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- https://chem.libretexts.org/Bookshelves/General_Chemistry/General_Chemistry_Supplement_(Eames)/Quantum_Chemistry/Particle_in_a_Boxwhere H is the Hamiltonian operator, which finds the total energy of the system E. (This approach uses the linear algebra concept of an eigenfunction and eigenstate, but don't worry if you don't know ...where H is the Hamiltonian operator, which finds the total energy of the system E. (This approach uses the linear algebra concept of an eigenfunction and eigenstate, but don't worry if you don't know what these are.) Kinetic energy KE is given by The kinetic energy is calculated using the equation above, using the square of the momentum operator (thus, the first derivative in the momentum operator becomes a second derivative when the operator is squared).
- https://chem.libretexts.org/Courses/Westminster_College/CHE_180_-_Inorganic_Chemistry/01%3A_Chapter_1_-_Electronic_Structure_of_the_Atom/1.3%3A_Development_of_Quantum_TheoryMacroscopic objects act as particles. Microscopic objects (such as electrons) have properties of both a particle and a wave. but their exact trajectories cannot be determined. The quantum mechanical m...Macroscopic objects act as particles. Microscopic objects (such as electrons) have properties of both a particle and a wave. but their exact trajectories cannot be determined. The quantum mechanical model of atoms describes the 3D position of the electron in a probabilistic manner according to a mathematical function called a wavefunction, often denoted as ψ. The squared magnitude of the wavefunction describes the distribution of the probability of finding the electron in a particular region in
- https://chem.libretexts.org/Courses/CSU_San_Bernardino/CHEM_2200%3A_General_Chemistry_II_(Mink)/06%3A_Electronic_Structure_and_Periodic_Properties/6.04%3A_Development_of_Quantum_TheoryMacroscopic objects act as particles. Microscopic objects (such as electrons) have properties of both a particle and a wave. but their exact trajectories cannot be determined. The quantum mechanical m...Macroscopic objects act as particles. Microscopic objects (such as electrons) have properties of both a particle and a wave. but their exact trajectories cannot be determined. The quantum mechanical model of atoms describes the 3D position of the electron in a probabilistic manner according to a mathematical function called a wavefunction, often denoted as ψ. The squared magnitude of the wavefunction describes the distribution of the probability of finding the electron in a particular region in
- https://chem.libretexts.org/Courses/National_Yang_Ming_Chiao_Tung_University/Chemical_Principles_for_Medical_Students/01%3A_Electronic_Structure_of_Atoms/1.05%3A_Quantum_Mechanics_and_Atomic_OrbitalsThere is a relationship between the motions of electrons in atoms and molecules and their energies that is described by quantum mechanics. Because of wave–particle duality, scientists must deal with t...There is a relationship between the motions of electrons in atoms and molecules and their energies that is described by quantum mechanics. Because of wave–particle duality, scientists must deal with the probability of an electron being at a particular point in space. To do so required the development of quantum mechanics, which uses wavefunctions to describe the mathematical relationship between the motion of electrons in atoms and molecules and their energies.
- https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Book%3A_Quantum_States_of_Atoms_and_Molecules_(Zielinksi_et_al)/03%3A_The_Schrodinger_Equation/3.07%3A_Meaning_of_the_WavefunctionThe square is used, rather than the modulus itself, just like the intensity of a light wave depends on the square of the electric field. The Born interpretation therefore calls the wavefunction the pr...The square is used, rather than the modulus itself, just like the intensity of a light wave depends on the square of the electric field. The Born interpretation therefore calls the wavefunction the probability amplitude, the absolute square of the wavefunction is called the probability density, and the probability density times a volume element in three-dimensional space (dτ) is the probability.
- https://chem.libretexts.org/Courses/Louisville_Collegiate_School/General_Chemistry/LibreTexts_Louisville_Collegiate_School_Chapters_06%3A_Electronic_Structure_and_Periodic_Properties_of_Elements/LibreTexts%2F%2FLouisville_Collegiate_School%2F%2FChapters%2F%2F06%3A_Electronic_Structure_and_Periodic_Properties_of_Elements%2F%2F6.3%3A_Development_of_Quantum_TheoryMacroscopic objects act as particles. Microscopic objects (such as electrons) have properties of both a particle and a wave. but their exact trajectories cannot be determined. The quantum mechanical m...Macroscopic objects act as particles. Microscopic objects (such as electrons) have properties of both a particle and a wave. but their exact trajectories cannot be determined. The quantum mechanical model of atoms describes the 3D position of the electron in a probabilistic manner according to a mathematical function called a wavefunction, often denoted as ψ. The squared magnitude of the wavefunction describes the distribution of the probability of finding the electron in a particular region in
- https://chem.libretexts.org/Courses/Widener_University/CHEM_175_-_General_Chemistry_I_(Van_Bramer)/05%3A_Electronic_Structure_and_Periodic_Properties/5.02%3A_Development_of_Quantum_TheoryMacroscopic objects act as particles. Microscopic objects (such as electrons) have properties of both a particle and a wave. but their exact trajectories cannot be determined. The quantum mechanical m...Macroscopic objects act as particles. Microscopic objects (such as electrons) have properties of both a particle and a wave. but their exact trajectories cannot be determined. The quantum mechanical model of atoms describes the 3D position of the electron in a probabilistic manner according to a mathematical function called a wavefunction, often denoted as ψ. The squared magnitude of the wavefunction describes the distribution of the probability of finding the electron in a particular region in
- https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Physical_Chemistry_(LibreTexts)/03%3A_The_Schrodinger_Equation_and_a_Particle_in_a_BoxThis page discusses the particle in a box model, highlighting the distinction between classical and quantum mechanics through confined energy levels. It emphasizes Schrödinger's equation as essential ...This page discusses the particle in a box model, highlighting the distinction between classical and quantum mechanics through confined energy levels. It emphasizes Schrödinger's equation as essential for understanding wave-particle duality, energy quantization, and the probabilistic nature of wavefunctions. The model can be extended to three dimensions, introducing degeneration, and serves to simplify complex quantum phenomena, establishing its significance in undergraduate physics education.
- https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Physical_Chemistry_(LibreTexts)/14%3A_Nuclear_Magnetic_Resonance_Spectroscopy/14.03%3A_Proton_NMR_Spectrometers_Operate_at_Frequencies_Between_60_MHz_and_750_MHzThis page discusses the Heisenberg Uncertainty Principle, established by Werner Heisenberg in the mid-1920s. It states that more precise measurement of an electron's position results in greater uncert...This page discusses the Heisenberg Uncertainty Principle, established by Werner Heisenberg in the mid-1920s. It states that more precise measurement of an electron's position results in greater uncertainty in its momentum, due to the wave nature of matter. This principle illustrates fundamental limits in quantum measurements by mathematically expressing the relationship between uncertainties in position and momentum.
- https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Physical_Chemistry_(LibreTexts)/14%3A_Nuclear_Magnetic_Resonance_Spectroscopy/14.01%3A_Nuclei_Have_Intrinsic_Spin_Angular_MomentaThis page discusses the concept of spin in fundamental particles, highlighting that electrons, protons, and neutrons have a spin quantum number of 12, marking them as fermions, while bo...This page discusses the concept of spin in fundamental particles, highlighting that electrons, protons, and neutrons have a spin quantum number of 12, marking them as fermions, while bosons like photons have integer spins. The significance of spin lies in its role in defining electronic states of atoms and molecules, with a complete understanding requiring principles of relativistic quantum mechanics.
- 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.
