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?
The spin expectation value for one particle is a theoretical prediction of the average value of the spin of a particle in a given state. It is calculated using the quantum mechanical formalism and is represented by the symbol ⟨S⟩.
The spin expectation value is calculated by taking the inner product of the state vector with the operator for spin, which is represented by the symbol S. This calculation involves complex numbers and matrix operations.
The spin expectation value is significant because it provides information about the properties of a particle in a given state. It can help predict the outcome of a measurement of the spin of the particle.
The spin expectation value is a theoretical prediction, while actual measurement is the result of a physical measurement of the spin of a particle. The two may differ due to uncertainties and limitations in the measurement process.
No, the spin expectation value cannot predict the exact spin measurement of a particle. It can only give a probabilistic prediction of the average value of the spin. The actual measurement may vary due to the inherent randomness of quantum mechanics.