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This theorem helps us understand quantum mechanics in an intuitive way, and even to visualise it like we can visualise classical physics!
Hey everyone! I'm back with Part 2 of this mini-series on the Ehrenfest theorem! In this video, we learn about what expectation values are, as well as how important they are in the mathematics of quantum physics. We then take everything we learnt about operators and commutators in Part 1 (check it out here: • Ever heard of Quantum ... ) as well as our newly-understood expectation values, and apply all this knowledge to Ehrenfest's theorem.
Firstly, expectation values are useful because they are essentially the weighted average of a probability distribution. For example, the wave function of a quantum system is directly related to the probability distribution of finding particles in that system at different points in space. This is the probability distribution when a measurement is made using the position operator. The expectation value is simply the weighted average (the mean) of this distribution. It's not necessarily the position at which we're most likely to find the particle(s), which is the modal position. But if that's the case, then why do we even care about the expectation value?
Well, for a couple of reasons really. Firstly, the expectation value can act as a nice (but primitive) indicator of how the probability distribution is behaving. It's exactly like using an average value as some sort of representative of the entire data set. For example, we might say that the mean height of a class full of students is 5 feet 6 inches (or 1.68 m). This gives us some information (though not as much as knowing the height of each student in the class) because we can use this mean to compare to the mean student height of other classes. In the same way, we can find the expectation value of an electron's position in a system.
The second (more compelling) reason to use expectation values is because they often behave in a rather surprisingly classical way. Although quantum mechanics is probabilistic and difficult to comprehend, the expectation values of certain probability distributions can behave like their classical counterparts. We see this through Ehrenfest's theorem. When we substitute the position operator into Ehrenfest's theorem, we find that the mass of our particle multiplied by the rate of change of the expectation value of the position is equal to the expectation value of the particle's momentum. This might be difficult to understand in text form, but basically this is almost identical to the classical physics definition of momentum! The mass of a classical object, multiplied by its rate of change of position (i.e. its velocity) is equal to its momentum.
And this tells us something interesting. Quantum systems behave in a specific way so that their probability distributions change with time, in a way that their expectation values behave surprisingly classically.
We can even substitute the momentum operator into Ehrenfest's theorem, and follow the mathematics through. This one's a bit more fiddly and needs a few more assumptions, but under the right conditions the expectation value of the momentum of a system has a relationship very similar to Newton's Second Law of Motion - the fundamental classical relationship! We find that the rate of change of the expectation value of the momentum of the system is equal to (the equivalent) of the force exerted on the system. And Newton's Second Law says that the rate of change of a classical object's momentum is equal to the force exerted on it. Pretty cool right?
Ehrenfest's theorem of course has lots of operators in it, as well as a commutator and an expectation value. For this reason, it was important to split this particular explanation up into two. I hope you enjoyed both these videos, because I really enjoyed making them. I really like breaking down a complex looking equation into small bits and explaining them to you guys. Thanks so much for watching my ramblings lol.
Again, thanks for all your support, and big thanks to SkillShare for sponsoring this video! I have a second channel, Parth G's Shenanigans here on RU-vid, where I post my own original music. I also have an Instagram @parthvlogs. Feel free to follow me on there, and subscribe here for more fun physics content! I'll see you really soon :)
2 авг 2024
