The Math behind Schrödinger Equation: The Wave-particle duality and the Heat equation.

History and semiformal derivation of the Schrödinger Equation.

In quantum mechanics there is no certainty about the outcome of an event, as it is in classical physics. This is because each event has associated a probability, and that is what the Schrödinger Equation tell us about a particle. But, how does Erwin Schrödinger got to solve this? Richard Feyman once said:

It isn’t possible to derive it from something you know, it just simply came out from the mind of Schrödinger.

However, we would see that it has a relationship with the heat equation proposed by Joseph Fourier in 1822 (a diffusion equation à la Fourier) and that the particle-wave postulates of Albert Einstein and Louis de Broglie give us a hint to a semi-formal derivation.

Figure 1: The Heat Equation. Source: Author

1. The Wave-particle duality

In 1897 it was proposed that electrons are emitted from atoms when they absorb energy from light, this phenomenon is known as the photoelectric effect. Later, in 1905 Einstein provided an explanation to this event by postulating the existence of quanta or discrete units of energy.

Figure 2. Einstein's proposal. Source: Author

The relationship in Figure 2 tells us that the energy E of a particle can take multiples of the plank constant h only in relation to the frequency v of the wave. In other words, Einstein proposed that a wave behave as a particle in certain processes because it is only necessary to know the frequency of it.

At the beginning of the 20's, it remained a desideratum the possibility of diffracting the electron beams. In fact, it could be expected that the distortion in the wave structure of an electron near an aperture (comparable in size to the wavelength) could diffuse it. Until 1924 when Louis de Broglie, under Jean Perrin and Paul Langevin doctorate thesis direction, gave an expression that explicitly calculates the wavelength of a wave

Figure 3. De Broglie’s proposal. Source: Author

where λ and p represent respectively the length and momentum of the particle. Now, how can we get to the equation in Figure 7(below) from the prior postutales? In three words: energy, space and time, specifically:

Figure 4. Total energy of a particle. Source: Author

• Energy Requirement. The total energy (kinetic and potential) of a particle allows us to find a relationship with the wave-particle duality. In other words, we need to satisfy an energy requirement to be able to model the behavior of the particle.

Figure 5. Change with respect to space. Source: Author

• Behavior in space. The spatial frequency, k,describes the number of waves by space unit. Moreover, the change with respect to space (how does a magnitude changes if it changes of place) show us how does the initial state of the wave evolves in spatial frequency terms.

Figure 6. Change with respect to time. Source: Author

• Behavior in time. The angular frequency, ω, describes the number of waves by time unit. Furthermore, the change with respect to time (how does a magnitude changes over time) of the initial state of the wave led us to express it in terms of the angular frequency.

Thus, we get to the Time Dependent Schrödinger Equation (Figure 7). If you’re interested in the technical details see chapter 1 of [4].

Figure 7. The Time Dependent Schrödinger Equation. Source: Author

2. The heat equation

By the time Fourier started working on this equation (around 1802), the science of heat and the theory of differential equations were in the early stages of development. Specifically, it has already been established heat conduction by temperature differences and heat storage. For example, the work of Jean Baptiste Biot on heat conduction among discontinuous bodies in 1804 or the measurement of heat capacity by Lavoisier and Laplace in 1783.

Figure 8. Heat diffusion in metal teapots. Photo by Steve Cook via pixabay (source)

Moving away from Biot’s perspective, Fourier formulated the heat conduction in terms of a partial differential equation in continuous bodies (like a metal lamina). Therefore, he could think about the problem with three components:

1. The heat transport in space.
2. The heat storage within elements of the body.
3. Boundary conditions.

The boundary conditions are what allows the interaction of the exterior with the interior of the element or body, mathematically they are represented as the second equation, u(x,0), in the figure below (Figure 9).

Figure 9. The heat equation with initial value conditions. Source: Author

Let’s breakdown the problem above to understand it:

• The first equation is what Joseph Fourier formulated, that is, that the change of heat with respect to time (∂ u / ∂ t) equals the acceleration heat (∂² u / ∂ x²) through the body multiplied by some constant m.
• The second equation is how the heat is going to behave initially, and I mean initially because it is establishing what is known in mathematical jargon as initial value condition, the behavior at time t = 0.

Now, we could ask ourselves how to solve this problem, that is, how does the heat evolve in the continuous body through space x and time t ? Let’s go back to Figure 8 and imagine that the bottom of the teapot is our initial value condition. Then, a couple of natural questions to ask are

1. Is it going to be warmer at the lid or at the bottom of the pot?
2. How does the diminishment of heat happens, is it fast or slow?

Mathematics is our answer, particularly, what is known as the Heat kernel (Figure 10), it is the core behind the solution of the problem in Figure 9.

Figure 10. The heat Kernel. Source: Author

If you’re familiar with the normal distribution, this function may seem recognizable and that’s because the root behind them is the gaussian function. Now, if you once took a probability course you could remember that most of the values are centered around the mean μ and that the decay is explained by the variance σ². In a similar way, for the heat kernel, the diffusion of heat depends on the time t and the constant m. Therefore, the decay of this function explains to us how heat is diffused through time and space. For the teapot (Figure 8), how the heat coming from the burner is diffusing while it reaches the lid and time pass.

But, does this equation assure the boundary conditions? Yes, and here’s the trick integration. As one also remembers from a probability course, a probability distribution must satisfy that it integrates to one. In teapot terms that one could measure distinct temperatures along the pot. So, assuming that the heat kernel fulfill this property, one could proof mathematically that it integrates to f(x). Try it out, integrate the function in Figure 11, as a hint you could check out the concept of the Dirac Delta function.

Figure 11. Solution to the initial value problem of the heat equation. Source: Author

Finally, we said before about a relationship between the heat equation and the time-dependent Schrödinger equation, to refresh our memory let’s see again Figure 1 and Figure 7. At a first glance there appears to be no relationship at all, but if we take into account the common terms in the two equations and by setting V(x) = 0 (there is no force acting on the particle), we can reformulate the equation in Figure 7 as follows.

Figure 12. The free Schrödinger equation. Source: Author

Thus, the Schrödinger equation is the heat equation but with an imaginary constant, see the concept of imaginary number.

3. What is the Schrödinger equation?

After the publication of the seminal work of Louis de Broglie, no one could understand it. In fact, Erwin Schrödinger rejects his ideas at the beginning, until Victor Henri replied him “You ought to read de Broglie’s thesis again; Langevin thinks this is a very good work” and so Erwin began to develop his ideas. Moreover, around the time Peter Debye and Erwin Madelung were trying to devise a wave theory of the atomic levels.

Schrödinger needed two attempts to set the foundations of what is now know as non-relativistic wave mechanics. In the first one, he tried to generalize De Broglie’s waves to the electron on the hydrogen atom (bound particles). Nevertheless, the results didn’t agree with actual observations because he was not taking into account the spin of the electron. A few months later, on his second attempt he noticed that by treating the electron in a non-relativistically way his approach agreed with observation. Therefore, publishing his remarkable article “Quantization as a problem of proper values” in 1926.

Finally, what is the Schrödinger equation? It is a formulation which describes how atoms or subatomic particles will behave over time based on its current state or initial state (initial value condition).

4. Further topics

In the second part of this series of articles about the Schrödinger equation, we’re going to develop the concepts of a wave packet as initial value condition and the time independent case to study the Quantum Tunneling effect. For technical details of the mathematical developments in this article, see Chapter 1 and 2 of [4].

References

[1] Eisberg, Robert. Quantum Physics. John Wiley & Sons, 1985.

[2] Jammer, Max. The Conceptual Development of Quantum Mechanics. The American Institute of Physics, 1989.

[3] Naramsinhan, T.N. Fourier’s Heat conduction equation: History, Influence and Connections.

[4] Trejo, Miguel. Una interpretación matemática de la ecuación de Schröedinger (Tesis). URL: https://github.com/TremaMiguel/Thesis-Schrodinger-Equation

[5] Wheaton, Bruce. The Tiger and the Shark. Cambridge University Press, 1983.