- In the classical world, we are used to things working usually. Take a flying cannonball. In time, we can describe its position xx and tell its velocity or momentum pp (same thing as velocity, for all practical intents and purposes.) The quantities xx and pp are numbers (maybe sets of numbers, i.e., coordinates or coordinate components, but expressed using numbers nonetheless.) Because they are numbers, we can multiply them in any order: the products p⋅xp⋅x and x⋅px⋅p represent the same thing.
- While we are accustomed to things working in a certain way in the classical world, nature often surprises us. Particles, unlike miniature cannonballs, do not have positions and momenta. This unexpected nature of particles is a fascinating aspect of quantum physics. They have stated. When interpreted as positions or momenta under the right circumstances, these states can be represented in mathematics by operators like ^xx^ or ^p,p^ acting on a state ψ.ψ.
- Moreover, this has two consequences.
- The quantum system’s state, ψ, is governed by the famous Schrödinger equation, a “wave equation.” This means that, under most circumstances, it will give some periodic solution. As a result, the state of this quantum system will exhibit wavelike behavior, including constructive and destructive interference, a phenomenon that is truly awe-inspiring and something cannonballs never do. Second, those operators I mentioned for positions and momenta yield measurable numbers under the right circumstances. So, with the proper experiment, that quantum system still exhibits behavior that makes it appear like a miniature cannonball.
- So when you fire, say, an electron, while the electron is en route, it does not have a position or momentum in the classical sense. The state describes its existence ψ and its wave function. However, when it affects, say, the fluorescent screen of a CRT, that interaction confines it to a well-defined position. Its position operator ^x, acting on its state, produces a set of coordinates, the location where you see a flash on the screen. This is similar to how a camera captures a moving object, freezing it in a specific position at a specific time. If this explanation appears abstract and complex to make sense of, that is no accident. Quantum physics cannot be intuited. To use a stupid analogy, almost everyone had a chance to play with language models like ChatGPT in the past year. Ignoring their more recent attempts to integrate language and visual capabilities, the original language model has no visual experiences. No ability to intuit or visualize anything related to space or geometry. It can still reason about these things, but it often struggles, as its words are not connected to any experience it would have at its disposal. We humans are in the same situation as quantum physics. Just as language models have no visual cortex, our brains lack a “quantum physics cortex” that would allow us to intuit the quantum world. Therefore, like the language models, we must resort to painful, slow, step-by-step reasoning, relying on the reasonably precise language of mathematics. However painful this feels, it is essential to remember that any attempt to visualize quantum physics using our intuition rooted in classical physics is doomed to fail and will probably become an obstacle to understanding.
