# Heat Equation and Schrödinger Equation

There is an intriguing relationship between Schrödinger equation for a free particle and homogeneous heat equation.

1-dimentional Schrödinger equation for a free particle is

\label{eq:se}
i\hbar\frac{\partial\psi(x,t)}{\partial t}=-\frac{\hbar^2}{2m}\frac{\partial^2\psi(x,t)}{\partial x^2}.

Take the Wick rotation $t\mapsto \tau=it$. Then the Schrödinger equation \eqref{eq:se} turns into

\label{eq:hheq}
\frac{\partial\phi(x,\tau)}{\partial\tau}=\frac{\hbar}{2m}\frac{\partial^2\phi(x,\tau)}{\partial x^2},

where $\phi(x,\tau)=\psi\left(x,\frac{\tau}{i}\right)$. \eqref{eq:hheq} is a homogeneous heat equation with diffusion coefficient $\alpha^2=\frac{\hbar}{2m}$. Conversely, apply the Wick rotation $t\mapsto\tau=-it$ to the 1-dimensional homogeneous heat equation

\label{eq:hhe2}
\frac{\partial u(x,t)}{\partial t}=\alpha^2\frac{\partial^2 u(x,t)}{\partial x^2}.

Then the resulting equation is

\label{eq:se2}
i\hbar\frac{\partial w(x,\tau)}{\partial t}=-\alpha^2\hbar\frac{\partial^2 w(x,\tau)}{\partial x^2},

where $w(x,\tau)=u\left(x,-\frac{\tau}{i}\right)$. \eqref{eq:se2} is a Schrödinger equation for a free particle with $m=\frac{1}{2\alpha^2}$. The solution of \eqref{eq:hhe2} with homogeneous boundary conditions takes the form
$$u(x,t)=\sum_{n=0}^\infty A_ne^{-\lambda_n^2\alpha^2 t}X_n(x).$$
Its Wick rotated solution is
\begin{align*}
w(x,\tau)&=\sum_{n=0}^\infty w_n(x,\tau)\\
&=\sum_{n=0}^\infty A_ne^{-i\lambda_n^2\alpha^2\tau}X_n(x).
\end{align*}
$i\hbar\frac{\partial w(x,\tau)}{\partial\tau}=\lambda_n^2\alpha^2\hbar w(x,\tau)$. Thus for each $n=0,1,2,\cdots$, $E_n=\lambda_n^2\alpha^2\hbar$ is the energy and $\omega_n=\lambda_n^2\alpha^2$ is the frequency of the wave $w_n(x,\tau)$.