![]() Now that he’s gone, death sits amid the onrush of my thoughts like a black stone, opaque and immovable. My sister Martha chokes up when she calls to tell me, and I choke up too, mirror neurons or whatever doing their job. I expected Dad’s death for years, even hoped for it, for Lois’s sake, and his. When Emily and I visited him and Lois, he thrust his fist up, grinning rakishly, and exclaimed in a hoarse whisper, “Hooray for you!” Dad couldn’t really converse, but his spirits were still high. He lived with Lois, my stepmom, at their home in Connecticut, but she had to hire someone to help care for him. The last few years were tough, after strokes battered his old brain. So, this operator, when applied to a wave function, yields the value of what we’d find in an experiment.Dad died. This is the real-life equivalent of measuring the particle. ![]() Since the system is described by a function, we can apply a “measurement operator” to that function. But in quantum physics, the maths primarily deals with wavefunctions. ![]() We can describe momentum with certainty - the product of mass and velocity. So a question we can ask is, how do we mathematically deal with the idea of probabilistic momentum? In classical physics, we don’t need probabilities. With a simple mathematical transformation (usually not that simple), we can calculate the probability of finding our particle with different values of momentum in space. Unsurprisingly, the wave function is also directly linked with momentum. So with the quantum mechanical difference that we now use wave functions instead of definite positions, can we still say that momentum is a product of mass and velocity? Well, no. It could simply be 3, or 5, maybe even 9. This value need not necessarily be the position that has the greatest probability. It’s only when we make a measurement on the particle, that we cause a collapse in the wave function, and thus know a certain value for the particle’s position. All that matters is that if we wanted to, we could find the Ψ of our system. Now there are a few subtleties to this but for our purpose, all we care about is that the Ψ² function is telling us the probability distribution of finding our electron in our one–dimensional system. Whereas at 4, Ψ² is small and so we’re least likely to find our electron. Specifically, this function is telling us - Ψ² is telling us - that at r=2 because Ψ² is large, we’re most likely to find our electron. It’s this function like we established, that is the important one. If you want to find out more about wave functions, check this out: It’s important to note that the wave function, when squared, can be used to calculate the probability of finding a particle at different positions in space. The wave function of a system is a mathematical function that contains all the information we can know about a system. Therefore, a particle’s position, through space and time, is defined using a wave function. We can, with certainty, only describe the probability of a particle being found at a particular point in space. ![]() The first difference I think you would’ve already taken note of is that in quantum theory, we deal with probabilistic positions - not definite ones. But how do we deal with momentum in quantum mechanics? Is it again, just mass and velocity? Or is something else altogether? In classical physics, we calculate an object’s momentum by multiplying its mass by its velocity. “How does something complex describe something real?” The imaginary number crept its way into equations and showed up virtually everywhere. It’s a completely new idea that replaces our classical understanding.Īround when Schrodinger was messing with wave functions, it was well known that quantum mechanics, to be successful, required complex systems. The list goes on but what I’d like for you to take away from this is that quantum mechanics is not an improvement on classical mechanics. The idea of an objective reality seems foolish replaced with notions like probability distributions rather than deterministic outcomes, wavefunctions rather than positions and momenta, Heisenberg uncertainty relations rather than individual properties, complex rather than real numbers. And since it’s quantum physics, new rules are needed, and to describe them, new counterintuitive equations. We throw classical mechanics out the window and quantum physics takes over. When things get down to the size of an atom, the physics that governs us no longer applies. On the smallest scale, the world is weird. ![]()
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