Theorems of Quantum Mechanics
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Schrödinger Equation: A fundamental equation in quantum mechanics that describes how the quantum state of a physical system changes over time. Hamiltonian: An operator corresponding to the total energy of the system, including both kinetic and potential energies. Variation Method: An approximation method used to find the ground state energy of a quantum system by minimizing the energy expectation value. Perturbation Theory: A method used to find an approximate solution to a problem by starting from the exact solution of a related, simpler problem and adding corrections. Bracket Notation: A notation introduced by Dirac, used to represent the inner product of two functions in quantum mechanics. Matrix Element: The integral of an operator sandwiched between two functions, representing the transition amplitude between states. Hermitian Operator: An operator that is equal to its own conjugate transpose, ensuring that its eigenvalues are real numbers. Eigenvalues: The possible outcomes of a measurement of a physical quantity, represented by the operator. Eigenfunctions: The functions that correspond to the eigenvalues of an operator, representing the state of the system. Orthogonality: A property of functions where their inner product is zero, indicating that they are independent. Completeness: A property of a set of functions where any well-behaved function can be expanded as a linear combination of the set. Parity Operator: An operator that replaces each Cartesian coordinate with its negative, used to determine the symmetry properties of wave functions. Commuting Operators: Operators that can be applied in any order without changing the result, indicating that the corresponding physical quantities can be simultaneously measured. Uncertainty Principle: A principle stating that certain pairs of physical properties, like position and momentum, cannot be simultaneously measured with arbitrary precision. Superposition: The principle that a quantum system can exist in multiple states at once, and the overall state is a combination of these states. Reduction of the Wave Function: The process by which a quantum system's state becomes definite upon measurement, also known as wave function collapse. Dirac Delta Function: A function that is zero everywhere except at a single point, where it is infinitely high and integrates to one, used to represent point particles. Heaviside Step Function: A function that is zero for negative arguments and one for positive arguments, used to represent sudden changes. Time-Dependent Schrödinger Equation: An equation describing how the quantum state of a system evolves over time. Stationary State: A quantum state with all observables independent of time, typically an eigenstate of the Hamiltonian.