Journal Club: From Baking a Cake to Solving the Schrödinger Equation

This latest installment of the journal club isn't exactly medical physics related, but it was something that I thought was interesting more from a topical point of view rather than the actual research itself.

The title of the paper, From Baking a Cake to Solving the Schrödinger Equation, may give the impression of being just another trivial waste of time and a candidate for an Ig Nobel award.

Look a little deeper and it actually turns out to be much more.

Physics is all about developing, describing and modeling systems or processes, which is exactly what this paper is about. Once you have a decent model, you use it to make predictions and design experiments to verify those predictions. The author takes a seemingly trivial process (how changes in cake dimension and volume affect baking time) and attempts to describe the process mathematically using well known equations and experimental results.

Found via Talk Like A Physicist

First comes the initial model. The authors begin with the diffusion equation.

Along with some known initial conditions, the equation can be solved to produce a solution that approximates the cake baking process. In the case of the paper, the solution provides the baking time given the dimensions of the cake.

Once the solutions have been obtained, it's now possible to visualize the theoretical behaviour of the system. However, this still needs to be correlated to the actual observed behaviour, which is where the experimental part comes in. If the experimental results deviate from the expected theoretical results, it probably means that some of the assumptions in the model were incorrect and need to be modified. Usually it's possible to figure out how the model needs to be modified by studying the differences between the theoretical and experimental results. With the modified model, a new set of solutions can be created and then verified against experimental results.

With enough iterations, the model becomes accurate enough to make predictions that can be verified experimentally. Often new experiments need to be designed in order to verify any predictions made. Sometimes current technology is insufficient and verifying predictions must wait years or decades before it can be done. If you're really clever, you notice that the solutions of the model can be applied to other systems, or you notice that the solution or equations resemble a process in a completely unrelated field. In the paper, the author notes the similarity between the diffusion equation and Shrödinger's equation and analyzes not only what the solutions mean when applied to the Shrödinger equation, but also the limitations of the solutions.

Thus, rather than being an apparently trivial paper, this paper is really a very impressive study of what physics and the process of doing physics is all about.

Abstract:

The primary emphasis of this study has been to explain how modifying a cake recipe by changing either the dimensions of the cake or the amount of cake batter alters the baking time. Restricting our consideration to the génoise, one of the basic cakes of classic French cuisine, we have obtained a semi-empirical formula for its baking time as a function of oven temperature, initial temperature of the cake batter, and dimensions of the unbaked cake. The formula, which is based on the Diffusion equation, has three adjustable parameters whose values are estimated from data obtained by baking génoises in cylindrical pans of various diameters. The resulting formula for the baking time exhibits the scaling behavior typical of diffusion processes, i.e. the baking time is proportional to the (characteristic length scale) of the cake. It also takes account of evaporation of moisture at the top surface of the cake, which appears to be a dominant factor affecting the baking time of a cake. In solving this problem we have obtained solutions of the Diffusion equation which are interpreted naturally and straightforwardly in the context of heat transfer; however, when interpreted in the context of the Schrödinger equation, they are somewhat peculiar. The solutions describe a system whose mass assumes different values in two different regions of space. Furthermore, the solutions exhibit characteristics similar to the evanescent modes associated with light waves propagating in a wave guide. When we consider the Schrödinger equation as a non-relativistic limit of the Klein-Gordon equation so that it includes a mass term, these are no longer solutions.