A Casual Chat
Last month, I bought three relativity textbooks and an Introduction to Quantum Mechanics online. I originally intended to study the mathematical framework of relativity thoroughly. However, after the books arrived, I found myself deeply captivated by quantum mechanics and have been reading it continuously. Moreover, while studying quantum mechanics, my knowledge of linear algebra and differential equations has increased significantly, which was quite unexpected. In my view, whether it is special or general relativity, it is essentially a geometric theory; you always have to imagine what is observed from one reference frame and then what is seen from another. Quantum mechanics, although entirely new to me, is more mathematical in nature, primarily involving the solution and understanding of differential equations. I think this is why I am more interested in quantum mechanics, as I am better at algebra than geometry.
The content in quantum mechanics that fascinates me the most is the path integral formulation invented by Feynman. Records show that Feynman used his method to calculate results in a single night that took others months to achieve, demonstrating the superiority of the path integral approach. Of course, I am also aware that path integrals are not simple; they involve functional integration, a very advanced topic. For a student like me who hasn’t even mastered mathematical analysis yet, functionals are difficult to grasp. Nevertheless, I try my best to get closer to it. To this end, I have come across quite a bit of exciting content, such as the derivation of the Schrödinger equation, the mechanics-optics analogy, the Jacobi equation, and so on.
Regrettably, in orthodox quantum mechanics textbooks, these exciting topics are rarely covered; if they are, they are usually just mentioned in passing. Most quantum mechanics courses do not teach path integrals, or if they do, they are only included as an appendix. The derivation of the Schrödinger equation is often not covered either. This has led me to develop a habit: the most interesting things in a book are often in the appendices. Therefore, I generally have no interest in the formal, standard content of textbooks; the appendix material is what I love to read most. However, those exciting topics are not necessarily difficult. For instance, the heuristic derivation of the Schrödinger equation below is not only easy but also intuitive.
The Schrödinger Equation
Before the birth of quantum mechanics, scientists had already discovered through experiments that light possesses both wave-like and particle-like properties. Louis de Broglie proposed that matter also possesses both wave and particle properties, which laid the foundation for quantum mechanics. According to quantum theory, the energy of a photon can be expressed as E=h\nu=\hbar (2\pi \nu), where \nu is the frequency, \hbar=\frac{h}{2\pi}, and h is Planck’s constant. It is customary to denote \omega=2\pi \nu, so E=\hbar \omega.
At the same time, a photon also possesses momentum, the magnitude of which is p=\frac{h}{\lambda}=\hbar k, where k=\frac{2\pi}{\lambda}. These properties are interchangeable between waves and particles due to their wave-particle duality.
For the simplest one-dimensional simple harmonic wave, we can write its equation as y=A\cos(kx-\omega t). We can understand it this way: at time t=t_0, the shape of the wave is y=A\cos(kx-\omega t_0); at position x=x_0, the amplitude varies according to the law y=A\cos(kx_0-\omega t). For mathematical convenience, we convert the wave into a complex form, extending it to a two-dimensional wave, i.e., y=Ae^{i(kx-\omega t)}. It is evident that its real part is the equation of the one-dimensional simple harmonic wave.
Since physical particles have wave properties, we should assign a wave equation to the particle. Considering the simplest form of a harmonic wave, we have \Psi(x,t)=Ae^{i(kx-\omega t)}. The following process is straightforward:
\begin{aligned} \frac{\partial \Psi}{\partial t} &=& -i\omega \Psi \\ \hbar \frac{\partial \Psi}{\partial t} &=& -i\hbar \omega \Psi = -i E \Psi \\ i\hbar \frac{\partial \Psi}{\partial t} &=& E \Psi \end{aligned}
By analogy with classical mechanics, considering a conservative system, we have "Energy = Kinetic Energy + Potential Energy", i.e., E=T+U=\frac{p^2}{2m}+U=\frac{{\hbar}^2 k^2}{2m}+U. Therefore, the above equation becomes: i\hbar \frac{\partial \Psi}{\partial t}=\left(\frac{{\hbar}^2 k^2}{2m}+U\right)\Psi
Where: \frac{\partial \Psi}{\partial x}=ik \Psi, \quad \frac{\partial^2 \Psi}{\partial x^2}=-k^2 \Psi
Substituting these in, we obtain: i\hbar \frac{\partial \Psi}{\partial t}=-\frac{\hbar^2 }{2m}\frac{\partial^2 \Psi}{\partial x^2}+U\Psi
This is the Schrödinger equation, which is a linear partial differential equation.
Some sensitive readers might notice that from \frac{\partial \Psi}{\partial x}=ik \Psi, one could directly obtain k^2=-\frac{1}{\Psi^2}\left(\frac{\partial \Psi}{\partial x}\right)^2, and then by substitution get: i\hbar \frac{\partial \Psi}{\partial t}=-\frac{{\hbar}^2 }{2m}\frac{1}{\Psi}\left(\frac{\partial \Psi}{\partial x}\right)^2+U\Psi
Could this also be an equivalent form of the Schrödinger equation?
[Click to view Animation: Fourier Transform]
Of course not. Since the above is a "heuristic derivation," we considered the simplest harmonic wave, which is why we obtained different forms. However, from the knowledge of Fourier series, we know that any complex wave can be formed by the superposition of harmonic waves with different frequencies. Therefore, the equation we obtain must satisfy the condition that "the superposition of two solutions is still a solution to the original equation." That is to say, only the linear form of the Schrödinger equation: i\hbar \frac{\partial \Psi}{\partial t}=-\frac{{\hbar}^2 }{2m}\frac{\partial^2 \Psi}{\partial x^2}+U\Psi is the correct wave equation for quantum mechanics. Simply put, the wave equation must be linear.
Summary
Clearly, the above is merely a formal and mathematical treatment; we have not yet assigned physical meaning to it, such as the specific meaning of the wave function, and the correctness of the results remains to be verified. This is not a rigorous derivation, which is why I emphasize it is heuristic thinking. However, quantum mechanics textbooks tell us that this is correct. For an introduction to quantum mechanics, this is sufficient. Readers should also realize that this process is not difficult; it is even surprisingly simple. However, I have yet to find a quantum mechanics textbook that mentions this; they all seem to lack this creative, heuristic thinking and are instead filled with complex analytical calculations. Perhaps it is because I have read too few books. Or perhaps everyone’s learning approach is different. I just feel that while not everyone can master quantum mechanics, everyone can learn some quantum mechanics.
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