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Using the Fourier inversion theorem, the free particle wave function may be represented by a superposition of momentum eigenfunctions, or, wave packet: [4] (,) = ^ (()), where =, and ^ is the Fourier transform of a "sufficiently nice" initial wavefunction (,).
The wave function of an initially very localized free particle. In quantum physics, a wave function (or wavefunction) is a mathematical description of the quantum state of an isolated quantum system. The most common symbols for a wave function are the Greek letters ψ and Ψ (lower-case and capital psi, respectively). Wave functions are complex ...
The second term is the free particle propagator, corresponding to i times a diffusion process. To lowest order in ε they are additive; in any case one has with (1) : ψ ( y ; t + ε ) ≈ ∫ ψ ( x ; t ) e − i ε V ( x ) e i ( x − y ) 2 2 ε d x . {\displaystyle \psi (y;t+\varepsilon )\approx \int \psi (x;t)e^{-i\varepsilon V(x)}e^{\frac ...
The time-independent Schrödinger equation for the wave function is ^ = [+ ()] = (), where Ĥ is the Hamiltonian, ħ is the reduced Planck constant, m is the mass, E the energy of the particle. The step potential is simply the product of V 0 , the height of the barrier, and the Heaviside step function : V ( x ) = { 0 , x < 0 V 0 , x ≥ 0 ...
Consequently, the wave function also became a four-component function, governed by the Dirac equation that, in free space, read (+ (= )) =. This has again the form of the Schrödinger equation, with the time derivative of the wave function being given by a Hamiltonian operator acting upon the wave function.
The azimuthal wave functions in that case are identical to the energy eigenfunctions of the particle on a ring. The statement that any wavefunction for the particle on a ring can be written as a superposition of energy eigenfunctions is exactly identical to the Fourier theorem about the development of any periodic function in a Fourier series.
The simplest approach is to focus on the description in terms of plane matter waves for a free particle, that is a wave function described by =, where is a position in real space, is the wave vector in units of inverse meters, ω is the angular frequency with units of inverse time and is time.
Some trajectories of a particle in a box according to Newton's laws of classical mechanics (A), and according to the Schrödinger equation of quantum mechanics (B–F). In (B–F), the horizontal axis is position, and the vertical axis is the real part (blue) and imaginary part (red) of the wave function.