# Stern Gerlach measurements

#### Dominique

I've worked through a Stern Gerlach experiment for the Sx and Sz directions using the density matrix formalism to account for the environment. This shows a result which I was surprised at for Sx but I think it is correct but relies on decoherence to give the "actual" value. I'm not confident about the result though. Would anyone be kind enough to check my reasoning and conclusions please? Working is attached.

Thank you in anticipation of any help

Kind regards

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#### topsquark

Forum Staff
I've worked through a Stern Gerlach experiment for the Sx and Sz directions using the density matrix formalism to account for the environment. This shows a result which I was surprised at for Sx but I think it is correct but relies on decoherence to give the "actual" value. I'm not confident about the result though. Would anyone be kind enough to check my reasoning and conclusions please? Working is attached.

Thank you in anticipation of any help

Kind regards

The Mathematics of the solution look good but I am unclear about your kets |e> and |e(up arrow)>. What in the environment ket changes when you apply time evolution? (At least that's what I presume you are trying to designate.)

-Dan

#### Dominique

Thank you for your reply. The ket |e> is the ket for the environment just prior to measurement and can be thought of as the tensor product of a huge number of kets representing all the particle states that exist in the background universe. This is very delicate and can easily change because it depends on so many states and will change depending on the outcome of the SG experiment. This is what Deutsch calls the sphere of influence which spreads out from the site of the experiment ( at the speed of light if an outcome is designated by say an LED lighting up). Once this happens the tensor product string for the environment changes from |e> to either |e(up)> or |e(down)> very rapidly because |e> is very delicate anyway because its state depends on so many other states. This is just the process of decoherence which leads us to determine which of the two partitions of the multiverse we have ended up in.
<e(up)|e(down)> rapidly decays away but never goes to zero. The superposition is always there otherwise we wouldn't be talking about the Everett view. In a way the superposition (coherency) infects the environment.

Thanks Dan - does this help?

#### topsquark

Forum Staff
Thank you for your reply. The ket |e> is the ket for the environment just prior to measurement and can be thought of as the tensor product of a huge number of kets representing all the particle states that exist in the background universe. This is very delicate and can easily change because it depends on so many states and will change depending on the outcome of the SG experiment. This is what Deutsch calls the sphere of influence which spreads out from the site of the experiment ( at the speed of light if an outcome is designated by say an LED lighting up). Once this happens the tensor product string for the environment changes from |e> to either |e(up)> or |e(down)> very rapidly because |e> is very delicate anyway because its state depends on so many other states. This is just the process of decoherence which leads us to determine which of the two partitions of the multiverse we have ended up in.
<e(up)|e(down)> rapidly decays away but never goes to zero. The superposition is always there otherwise we wouldn't be talking about the Everett view. In a way the superposition (coherency) infects the environment.

Thanks Dan - does this help?
Yes, that does help, thank you. I haven't really studied "environmental" factors before. But kets are kets. I just needed to know the notation.

In that case I think everything in your first post works out okay.

-Dan

#### Dominique

I was a little surprised to find out the result of the Sx measurement came out to (h/2)cos(theta) rather than zero. However I conclude that in a real experiment the effect of the environmental decoherence would cause such a result, but theta would rapidly increase to (pi/2) hence reducing <A> to the zero value expected for such a test. Do you think my reasoning is correct here? c.f .
arXiv:1208.0904v1 [quant-ph] 4 Aug 2012 , p13
Many thanks for any help given.

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