The Schrödinger equation is a partial differential equation whose solution is the wave equation, which describes the probability density of a given particle over space. The general form of the Schrödinger equation is where i is the imaginary unit, ħ is the reduced Planck constant, Ψ is the wave function, and Ĥ is the Hamiltonian operator (representing the total energy of the system). In steady-state systems (where the wave equation does not depend on time, such as in an atomic or molecular orbital) this simplifies to
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| - The Schrödinger equation is a partial differential equation whose solution is the wave equation, which describes the probability density of a given particle over space. The general form of the Schrödinger equation is where i is the imaginary unit, ħ is the reduced Planck constant, Ψ is the wave function, and Ĥ is the Hamiltonian operator (representing the total energy of the system). In steady-state systems (where the wave equation does not depend on time, such as in an atomic or molecular orbital) this simplifies to
- In quantum mechanics, the Schrödinger equation is an equation that describes how the quantum state of a physical system changes with time. It was formulated in late 1925, and published in 1926, by the Austrian physicist Erwin Schrödinger. In the standard interpretation of quantum mechanics, the wavefunction is the most complete description that can be given to a physical system. Solutions to Schrödinger's equation describe not only molecular, atomic, and subatomic systems, but also macroscopic systems, possibly even the whole universe. __TOC__
- In quantum mechanics, the Schrödinger equation is the differential equation that dictates how the wavefunction of a quantum mechanical system evolves over time. The most general Schrödinger equation is This is similar to a statement of conservation of energy; the operator on the left hand side gives the energy of the wavefunction in terms of how it changes with time, and the operator on the right hand side is the Hamiltonian operator, which gives the total energy of the system in terms of how it changes with space.
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- The fuzzyness of a particle's position illustrated, which is not definite in quantum mechanics.
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- Increasing levels of wavepacket localization, meaning the particle has a more localized position.
- In the limit ħ → 0, the particle's position and momentum become known exactly. This is equivalent to the classical particle.
- Time development of wave packet, as described by the solution to Schrödinger’s equation for a 1-d step potential system, shown in slices of position-time coordinates . The particle is shown as the blue circles, the opacity corresponds to the probability density of the particle at the location shown. The step potential is the dotted line. The probability of transmission is greater than reflection, since the total energy E of the particle exceeds the potential energy V.
- Quantum tunneling through a barrier. A particle coming from the left does not have enough energy to climb the barrier. However, it can sometimes "tunnel" to the other side.
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| - In quantum mechanics, the Schrödinger equation is the differential equation that dictates how the wavefunction of a quantum mechanical system evolves over time. The most general Schrödinger equation is This is similar to a statement of conservation of energy; the operator on the left hand side gives the energy of the wavefunction in terms of how it changes with time, and the operator on the right hand side is the Hamiltonian operator, which gives the total energy of the system in terms of how it changes with space. For a single particle moving at non-relativistic speeds in a scalar potential, the form of the Hamiltonian operator is simple and it can be substituted in, giving Here, μ is the mass of the particle and V is the potential energy of the particle.
- The Schrödinger equation is a partial differential equation whose solution is the wave equation, which describes the probability density of a given particle over space. The general form of the Schrödinger equation is where i is the imaginary unit, ħ is the reduced Planck constant, Ψ is the wave function, and Ĥ is the Hamiltonian operator (representing the total energy of the system). In steady-state systems (where the wave equation does not depend on time, such as in an atomic or molecular orbital) this simplifies to where E is a constant. This is known as the time-independent Schrödinger equation. File:D5 orbital.png This quantum mechanics-related article contains minimal information concerning its topic. You can help the Physics Wiki by adding to it.
- In quantum mechanics, the Schrödinger equation is an equation that describes how the quantum state of a physical system changes with time. It was formulated in late 1925, and published in 1926, by the Austrian physicist Erwin Schrödinger. In classical mechanics, the equation of motion is Newton's second law, and equivalent formulations are the Euler-Lagrange equations and Hamilton's equations. In all these formulations, they are used to solve for the motion of a mechanical system, and mathematically predict what the system will do at any time beyond the initial settings and configuration of the system. In quantum mechanics, the analogue of Newton's law is Schrödinger's equation for a quantum system, usually atoms, molecules, and subatomic particles; free, bound, or localized. It is not a simple algebraic equation, but (in general) a linear partial differential equation. The differential equation encases the wavefunction of the system, also called the quantum state or state vector. In the standard interpretation of quantum mechanics, the wavefunction is the most complete description that can be given to a physical system. Solutions to Schrödinger's equation describe not only molecular, atomic, and subatomic systems, but also macroscopic systems, possibly even the whole universe. Like Newton's Second law, the Schrödinger equation can be mathematically transformed into other formulations such as Werner Heisenberg's matrix mechanics, and Richard Feynman's path integral formulation. Also like Newton's Second law, the Schrödinger equation describes time in a way that is inconvenient for relativistic theories, a problem that is not as severe in matrix mechanics and completely absent in the path integral formulation. __TOC__
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