Stern-Gerlach spin probability

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SUMMARY

The Stern-Gerlach experiment measures the z component of spin angular momentum for spin-half particles, yielding results of either 1/2ħ or -1/2ħ. Quantum mechanics predicts a 75% probability for measuring 1/2ħ, leading to a definitive 25% probability for measuring -1/2ħ. When sending 20,000 particles through the apparatus, 5,000 particles are expected to yield a z component of -1/2ħ. Proper polarization of incoming spins is essential to achieve these probabilities, as unpolarized spins would result in a 50-50 distribution.

PREREQUISITES
  • Understanding of quantum mechanics principles, particularly spin-half particles
  • Familiarity with the Stern-Gerlach experiment and its significance
  • Knowledge of probability theory as applied to quantum measurements
  • Basic grasp of angular momentum in quantum systems
NEXT STEPS
  • Study the mathematical foundations of quantum mechanics, focusing on spin states
  • Explore the implications of polarization in quantum experiments
  • Learn about the statistical interpretation of quantum measurement outcomes
  • Investigate advanced topics in quantum mechanics, such as entanglement and superposition
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Students and researchers in physics, particularly those focusing on quantum mechanics, experimental physicists working with spin systems, and educators teaching quantum concepts.

sweetdreams12
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So I've just been reading about the Stern-Gerlach apparatus/experiment and had some questions about the half spin probabilities...

First let's say a collection of spin-half particles are sent through a S-G apparatus (which I know measures the z component of particles spin angular momentum) it is known that the results are always either 1/2ħ or -1/2ħ. And though the result cannot be predicted before it has been calculated by quantum mechanics that the probability of getting 1/2ħ is 0.75. So then what is the probability of getting the result -1/2ħ? I saw some sites referring to cosine and sine while others said it was 1 so it confused me

So for example if say 20000 particles are sent through, how many will have a z component equal to -1/2ħ (I'm thinking 50%?) then what would be the average z component of spin for all the particles?
 
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It is a binary measurement. If you are not measuring +1/2 you are measuring -1/2 and the probability should add up to one. Thus, if you know that the probability of measuring 1/2 is 75%, then the probability of measuring -1/2 must be 25%.

Of course, in order to have such a setup, the incoming spins must already be polarized or the probabilities will be 50-50.
 

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