What is a quantum operator? And just how useful are quantum commutators? Find out how they help us understand the Ehrenfest Theorem!
Hi everyone, I'm back with a new video! This time it's the first in a two-part mini-series on one of the coolest theorems in quantum mechanics - Ehrenfest's Theorem. With a good grasp of high school level mathematics, I believe it is possible to understand this theorem on a basic level. However, we need to do some groundwork first.
(Check out part 2 of this mini series here: • Quantum Physics Become... )
A quantum operator is a mathematical object / entity that corresponds to taking a measurement on a quantum system in real life. Examples of quantum operators include the position operator, the momentum operator, and the Hamiltonian operator. The Hamiltonian is an extremely important operator in quantum mechanics, consisting of kinetic energy and potential energy terms. It's also worth knowing that the operators we will deal with in this video are known as "observables". Entities that respond to things we can observe in the real world (such as position and momentum).
But one of the most interesting takeaways here is that in quantum mechanics, making a measurement is a complicated process. Unlike in classical physics, making a measurement doesn't necessarily affect or alter the system in some way, quantum mechanics tells us that a measurement fundamentally changes the wave function of the system. This process (of taking a measurement, causing the wave function to change) is known as the "collapse of the wave function". And mathematically, this process of making a measurement is represented by "applying" a quantum operator to the wave function ket of a system. The wave function, remember, is just a mathematical representation of all the information we have about our system. It is directly related to the probability distribution of the system, meaning it tells us the likelihood of finding particles that part of our system, in different locations in space (and much more).
Now, because measurements are fundamentally system-altering in quantum mechanics, this means that the order in which we make two measurements can actually give us different results. For example, if we were to measure a quantum system's position, and then its momentum, we might get different results to if we measured its momentum first, and then its position. For this reason, it becomes useful to define a mathematical quantity known as the "commutator" between two measurement operators.
A commutator is just a measure of the difference in results when we measure one quantity (e.g. position) and then the other (e.g. momentum), as compared to when we measure them the other way round (i.e. momentum first, and then position). If switching the order gives us different results, then the commutator is not zero, and the order in which we make measurements needs to be considered carefully. However, in some cases the order does not matter. For such cases (and such pairs of operators), the commutator is zero. This is similar to saying 2 x 5 = 5 x 2. Multiplication is a commutative operation, meaning the order in which we multiply the two numbers does not matter.
Once we learn enough about operators and commutators, we can start to notice that they show up in Ehrenfest's theorem! In the finale of this two-part mini-series, we will be learning about expectation values (very important to QM) and then finally putting it all together to understand what Ehrenfest's Theorem is trying to tell us.
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2 авг 2024
