- #1
Vyurok
- 7
- 1
In the attached image, there is a passage from the textbook Faddeev, L.D., Yakubovskii, O.A. — Lectures on Quantum Mechanics for Mathematics Students, and I have the following two questions:
1)
It is clear what it means to specify the conditions of an experiment in classical mechanics so that the result of measuring any physical quantity (observable, which in classical mechanics is defined as a smooth function of ##n## coordinates and ##n## momenta) is well-defined at any moment in time. For example, one can fix ##n## values of coordinates and ##n## values of momenta at the initial moment; then the values of the coordinates and momenta at any subsequent time are uniquely determined, i.e., ##q(t) = q(t, \; q_{0}, \; p_{0})## and ##p(t) = p(t, \; q_{0}, \; p_{0})##. Therefore, the value of any observable at any subsequent time is uniquely determined, because any observable in classical mechanics is a function of coordinates and momenta, i.e., ##f(t) = f(q(t), \; p(t)) = f(q(t, \; q_{0}, \; p_{0}), p(t, \; q_{0}, \; p_{0}))##.
But what does it mean to specify the conditions of an experiment in quantum mechanics such that these conditions determine the state of the system? That is, so that when the experiment is repeated multiple times, the measurement of any physical quantity at any moment in time yields a probability distribution for the values of that physical quantity at that time? Incidentally, this raises another question: what if the specified experimental conditions are not sufficient to determine the state of the system? Then we begin measuring some physical quantity at a given moment and mark points on the real line – values that the measured quantity takes at that moment in each iteration of the experiment. That is, we run the experiment for the first time, wait for a time ##t_{0}##, measure the chosen physical quantity at that moment, and obtain some value ##A_{1}## – a point on the real line. Then we run the experiment a second time, wait the same amount of time ##t_{0}##, measure the same physical quantity again at that moment, and obtain some value ##A_{2}## – another point on the real line. And so on, until a picture of the distribution of these points on the real line begins to emerge. Looking at this picture, we see that some areas have a higher density of points than others - in other words, even in this case, we can obtain some probability distribution of the points on the real line, right? But then, how is this different from the first case?
2)
In classical mechanics, any measurable physical quantity is a function of coordinates and momenta, i.e., its value at a given time is uniquely determined by the values of the coordinates and momenta at that time. Therefore, observables in classical mechanics can be thought of as smooth functions on phase space, and we note that these can be treated as forming an algebra. But how should we think about observables in quantum mechanics? Nothing is written about this here. Yet somehow there is a claim that observables in quantum mechanics also form an algebra. How should this be understood?