Laplace operator has many uses, and it is also an important example of elliptic operator.
In physics, the commonly used mathematical models are Apollo equation, heat conduction equation and Helmholtz equation.
In electrostatics, the applications of Laplace equation and Poisson equation can be seen everywhere. In quantum mechanics, it represents the kinetic energy term in Schrodinger equation.
In mathematics, the function that is reduced to zero by Laplace operator is called harmonic function. Laplacian operator is the core of Hodge's theory and the result of homology on Drummon.
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In quantum mechanics, Hamiltonian (h) is a considerable measurement, which corresponds to the total energy of the system. Like all other operators, the spectrum of Hamiltonian operator is the set of all possible results when measuring the total energy of the system. Like other self-adjoint operators, the spectrum of Hamiltonian operators can be decomposed into pure points, absolute continuity and singularity by spectral measures. The pure point spectrum corresponds to the eigenvector, which in turn corresponds to the bound state of the system. Absolute continuous spectrum corresponds to free state. Singular spectrum is very interesting and consists of physically impossible results. For example, consider the case of a square well with finite depth, which allows a bound state with discrete negative energy and a free state with continuous positive energy.
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D 'Alembert operator is the form of Laplace operator in Minkowski space-time. The symbol of this operator is a square to indicate that it is in Minkowski space-time in four dimensions. D 'Alembert operators are generally expressed as or can be expressed as, and they are exactly the same.
D 'Alembert operator is mainly used in electromagnetism and special relativity. For example, the D 'Alembert operator is used in the Klein-Gordon equation.