Eigen functions applications
- 1. APPLICATIONS OF EIGEN FUNCTIONS IN QUANTUM MECHANICS B.V.C.M. BENARAGAMA (170070D) Department of Chemical & Process Engineering University of Moratuwa An operator acting on a function, changes functions into another function. Occasionally there are instances where, the received function is directly proportional to the function used. 𝐴(𝜓) ∝ 𝜓 In this type of a situation, a constant, 𝑎 can be defined, as the Eigenvalue of the function. The operator, 𝐴 is hence known as an Eigen operator. 𝐴(𝜓) = 𝑎𝜓 The 𝑎 eigenvalues represents the possible measured values of the 𝐴 operator. Classically, 𝑎 would be allowed to vary continuously, but in quantum mechanics, 𝑎 typically has only a sub-set of allowed values. Both time-dependent and time-independent Schrödinger equations are the `best-known instances of an eigenvalue equations in quantum mechanics, with its eigenvalues corresponding to the allowed energy levels of the quantum system. (“3.3: The Schrödinger Equation is an Eigenvalue Problem - Chemistry LibreTexts,” n.d.) Some Eigen operators used in quantum mechanics and what their eigen constants represent are given below. (“Operators in Quantum Mechanics,” n.d.)
- 2. The left side of the above table denotes the physical meaning of Eigen value obtained when relevant operator on the right side is applied on the wave equation giving the position of the a quantum particle. A famous example denoting the Energy of a particle when Hamiltonian operator is applied on the wave function, is given below. (“The Hamiltonian Operator | My Blog,” n.d.) REFERENCES 3.3: The Schrödinger Equation is an Eigenvalue Problem - Chemistry LibreTexts. (n.d.). Retrieved December 9, 2019, from https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Ma ps/Map%3A_Physical_Chemistry_(McQuarrie_and_Simon)/03%3A_The_Schrödinger_Equa tion_and_a_Particle_in_a_Box/3.03%3A_The_Schrödinger_Equation_is_an_Eigenvalue_Pro blem Operators in Quantum Mechanics. (n.d.). Retrieved December 9, 2019, from http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/qmoper.html The Hamiltonian Operator | My Blog. (n.d.). Retrieved December 9, 2019, from http://physciense.blogspot.com/2014/03/the-hamiltonian-operator.html