I Spin expectation value for one particle vs actual measurement

Ahmed1029
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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?
 
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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?
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.
 
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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.
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?
 
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?
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.

-Dan
 
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".
 
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I am not sure if this falls under classical physics or quantum physics or somewhere else (so feel free to put it in the right section), but is there any micro state of the universe one can think of which if evolved under the current laws of nature, inevitably results in outcomes such as a table levitating? That example is just a random one I decided to choose but I'm really asking about any event that would seem like a "miracle" to the ordinary person (i.e. any event that doesn't seem to...
Not an expert in QM. AFAIK, Schrödinger's equation is quite different from the classical wave equation. The former is an equation for the dynamics of the state of a (quantum?) system, the latter is an equation for the dynamics of a (classical) degree of freedom. As a matter of fact, Schrödinger's equation is first order in time derivatives, while the classical wave equation is second order. But, AFAIK, Schrödinger's equation is a wave equation; only its interpretation makes it non-classical...

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