Question

1. A Light Introduction Someone conducts a version of the Stern-Gerlachh experiment that can supposedly distinguish three different states for an electron spin: $|a\rangle = (|\uparrow\rangle + |\downarrow\rangle)/\sqrt{2}$, $|b\rangle = (|\uparrow\rangle - |\downarrow\rangle)/\sqrt{2}$, $|c\rangle = |\downarrow\rangle$. There are 3 lights corresponding to each state. If found in state $|a\rangle$, then light 1 activates, if $|b\rangle$ then 2, etc. (a) Suppose the electron is sent in the state $|\uparrow\rangle$. What are the possible outcomes and their probabilities? If sent multiple times, what conclusions could you draw about the state? Are there any problems? What about for the down state? (b) Answering the same questions in part (a), consider what would happen if the state sent was $(|\uparrow\rangle + |\downarrow\rangle)/\sqrt{2}$ (c) Finally, consider the operator: $A = a (|\uparrow\rangle\langle\uparrow| + |\uparrow\rangle\langle\downarrow| + |\downarrow\rangle\langle\uparrow| - |\downarrow\rangle\langle\downarrow|)$, in which a is real. Find the eigenvalues and eigenvectors for A. Don't forget to show your work!

          1. A Light Introduction
Someone conducts a version of the Stern-Gerlachh experiment that can
supposedly distinguish three different states for an electron spin:
$|a\rangle = (|\uparrow\rangle + |\downarrow\rangle)/\sqrt{2}$,
$|b\rangle = (|\uparrow\rangle - |\downarrow\rangle)/\sqrt{2}$,
$|c\rangle = |\downarrow\rangle$.
There are 3 lights corresponding to each state. If found in state $|a\rangle$, then
light 1 activates, if $|b\rangle$ then 2, etc.
(a) Suppose the electron is sent in the state $|\uparrow\rangle$. What are the
possible outcomes and their probabilities? If sent multiple times, what
conclusions could you draw about the state? Are there any
problems? What about for the down state?
(b) Answering the same questions in part (a), consider what
would happen if the state sent was $(|\uparrow\rangle + |\downarrow\rangle)/\sqrt{2}$
(c) Finally, consider the operator:
$A = a (|\uparrow\rangle\langle\uparrow| + |\uparrow\rangle\langle\downarrow| + |\downarrow\rangle\langle\uparrow| - |\downarrow\rangle\langle\downarrow|)$,
in which a is real. Find the eigenvalues and eigenvectors for A.
Don't forget to show your work!

1. A Light Introduction
Someone conducts a version of the Stern-Gerlachh experiment that can
supposedly distinguish three different states for an electron spin:
|a⟩ = (|↑⟩ + |↓⟩)/√(2),
|b⟩ = (|↑⟩ - |↓⟩)/√(2),
|c⟩ = |↓⟩.
There are 3 lights corresponding to each state. If found in state |a⟩, then
light 1 activates, if |b⟩ then 2, etc.
(a) Suppose the electron is sent in the state |↑⟩. What are the
possible outcomes and their probabilities? If sent multiple times, what
conclusions could you draw about the state? Are there any
problems? What about for the down state?
(b) Answering the same questions in part (a), consider what
would happen if the state sent was (|↑⟩ + |↓⟩)/√(2)
(c) Finally, consider the operator:
A = a (|↑⟩⟨↑| + |↑⟩⟨↓| + |↓⟩⟨↑| - |↓⟩⟨↓|),
in which a is real. Find the eigenvalues and eigenvectors for A.
Don't forget to show your work!

Added by William A.

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