The Schrödinger equation shares a direct mathematical connection with the heat equation.
the verdict
SUPPORTED
the evidence backs this
refutedsupported
the weight of evidence
2 sources for · 0 against
Peer-reviewed literature demonstrates direct mathematical formulations and Cauchy problems that span and link the diffusion (heat) equation and the Schrödinger equation, often through analytical continuation or limiting parameters.
We consider the limiting case λ→0 of the Cauchy problem ∂uλ/∂t= (λ/2μ) ∇2xuλ +[V (x)/λ]uλ, uλ(x,0) =exp[−S0(x)/λ]T0(x); S0, T0 independent of λ, for both real and pure imaginary λ. We prove two new theorems relating the limiting solution of the above Cauchy problem to the corresponding equations of classical mechanics μ (d2x/dτ2)(τ) =−∇xV[x (τ)], τ∈ (0,t). These relationships include the physical result quantum mechanics → classical mechanics as h/→0.
We consider the limiting case λ→0 of the Cauchy problem, ∂gλ(x,t)/∂t = (1/2) λΔxgλ(x,t)+(V(x)/λ) gλ( x,t), with gλ (x,0) = exp{−S0(x)/λ}T0(x), V, S0 being real-valued functions on N, T0 a complex-valued function on N; V, S0, T0 being independent of λ, Δx being the Laplace–Beltrami operator on N, some complete Riemannian manifold. We prove some new results relating the limiting behavior of the solution to the above Cauchy problem to the solution of the corresponding classical mechanical problem D2Z(s)/∂s2 = −∇ZV[Z(s)], s∈[0,t], with Z(t) = x and Z(0) = ∇S0(Z(0)).One of our results is equivalent to the fact that for short times Schrödinger quantum mechanics on the Riemannian manifold N tends to classical Newtonian mechanics on N as h/ tends to zero.
Everything we examined (2) — 1 independent source
This check searched the claim as stated. It did not run a separate search for evidence against it.