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- https://chem.libretexts.org/Courses/BethuneCookman_University/B-CU%3ACH-331_Physical_Chemistry_I/CH-331_Text/CH-331_Text/04._Postulates_and_Principles_of_Quantum_Mechanics/4.5%3A_The_Eigenfunctions_of_Operators_are_OrthogonalThe eigenvalues of operators associated with experimental measurements are all real; this is because the eigenfunctions of the Hamiltonian operator are orthogonal, and we also saw that the position an...The eigenvalues of operators associated with experimental measurements are all real; this is because the eigenfunctions of the Hamiltonian operator are orthogonal, and we also saw that the position and momentum of the particle could not be determined exactly. We now examine the generality of these insights by stating and proving some fundamental theorems. These theorems use the Hermitian property of quantum mechanical operators, which is described first.
- https://chem.libretexts.org/Courses/DePaul_University/Thermodynamics_and_Introduction_to_Quantum_Mechanics_(Southern)/08%3A_The_Postulates_of_Quantum_Mechanics/8.07%3A_Postulates_3_and_4_of_Quantum_Mechanics/8.7.01%3A_Eigenfunctions_of_Operators_are_OrthogonalThe eigenvalues of operators associated with experimental measurements are all real; this is because the eigenfunctions of the Hamiltonian operator are orthogonal, and we also saw that the position an...The eigenvalues of operators associated with experimental measurements are all real; this is because the eigenfunctions of the Hamiltonian operator are orthogonal, and we also saw that the position and momentum of the particle could not be determined exactly. We now examine the generality of these insights by stating and proving some fundamental theorems. These theorems use the Hermitian property of quantum mechanical operators, which is described first.
- https://chem.libretexts.org/Courses/Grinnell_College/CHM_364%3A_Physical_Chemistry_2_(Grinnell_College)/04%3A_Postulates_and_Principles_of_Quantum_Mechanics/4.05%3A_Eigenfunctions_of_Operators_are_OrthogonalThe eigenvalues of operators associated with experimental measurements are all real; this is because the eigenfunctions of the Hamiltonian operator are orthogonal, and we also saw that the position an...The eigenvalues of operators associated with experimental measurements are all real; this is because the eigenfunctions of the Hamiltonian operator are orthogonal, and we also saw that the position and momentum of the particle could not be determined exactly. We now examine the generality of these insights by stating and proving some fundamental theorems. These theorems use the Hermitian property of quantum mechanical operators, which is described first.
- https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Physical_Chemistry_(LibreTexts)/04%3A_Postulates_and_Principles_of_Quantum_MechanicsThis page outlines key principles of quantum mechanics, highlighting that a system's state is defined by a wavefunction that encapsulates all physical information. Observable quantities are linked to ...This page outlines key principles of quantum mechanics, highlighting that a system's state is defined by a wavefunction that encapsulates all physical information. Observable quantities are linked to operators, with measurable outcomes represented by eigenvalues. The time-dependent Schrödinger equation governs wavefunction behavior, while stationary states are derived from the time-independent equation.
- https://chem.libretexts.org/Courses/Pacific_Union_College/Quantum_Chemistry/03%3A_The_Schrodinger_Equation_and_a_Particle_in_a_Box/3.02%3A_Linear_Operators_in_Quantum_MechanicsAn operator is a generalization of the concept of a function. Whereas a function is a rule for turning one number into another, an operator is a rule for turning one function into another function.
- https://chem.libretexts.org/Courses/BethuneCookman_University/BCU%3A_CH_332_Physical_Chemistry_II/Text/7%3A_Approximation_Methods/7.02%3A_Linear_Variational_Method_and_the_Secular_DeterminantA special type of variation widely used in the study of molecules is the so-called linear variation function, a linear combination of N linearly independent functions (often atomic orbitals). Quite o...A special type of variation widely used in the study of molecules is the so-called linear variation function, a linear combination of N linearly independent functions (often atomic orbitals). Quite often a trial wavefunction is expanded as a linear combination of other functions (not the eigenvalues of the Hamiltonian, since they are not known) .
- https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/The_Live_Textbook_of_Physical_Chemistry_(Peverati)/21%3A_Operators_and_Mathematical_Background/21.01%3A_Operators_in_Quantum_MechanicsThe central concept in this new framework of quantum mechanics is that every observable (i.e., any quantity that can be measured in a physical experiment) is associated with an operator. To distinguis...The central concept in this new framework of quantum mechanics is that every observable (i.e., any quantity that can be measured in a physical experiment) is associated with an operator. To distinguish between classical mechanics operators and quantum mechanical ones, we use a hat symbol ^ on top of the latter.
- https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Mathematical_Methods_in_Chemistry_(Levitus)/15%3A_Matrices/15.08%3A_Hermitian_MatricesA Hermitian matrix (or self-adjoint matrix) is a square matrix with complex entries that is equal to its own conjugate transpose. Hermitian matrices are a generalization of the symmetric real matrices...A Hermitian matrix (or self-adjoint matrix) is a square matrix with complex entries that is equal to its own conjugate transpose. Hermitian matrices are a generalization of the symmetric real matrices we just talked about, and they also have real eigenvalues, and eigenvectors that form a mutually orthogonal set.
- https://chem.libretexts.org/Courses/Grinnell_College/CHM_364%3A_Physical_Chemistry_2_(Grinnell_College)/07%3A_Approximation_Methods/7.02%3A_Linear_Variational_Method_and_the_Secular_DeterminantA special type of variation widely used in the study of molecules is the so-called linear variation function, a linear combination of N linearly independent functions (often atomic orbitals). Quite o...A special type of variation widely used in the study of molecules is the so-called linear variation function, a linear combination of N linearly independent functions (often atomic orbitals). Quite often a trial wavefunction is expanded as a linear combination of other functions (not the eigenvalues of the Hamiltonian, since they are not known) .
- https://chem.libretexts.org/Courses/Lebanon_Valley_College/CHM_311%3A_Physical_Chemistry_I_(Lebanon_Valley_College)/02%3A_Foundations_of_Quantum_Mechanics/2.10%3A_Eigenfunctions_of_Operators_are_OrthogonalThe eigenvalues of operators associated with experimental measurements are all real; this is because the eigenfunctions of the Hamiltonian operator are orthogonal, and we also saw that the position an...The eigenvalues of operators associated with experimental measurements are all real; this is because the eigenfunctions of the Hamiltonian operator are orthogonal, and we also saw that the position and momentum of the particle could not be determined exactly. We now examine the generality of these insights by stating and proving some fundamental theorems. These theorems use the Hermitian property of quantum mechanical operators, which is described first.
- https://chem.libretexts.org/Courses/Lebanon_Valley_College/CHM_311%3A_Physical_Chemistry_I_(Lebanon_Valley_College)/02%3A_Foundations_of_Quantum_Mechanics/2.06%3A_Linear_Operators_in_Quantum_MechanicsAn operator is a generalization of the concept of a function. Whereas a function is a rule for turning one number into another, an operator is a rule for turning one function into another function.
