How Does the Stern-Gerlach Experiment Illustrate Quantum Spin Measurement?

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Discussion Overview

The discussion revolves around the Stern-Gerlach experiment and its implications for understanding quantum spin measurement, particularly focusing on the mathematical representation of spin states and the interpretation of specific equations from a referenced text. Participants explore the relationship between spin states along different axes and the conditions under which these states can be considered eigenstates of various spin operators.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant expresses confusion regarding the matrix representation of the spin matrix along a direction and its relation to the eigenstates of the Sz matrix, questioning the validity of the equation presented in the reference.
  • Another participant asks for clarification on the specific equations causing confusion, particularly focusing on equation 6.114.
  • Concerns are raised about whether the state represented by the vector (1, 0) can be considered an eigenstate of the spin matrix along a different direction, leading to further questioning of the implications of the equations.
  • Participants discuss the conditions under which a linear combination of eigenvectors can be an eigenvector of another operator, with one participant asserting that the assumption made about eigenvectors is incorrect.
  • There is a clarification that the equality in the equations holds under specific conditions for the coefficients a and b, as outlined in equation 6.117.
  • Questions arise regarding the purpose of determining which linear combinations of |z+> and |z-> are eigenvectors of the operator \hat n·\vec S, with an emphasis on understanding the underlying reasoning rather than simply accepting assumptions.

Areas of Agreement / Disagreement

Participants express differing views on the interpretation of the equations and the nature of eigenstates, indicating that multiple competing perspectives remain unresolved. There is no consensus on the implications of the equations or the validity of the assumptions made.

Contextual Notes

Participants reference specific equations and their implications without fully resolving the mathematical steps involved. The discussion highlights the complexity of interpreting quantum spin states and the conditions necessary for eigenstate representation.

rabbit44
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Hi, I've been trying to understand the Stern-Gerlach experiment. I've been reading this book which explains how to find the amplitudes of measuring a particle to have a spin 1/2 lined up along some direction.

Here is the book:

http://www-thphys.physics.ox.ac.uk/people/JamesBinney/QBc6.pdf

It's page 107 that confused me: that first matrix equation. The matrix representation of the spin matrix along theta seems to be in the basis of eigenstates of the Sz matrix. But then the object (1, 0) would be the state of the particle having spin along the z axis, in which case the equation doesn't seem to make sense, as it would suggest that the state of having spin along the z axis is an eigenstate of the spin matrix along theta.

Can anyone help here?
 
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Can You explain your trouble more explicitly?
Which equations are initial and where is misunderstanding?
(6.114)?
 
rabbit44 said:
It's page 107 that confused me: that first matrix equation. The matrix representation of the spin matrix along theta seems to be in the basis of eigenstates of the Sz matrix. But then the object (1, 0) would be the state of the particle having spin along the z axis, in which case the equation doesn't seem to make sense, as it would suggest that the state of having spin along the z axis is an eigenstate of the spin matrix along theta.
The equation would imply that if it had been true for any values of a and b, or at least specifically for a=1,b=0, but it isn't. See (6.117).
 
Minich said:
Can You explain your trouble more explicitly?
Which equations are initial and where is misunderstanding?
(6.114)?

Thanks for replying

Yes 6.114, I don't really get where this comes from. I think that

|+,z> = a*|+, p> + b*|-,p>

(where p is theta)

But I think that the matrix representation of sigma.n in 6.114 is in the basis of eigenstates of Sz, i.e. of |+,z> and |-,z>. I think this because the paragraph before 6.114 references 6.111, which is in the basis of eigenkets of Sz.

So if this is the case, is (1, 0) (a column vector), the representation of |+,z>? If this is the case then the left hand side of the equation surely reads:

sigma.n [ a|+,z> + b|-,z>] which isn't the same as the right hand side of [ a|+,z> + b|-,z>] as |+,z> isn't an eigenket of sigma.n

??ARGH?

Thanks
 
Fredrik said:
The equation would imply that if it had been true for any values of a and b, or at least specifically for a=1,b=0, but it isn't. See (6.117).

Sorry would you mind elaborating, I don't fully understand this. Sorry if I'm being very slow!
 
What 6.114-6.117 says is that if (a,b) is an eigenvector, then a and b must be specifically those numbers specified by 6.117. So (1,0) is clearly not an eigenvector (of \hat n\cdot\vec\sigma). (You said in #1 that the equation suggests that it is).

What you said in #4 is correct until the last statement before the argh. The two sides of the equations are the same, if a and b are as in 6.117. You seem to be assuming that a linear combination of two eigenvectors with different eigenvalues can't be an eigenvector of some other operator. That assumption is wrong.
 
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Fredrik said:
What 6.114-6.117 says is that if (a,b) is an eigenvector, then a and b must be specifically those numbers specified by 6.117. So (1,0) is clearly not an eigenvector (of \hat n\cdot\vec\sigma). (You said in #1 that the equation suggests that it is).

What you said in #4 is correct until the last statement before the argh. The two sides of the equations are the same, if a and b are as in 6.117. You seem to be assuming that a linear combination of two eigenvectors with different eigenvalues can't be an eigenvector of some other operator. That assumption is wrong.

Ah OK so what I said where the two sides is correct, but I was incorrect in saying that they can't equal?

What is the reason for setting (a,b) as an eigenvector of sigma.n? (last question I promise!)

Thanks
 
rabbit44 said:
Ah OK so what I said where the two sides is correct, but I was incorrect in saying that they can't equal?
If the word "where" is supposed to be "were", then yes. That's what the equation says, and the equality holds.

rabbit44 said:
What is the reason for setting (a,b) as an eigenvector of sigma.n? (last question I promise!)
Because he wants to find out which linear combinations of |z+> and |z-> are eigenvectors of \hat n\cdot\vec S. He's not "setting" (a,b) as an eigenvector. He's determining what the eigenvectors are.
 

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