Represent Action of Stern Gerlach Operator as a Matrix

In summary: However, it is a bit strange to have a norm of a matrix smaller than 1. You can divide the matrix by its norm in order to make it a state again.
  • #1
6
0

Homework Statement



Given the series of three Stern-Gerlach devices:

2hnm34n.jpg


Represent the action of the last two SG devices as matrices ##\hat{A}## and ##\hat{B}## in the ##|+z\rangle, |-z\rangle## basis.

Homework Equations



##|+n\rangle = cos(\frac{\theta}{2})|+z\rangle + e^{i\phi}sin(\frac{\theta}{2})|-z\rangle##

The Attempt at a Solution



A Stern-Gerlach device with one beam blocked off can be represented as a projection of the incoming state onto the outgoing state. A projection operator can be written as the outer product of the selected basis state with itself, e.g. ##|+z\rangle\langle\,+\,z\,|## would be the projection operator for an SGz device with the minus size blocked.

The projection matrix of the first SGn is then:

##
\hat{A}\rightarrow{|+n\rangle\langle\,+\,n\,|} =
\begin{pmatrix}
cos(\frac{\theta}{2}) \\
e^{i\phi}sin(\frac{\theta}{2})
\end{pmatrix}
\begin{pmatrix}
cos(\frac{\theta}{2}) & e^{-i\phi}sin(\frac{\theta}{2})
\end{pmatrix} =
\begin{pmatrix}
cos^{2}(\frac{\theta}{2}) & e^{-i\phi}sin(\frac{\theta}{2})cos(\frac{\theta}{2}) \\
e^{i\phi}sin(\frac{\theta}{2})cos(\frac{\theta}{2}) & sin^{2}(\frac{\theta}{2})
\end{pmatrix}
##

Therefore, the beam coming out of the positive side of the SGn should be:

##
|+n\rangle = \hat{A}|+z\rangle \rightarrow
\begin{pmatrix}
cos^{2}(\frac{\theta}{2}) & e^{-i\phi}sin(\frac{\theta}{2})cos(\frac{\theta}{2}) \\
e^{i\phi}sin(\frac{\theta}{2})cos(\frac{\theta}{2}) & sin^{2}(\frac{\theta}{2})
\end{pmatrix}
\begin{pmatrix}
1 \\
0
\end{pmatrix} =
\begin{pmatrix}
cos^{2}(\frac{\theta}{2}) \\
e^{i\phi}sin(\frac{\theta}{2})cos(\frac{\theta}{2})
\end{pmatrix}
##

Should this be equal to the given ##|+n\rangle##?

I feel as though I'm missing something completely obvious or misunderstanding the question.

Thank you for any help you can provide.
 
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  • #2
dbrad5683 said:
Therefore, the beam coming out of the positive side of the SGn should be:

##
|+n\rangle = \hat{A}|+z\rangle \rightarrow
\begin{pmatrix}
cos^{2}(\frac{\theta}{2}) & e^{-i\phi}sin(\frac{\theta}{2})cos(\frac{\theta}{2}) \\
e^{i\phi}sin(\frac{\theta}{2})cos(\frac{\theta}{2}) & sin^{2}(\frac{\theta}{2})
\end{pmatrix}
\begin{pmatrix}
1 \\
0
\end{pmatrix} =
\begin{pmatrix}
cos^{2}(\frac{\theta}{2}) \\
e^{i\phi}sin(\frac{\theta}{2})cos(\frac{\theta}{2})
\end{pmatrix}
##

Should this be equal to the given ##|+n\rangle##?

I feel as though I'm missing something completely obvious or misunderstanding the question.

Thank you for any help you can provide.

Hello and welcome to PF!
Yes, the output of SGn should be |+n>, up to a normalization factor.
 
  • #3
So would ##\frac{1}{cos(\frac{\theta}{2})}## make for a suitable normalization factor? Is there a more formal way of determining this normalization factor?
 
  • #4
dbrad5683 said:
So would ##\frac{1}{cos(\frac{\theta}{2})}## make for a suitable normalization factor?
Yes
Is there a more formal way of determining this normalization factor?
Not that I can think of at the moment. The usual way to find the normalization factor for a state |ψ>is to just look at <ψ|ψ>. But, in your example, you can spot the normalization factor by inspection.
 
  • #5
TSny said:
Yes

Not that I can think of at the moment. The usual way to find the normalization factor for a state |ψ>is to just look at <ψ|ψ>. But, in your example, you can spot the normalization factor by inspection.
Thanks for you help. I still don't see how ##\langle\,+\,n\,|+n\rangle## would give me the normalization factor though. I'm getting ##cos^{2}(\frac{\theta}{2}) + sin^{2}(\frac{\theta}{2}) = 1## which is what is expected.
 
  • #6
Let |ψ> be the output of GSn.
 
  • #7
Ahh I see and I set this equal to 1. Thank you for your help!
 
  • #8
TSny said:
Let |ψ> be the output of GSn.
This didn't work. I got a factor of ##\frac{1}{cos^{2}(\frac{\theta}{2})}## when evaluating ##\langle\psi|\psi\rangle## with ##|\psi\rangle## being the output of SGn.

##
\begin{pmatrix}
cos^{2}(\frac{\theta}{2}) &
e^{-i\phi}sin(\frac{\theta}{2})cos(\frac{\theta}{2})
\end{pmatrix}
\begin{pmatrix}
cos^{2}(\frac{\theta}{2}) \\
e^{i\phi}sin(\frac{\theta}{2})cos(\frac{\theta}{2})
\end{pmatrix} =
cos^{2}(\frac{\theta}{2})[cos^{2}(\frac{\theta}{2}) + sin^{2}(\frac{\theta}{2})] = cos^{2}(\frac{\theta}{2})
##
 
Last edited:
  • #9
The normalization factor gets applied to both occurences of ψ, so you need the square root of ##\langle\psi|\psi\rangle## as normalization factor.
 
  • #10
mfb said:
The normalization factor gets applied to both occurences of ψ, so you need the square root of ##\langle\psi|\psi\rangle## as normalization factor.
Thank you. How does this then relate back to the matrix representation? Would it just be ##\frac{1}{cos(\frac{\theta}{2})}\begin{pmatrix}cos^{2}(\frac{\theta}{2}) & e^{-i\phi}sin(\frac{\theta}{2})cos(\frac{\theta}{2}) \\ e^{i\phi}sin(\frac{\theta}{2})cos(\frac{\theta}{2}) & sin^{2}(\frac{\theta}{2})\end{pmatrix}##
 
  • #11
I don't think it makes sense to normalize the matrix. The norm smaller than 1 indicates that some particles get lost, which has a physical meaning.
 

1. What is the Stern Gerlach operator and how does it relate to quantum mechanics?

The Stern Gerlach operator is a mathematical representation of a physical experiment in which a beam of particles is passed through a magnetic field and their spin orientations are measured. It is an important concept in quantum mechanics because it demonstrates the discrete nature of particle spin and the concept of quantization.

2. How is the Stern Gerlach operator represented as a matrix?

The Stern Gerlach operator is represented as a 2x2 matrix, with each element corresponding to the different spin orientations of the particles. For example, the element in the first row and first column represents particles with spin up, while the element in the second row and second column represents particles with spin down.

3. What is the significance of the eigenvalues and eigenvectors of the Stern Gerlach operator?

The eigenvalues and eigenvectors of the Stern Gerlach operator represent the possible outcomes of the experiment. The eigenvalues correspond to the different possible spin orientations that particles can have, while the eigenvectors represent the states in which the particles are measured to have those spin orientations.

4. How does the Stern Gerlach operator demonstrate the principle of superposition?

The Stern Gerlach operator allows for the concept of superposition to be applied to particle spin. This means that particles can exist in a combination of different spin states at the same time, and their final state after measurement is determined by the probability amplitudes of these different states.

5. Can the Stern Gerlach operator be applied to other physical systems besides particles with spin?

Yes, the Stern Gerlach operator can be generalized to other physical systems with discrete properties, such as angular momentum or energy. It can also be extended to systems with more than two possible outcomes, by using a higher dimensional matrix representation.

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