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Second, it is necessary to Fourier transform back and forth because the linear step is made in the frequency domain while the nonlinear step is made in the time domain. An example of usage of this method is in the field of light pulse propagation in optical fibers, where the interaction of linear and nonlinear mechanisms makes it difficult to ...
The Klein–Gordon equation, + =, was the first such equation to be obtained, even before the nonrelativistic one-particle Schrödinger equation, and applies to massive spinless particles. Historically, Dirac obtained the Dirac equation by seeking a differential equation that would be first-order in both time and space, a desirable property for ...
The Fourier transform takes functions in the above space to elements of l 2 (Z), the space of square summable functions Z → C. The latter space is a Hilbert space and the Fourier transform is an isomorphism of Hilbert spaces. [nb 10] Its basis is {e i, i ∈ Z} with e i (j) = δ ij, i, j ∈ Z.
Joseph Fourier introduced sine and cosine transforms (which correspond to the imaginary and real components of the modern Fourier transform) in his study of heat transfer, where Gaussian functions appear as solutions of the heat equation. The Fourier transform can be formally defined as an improper Riemann integral, making it an integral ...
In quantum mechanics, the Schrödinger equation describes how a system changes with time. It does this by relating changes in the state of the system to the energy in the system (given by an operator called the Hamiltonian). Therefore, once the Hamiltonian is known, the time dynamics are in principle known.
In the generalized version of the Bloch theorem, the Fourier transform, i.e. the wave function expansion, gets generalized from a discrete Fourier transform which is applicable only for cyclic groups, and therefore translations, into a character expansion of the wave function where the characters are given from the specific finite point group.
The Schrödinger equation for a particle in a spherically-symmetric three-dimensional harmonic oscillator can be solved explicitly by separation of variables. This procedure is analogous to the separation performed in the hydrogen-like atom problem, but with a different spherically symmetric potential V ( r ) = 1 2 μ ω 2 r 2 , {\displaystyle ...
Note that because the initial Fourier transformation contained Lorentz invariant quantities like = only, the last expression is also a Lorentz invariant solution to the Klein–Gordon equation. If one does not require Lorentz invariance, one can absorb the 1 / 2 E ( p ) {\displaystyle 1/2E(\mathbf {p} )} -factor into the coefficients A ( p ...