In a previous blog entry, I show the “derivation” of the Schrödinger equation, which governs the behavior of the probabilistic wave function of a quantum particle. In this entry, I will show how this is related to the heat flow and temperature gradients we experience in our everyday lives.

**Deriving the Heat Equation**

We have an intuitive understanding that heat flows from hot objects to colder objects nearby. This is the second law of thermodynamics. It also applies to a single object like a sheet of metal, a cooking pan, or a wire. If there is a spot on that object that is hotter than the rest of the object, perhaps from a heat source, heat from that spot will spread out through the object over time. Here I will derive the heat equation, which was developed by Joseph Fourier.

First, start with Fourier’s law of heat transfer, which is an experimental result. It models the relationship between heat conduction or flow rate in a material and the temperature gradient along the direction of heat flow through that material. Imagine we are dealing with a roughly 1-dimensional object like a wire. The law of heat transfer in that wire will be:

Where:

Is the heat flow.

Is the temperature gradient along the wire length .

Is the thermal conductivity of a material.

In the visualization above, the heat going into segment at is . The heat going out of segment at is

Energy conservation (the first law of thermodynamics) demands the net amount of heat flow in the segment is the difference between the flow in (at ) and flow out (at ).

Applying Fourier’s law of heat transfer to this equation we have:

The amount of heat flowing in/out of the material will raise/lower its temperature according to the material’s mass and heat capacity .

For a flow rate , a total amount of heat flows through after a time duration .

Applying these substitutions to the energy conservation equation we have:

Note that the material here is a wire, so we can consider a 1-D density :

Doing some rearranging we have:

Shrinking these differences to infinitesimal differences gives us a partial differential equation: namely the heat equation!

With a heat source term we have:

Note thermal diffusivity of the material here:

This can be extended to include more dimensions to model how the temperature gradient changes as heat flows through a 2-d (plate) or 3-d (block) material.

**Transformation**

We can transform the Heat equation into the Schrödinger equation by making some variable substitutions. Both equations are partial differential equations with first-order derivatives with respect to time and second-order derivatives with respect to space. To perform this transformation follow the steps below:

Turn temperature into quantum wave function (probability density) multiplied by Planck’s constant :

So one can imagine quantum waves behaving like temperature.

We can turn thermal diffusivity into quantum spin unit divided by mass :

Turn the heat source term into the potential energy times the quantum wave function multiplied by :

So this is interesting, it means the “heat source” is dependent on the “temperature”, which is distinct from what we encounter in our day-to-day.

Turn time into imaginary time (This is known as a Wick rotation):

The presence of imaginary numbers here implies oscillation. This is how real “temperature” becomes complex probability waves, with real and imaginary components. We then have the Schrödinger Equation:

**1-D Case**

We can see how the behavior of temperature and quantum waves compare. We will consider the sourceless 1-D case for simplicity: the 1-D wire and 1-D square well.

The Heat/Schrödinger equation is:

Specify an initial condition ()

and a boundary condition at ( and )

The key to this comparison is the Wick rotation factor :

By changing we can go from real to imaginary, from temperature-like to wave-like, and the strange realm in-between.

Solve using separation of variables: split time and space dependence.

We then have two equations to solve:

For the space component:

Time component

Put time and space dependence back together:

There is a solution for each , so put them together for completeness:

Initial condition can be written more explicitly:

Use Fourier transformation to model the evolution of temperature/wave over time:

In the visualizations below, the initial condition is delta function at a point between and . Both the real and imaginary components are shown.

The black line is temperature-like with . Notice it flattens out and stays real.

The red line is quantum wave-like with . Notice the oscillations spread out and leak into the imaginary component.

The other lines, magenta, cyan and blue, are hybrids with . You can see a mix of the behaviors.