The discussion centers on the relationship between the expectation value of spin in quantum mechanics and actual measurements of multiple particles in the same state. When the expectation value of spin in the z direction for a single spin-1/2 particle is zero, measuring an even number of identical particles results in approximately half being spin up and half being spin down. This statistical behavior aligns with the concept of expectation values as limits of relative frequency, particularly in large samples. However, this relationship may not hold in systems with more than two states, as demonstrated with the example of an electron in a hydrogen atom.
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No more than if you toss an even number of coins you get half heads and half tails.Ahmed1029 said:When the expectation value of spin in the z direction for one particle is zero and I make measurements for an even number of particles in the same state, do I get exactly half to be spin up and half to be spin down along the z direction? More generally, what does spin expectation value for one particle say about measurement of many particles in the same state?
Not exactly, but I kind of get the idea as I was exposed to the notion of a probability ensemble before. My guess is that the expectation value tells me that if I have infinite identical systems and measure the average value of Z spin after measurement it will be the same as the expectation value. Am I right?PeroK said:No more than if you toss an even number of coins you get half heads and half tails.
Expectation value is a statistical concept. One way to look at a probability is as the limit of relative frequency. In that sense, the average value of a sample tends to the expectation value as the size of the sample increases without bound.
If these concepts are unfamiliar to you, you need a course in basic probability theory.
If you are dealing with a two state system, such as (potential free) spin 1/2 system, then yes. But if you have more than two states then this may not be true. The ensemble average of a three state system may not be the average of the individual states. For example, if we have an electron in a hydrogen atom the ensemble average of the electron's energy will not simply be the average of the energies of each state. The electron has a greater probability of being in the n = 1 state so the ensemble average will be reasonably close to the n = 1 energy.Ahmed1029 said:Not exactly, but I kind of get the idea as I was exposed to the notion of a probability ensemble before. My guess is that the expectation value tells me that if I have infinite identical systems and measure the average value of Z spin after measurement it will be the same as the expectation value. Am I right?
More or less. Although "limit of relative frequency" is more mathematically well-defined than "an infinite number of systems".Ahmed1029 said:Not exactly, but I kind of get the idea as I was exposed to the notion of a probability ensemble before. My guess is that the expectation value tells me that if I have infinite identical systems and measure the average value of Z spin after measurement it will be the same as the expectation value. Am I right?