You also have to know that under the diffusion equation, sine waves remain sine waves for all time, except they shrink; and the faster they wave, the faster they shrink. The rate of shrinking is quadratic in wave number, so $\sin(2x)$ shrinks four times as fast as $\sin(x)$ .

I don't know whether Schrödinger proved or guessed the equation with his name, but this equation can be derived similarly with the diffusion equation - see Gordon Baym, "Quantum Mechanics". However, differently from the diffusion equation, the diffusion coefficient in the Schrodinger equation is imaginary. That tells us that we have to ...

It seems like one can transform the diffusion equation to an equation that can replace the wave equation since the solutions are the same. I think replacing a real constant to an imaginary constant is deceivingly simple: even though the equations and solutions are written the same, they encode different information.

2018年6月12日 · Please see the term here: A measure of the tendency of a magnetic field to diffuse through a conducting medium at rest; it is equal to the partial derivative of the magnetic field strength with respect to time divided by the Laplacian of the magnetic field, or to the reciprocal of 4πμσ, where μ is the magnetic permeability and σ is the conductivity in …

2019年7月23日 · The solution of diffusion equation $$ \partial_t\rho=D\nabla^2\rho$$ with a point source $$ \rho(0,z) ...

2019年2月15日 · The conjugate field ψ∗ is but the complex conjugate of ψ, so an extra degree of freedom to expedite derivation of the diffusion equation, $$ \nabla^2 \psi = a^2 \partial_t \psi , $$ analogous to the Lagrangian of the free Schroedinger equation, real in that case--only.

The difference is typically the diffusion coefficient: \begin{align} \frac{\partial \psi}{\partial t}&=\nabla\cdot\left(\kappa\nabla\psi\right)\tag{diffusion}\\ \frac{\partial \psi}{\partial t}&=\kappa\nabla^2\psi\tag{heat} \end{align} Under the diffusion equation, we typically take $\kappa$ to be a spatially-dependent variable whereas in the heat equation it is a uniform …

2022年8月29日 · i.e. the discrete version of the diffusion equation using time-explicit Euler scheme for the first order time derivative and centered scheme for the second order space derivative. From probability transition to PDEs: general equation. If you add "extra terms" in the probability transition relation:

2021年1月7日 · Since this is clearly false, the diffusion equation is not time-reversal symmetric, while equations like the elementary wave equation are. On the other hand, in quantum mechanics we define time-reversal symmetry to be implemented by an antiunitary operator, which implies that the wave function be mapped to its complex conjugate.

2018年10月24日 · The visual idea is to describe the diffusion of some dilute chemical around a spherical sink or a sink at some point. It would be nice to obtain a time evolution when starting with a uniform density (this is only possible in problems 1) and 5)), but I would already be satisfied with a "nice" steady-state solution.