Measuring Spin - wave function collapse

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SUMMARY

The discussion centers on the measurement of spin components in quantum mechanics, specifically the behavior of spin operators S_x, S_y, and S_z. It is established that measuring S_z followed by S_y results in a particle in an eigenstate of S_y, while measuring S_y followed by S_x yields an expectation value of zero for S_x. This is attributed to the nature of eigenstates and the preferred measurement direction, where S_z exhibits non-zero values in certain states. The distinction between measurement outcomes and expectation values is emphasized, clarifying that expectation values can be zero even when measurements yield definite results.

PREREQUISITES
  • Understanding of quantum mechanics principles, particularly spin operators.
  • Familiarity with eigenstates and their implications in quantum measurements.
  • Knowledge of expectation values in quantum systems.
  • Basic grasp of Stern-Gerlach experiments and their significance in demonstrating quantum behavior.
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  • Study the implications of measuring spin in quantum mechanics, focusing on S_x, S_y, and S_z operators.
  • Explore the concept of eigenstates in quantum mechanics and their role in measurement outcomes.
  • Learn about expectation values and their calculation in quantum systems.
  • Investigate the Stern-Gerlach experiment and its relevance to understanding spin measurements.
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benbenny
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It is my understanding that a measurement of S_z followed by a measurement of S_y will result in a particle which is in an eigenstate of S_y. But it appears that a measurement of say S_y followed by a measurement of S_x results in zero. I see this from a question in which I am asked to find the expectation value of S_x for a particle in an eigenstate of S_y and my result is zero. But for S_z it is non zero.
I don't understand why this is so. Could someone help me understand why we find non zero S_z values but zero S_x values for a partilce in an S_y eigenstate.

Thanks in advance.

B
 
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Taking a measurement is not the same as finding the expectation value. If you take a measurement of Sx or Sz, you'll always get 1/2 or -1/2 (in units of h-bar). The expectation value is what you'd expect if you took many measurements and averaged the results.
 
vela said:
Taking a measurement is not the same as finding the expectation value. If you take a measurement of Sx or Sz, you'll always get 1/2 or -1/2 (in units of h-bar). The expectation value is what you'd expect if you took many measurements and averaged the results.

Ok. I understand now. Thank you Vela.

It makes sense that the expectation value in the S_z direction is NOT zero even if we are in a quantum state of S_x or S_y since it is the "preferred" direction I suppose, while S_x and S_y should exhibit zero expectation if we are in the S_z eigenstate as they sort of rotate around S_z.
or something like that. cheers.
 
If you're in an eigenstate of Sx, the preferred direction is along the x-axis. You should find the expectation values of Sy and Sz to be zero.

Keep in mind that x, y, and z are just labels we use to keep things straight. There's nothing intrinsically special about the z-axis.
 
vela said:
If you're in an eigenstate of Sx, the preferred direction is along the x-axis. You should find the expectation values of Sy and Sz to be zero.

Keep in mind that x, y, and z are just labels we use to keep things straight. There's nothing intrinsically special about the z-axis.

oh ok. now I think I really do understand. Source of confusion: hypothetical triple stern-gerlach experiment which exhibits that taking a measurement of Sx on an Sz eigenstate beam would produce 2 more beams in eigenstates of Sx. Obviously 2 resulting beams in the up and down eigenstates of Sx still amount to a zero expectation value.

Thanks again Vela, and if your interested in helping me with another unrelated question regarding GR and the cylinder condition that would be great as well : https://www.physicsforums.com/showthread.php?t=399738

cheers.

B
 

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