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| Much insight in [[quantum mechanics]] can be gained from understanding the solutions to the time-dependent non-relativistic [[Schrödinger equation]] in an appropriate [[configuration space]]. In vector Cartesian coordinates <math>\mathbf{r}</math>, the equation takes the form
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| :<math>
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| H \psi\left(\mathbf{r}, t\right) = \left(T + V\right) \, \psi\left(\mathbf{r}, t\right) =
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| \left[ - \frac{\hbar^2}{2m} \nabla^2 + V\left(\mathbf{r}\right) \right] \psi\left(\mathbf{r}, t\right) = i\hbar \frac{\partial\psi\left(\mathbf{r}, t\right)}{\partial t}
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| </math>
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| in which <math>\psi</math> is the [[wavefunction]] of the system, H is the [[Hamiltonian operator]], and T and V are the operators for the [[kinetic energy]] and [[potential energy]], respectively. (Common forms of these operators appear in the square brackets.) The quantity ''t'' is the time. [[Stationary state]]s of this equation are found by solving the [[eigenvalue]]-[[eigenfunction]] (time-independent) form of the Schrödinger equation,
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| :<math>
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| \left[ - \frac{\hbar^2}{2m} \nabla^2 + V\left(\mathbf{r}\right) \right] \psi\left(\mathbf{r}\right) = E \psi \left(\mathbf{r}\right)
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| </math> | |
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| or any equivalent formulation of this equation in a different coordinate system other than Cartesian coordinates. For example, systems with spherical symmetry are simplified when expressed with [[spherical coordinates]]. Very often, only numerical solutions to the Schrödinger equation can be found for a given physical system and its associated potential energy. Fortunately, there exists a subset of physical systems for which the form of the eigenfunctions and their associated energies can be found. These '''quantum-mechanical systems with analytical solutions''' are listed below, and are quite useful for teaching and gaining intuition about quantum mechanics.
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| == Solvable systems ==
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| *The [[free particle]]
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| *The [[delta potential]]
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| *The [[particle in a box]] / [[infinite potential well]]
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| *The [[finite potential well]]
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| *The [[Airy function|One-dimensional triangular potential]]
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| *The [[particle in a ring]] or [[ring wave guide]]
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| *The [[particle in a spherically symmetric potential]]
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| *The [[quantum harmonic oscillator]]
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| *The [[hydrogen atom]] or [[hydrogen-like atom]]
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| *The [[particle in a one-dimensional lattice (periodic potential)]]
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| *The [[Morse potential]]
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| *The [[step potential]]
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| *The [[Rigid_rotor#Quantum_mechanical_linear_rigid_rotor|linear rigid rotor]]
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| *The [[Rigid_rotor#Quantum_mechanical_rigid_rotor|symmetric top]]
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| *The [[Hooke's atom]]
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| *The [[Spherium]]
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| *Zero range interaction in a harmonic trap<ref>http://www.springerlink.com/content/k86t52r653522rk6/</ref>
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| *The [[Quantum pendulum]]
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| *The [[Rectangular potential barrier]]
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| ==References==
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| {{reflist}}
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| == See also ==
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| * [[List of quantum-mechanical potentials]] – a list of physically relevant potentials without regard to analytic solubility
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| * [[List of integrable models]]
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| == Reading materials ==
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| * {{cite book
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| | last = Mattis
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| | first = Daniel C.
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| | authorlink = Daniel C. Mattis
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| | title = The Many-Body Problem: An Encyclopedia of Exactly Solved Models in One Dimension
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| | publisher = [[World Scientific]]
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| | date = 1993
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| | isbn = 981-02-0975-4}}
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| [[Category:Quantum mechanics]]
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| [[Category:Quantum models]]
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| [[Category:Physics-related lists|Quantum-mechanical systems with analytical solutions]]
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29 year-old Florist Colton Crochet from Spruce Grove, really likes comics, como ganhar dinheiro na internet and papercraft. Recently had a family voyage to Old City of Sana'a.
Look at my blog; como conseguir dinheiro