum mechanics for describing physical quantities and their interactions 1. Energy Operator: An operator that, when applied to the wave function, returns the wave function multiplied by an allowed value of the energy 1. Differentiation Operator: An operator that differentiates a function with respect to a variable, often denoted with a circumflex (e.g., ( \hat{D}_n )) 1. Sum and Difference of Operators: The sum of two operators ( \hat{A}_n ) and ( \hat{B}_n ) applied to a function ( f(x) ) is defined as ( (\hat{A}_n + \hat{B}_n)f(x) = \hat{A}_n f(x) + \hat{B}_n f(x) ). The difference is similarly defined 1. Product of Operators: The product of two operators ( \hat{A}_n ) and ( \hat{B}_n ) applied to a function ( f(x) ) is defined as ( \hat{A}_n \hat{B}_n f(x) = \hat{A}_n (\hat{B}_n f(x)) ) 1. Operator Algebra: A set of rules and operations for manipulating operators, including the associative law of multiplication and commutative properties 1. Commutator: The commutator of two operators ( \hat{A}_n ) and ( \hat{B}_n ) is defined as ( [\hat{A}_n, \hat{B}_n] = \hat{A}_n \hat{B}_n - \hat{B}_n \hat{A}_n ). If the commutator is zero, the operators are said to commute 1. Eigenfunction: A function ( f(x) ) that, when operated on by a linear operator ( \hat{A}_n ), results in the function being multiplied by a constant ( k ). This constant is called the eigenvalue 1. Eigenvalue: The constant ( k ) that results from operating on an eigenfunction with a linear operator. It represents a characteristic value associated with the operator 1. Hamiltonian Operator: The operator corresponding to the total energy of a system, often used in the Schrödinger equation to find the energy eigenvalues and eigenfunctions 1. Linear Operator: An operator ( \hat{A}_n ) that satisfies the properties ( \hat{A}_n (f(x) + g(x)) = \hat{A}_n f(x) + \hat{A}_n g(x) ) and ( \hat{A}_n (cf(x)) = c\hat{A}_n f(x) ), where ( f ) and ( g ) are functions and ( c ) is a constant 1. Laplacian Operator: An operator denoted by ( \nabla^2 ) (del squared), which is the sum of the second partial derivatives with respect to each spatial coordinate 1. Expectation Value: The average value of a physical property ( B ) for a system in state ( \psi ), calculated as ( \langle B \rangle = \int \psi^* \hat{B} \psi , d\tau ), where ( \hat{B} ) is the operator corresponding to ( B ) 1. Normalization: The process of adjusting the wave function ( \psi ) so that the total probability of finding the particle is 1. This ensures that ( \int |\psi|^2 , d\tau = 1 ) 1. Quadratically Integrable: A function ( \psi ) is quadratically integrable if ( \int |\psi|^2 , d\tau ) is finite. This property is necessary for the function to be normalized 1.