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

• rabbit44
In summary, the conversation is about trying to understand the Stern-Gerlach experiment and the equations involved. The confusion arises from the use of eigenstates and eigenvectors, specifically in regards to the matrix representation of the spin matrix along theta. The individual is seeking clarification on how the equations work and the reason for setting (a,b) as an eigenvector of sigma.n.
rabbit44
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)?

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.

## 1. What is the Stern-Gerlach experiment?

The Stern-Gerlach experiment is a physics experiment that was first conducted in 1922 by Otto Stern and Walther Gerlach. It involves passing a beam of particles through an inhomogeneous magnetic field, which causes the particles to be deflected in different directions depending on their spin orientation.

## 2. What is the significance of the Stern-Gerlach experiment?

The Stern-Gerlach experiment was one of the first experiments to demonstrate the quantization of angular momentum and the existence of spin in particles. It also played a crucial role in the development of quantum mechanics and our understanding of the behavior of subatomic particles.

## 3. How does the Stern-Gerlach experiment work?

In the Stern-Gerlach experiment, a beam of particles, usually silver atoms, is passed through a non-uniform magnetic field. The magnetic field causes the particles to split into two beams, one with particles of spin up and one with particles of spin down. This is due to the fact that particles with different spin orientations experience different magnetic forces in the field.

## 4. What was the significance of the results of the Stern-Gerlach experiment?

The results of the Stern-Gerlach experiment were significant because they provided evidence for the quantization of angular momentum and the existence of spin in particles. This challenged classical physics and led to the development of quantum mechanics, which revolutionized our understanding of the behavior of particles at the atomic and subatomic level.

## 5. How is the Stern-Gerlach experiment relevant today?

The Stern-Gerlach experiment continues to be relevant today as it is still used to study the properties of particles and their spin. It has also led to the development of technologies such as magnetic resonance imaging (MRI) which uses the principles of the Stern-Gerlach experiment to create detailed images of the human body.

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