Question

1. Suppose that 10,000 spin 1/2 particles with random spin direction are analyzed by a detector oriented along a direction \hat{n} = \frac{1}{\sqrt{2}}(\hat{x} + \hat{z}). The SPINS program denotes this as 45°. The half that are \"spin up\" in \hat{n} go into an analyzer oriented in the \hat{z}direction. The half that are \"spin down\" in \hat{n} go into an analyzer oriented in the \hat{z}direction: a) What is the quantum state of the \"spin up\" particles leaving the first analyzer (oriented along \hat{n})? Express it in the \hat{z} basis. b) What is the quantum state of the \"spin down\" particles leaving the first analyzer (oriented along \hat{n})? Express it in the \hat{z} basis. c) Use the projection operators (e.g. $|+\rangle_\hat{z}\langle+|\hat{z}$) to determine the quantum states of particles leaving each of the remaining two analyzers. d) Compute the fraction of particles expected at each of the four counters. Is the simulation result shown consistent with theory? Try it for 100,000 particles. Is the result better?

          1. Suppose that 10,000 spin 1/2 particles with random spin direction are analyzed by a detector oriented along a direction \hat{n} = \frac{1}{\sqrt{2}}(\hat{x} + \hat{z}).
The SPINS program denotes this as 45°. The half that are \"spin up\" in \hat{n} go into an analyzer oriented in the \hat{z}direction. The half that are \"spin
down\" in \hat{n} go into an analyzer oriented in the \hat{z}direction:
a) What is the quantum state of the \"spin up\" particles leaving the first analyzer (oriented along \hat{n})? Express it in the \hat{z} basis.
b) What is the quantum state of the \"spin down\" particles leaving the first analyzer (oriented along \hat{n})? Express it in the \hat{z} basis.
c) Use the projection operators (e.g. $|+\rangle_\hat{z}\langle+|\hat{z}$) to determine the quantum states of particles leaving each of the remaining two analyzers.
d) Compute the fraction of particles expected at each of the four counters. Is the simulation result shown consistent with theory? Try it for
100,000 particles. Is the result better?

$1. Suppose that 10,000 spin 1/2 particles with random spin direction are analyzed by a detector oriented along a direction n̂ = (1)/(√(2))(x̂ + ẑ). The SPINS program denotes this as 45°. The half that are s̈pin upïn n̂ go into an analyzer oriented in the ẑdirection. The half that are s̈pin downïn n̂ go into an analyzer oriented in the ẑdirection: a) What is the quantum state of the s̈pin upp̈articles leaving the first analyzer (oriented along n̂)? Express it in the ẑ basis. b) What is the quantum state of the s̈pin downp̈articles leaving the first analyzer (oriented along n̂)? Express it in the ẑ basis. c) Use the projection operators (e.g. |+⟩ẑ⟨+|ẑ) to determine the quantum states of particles leaving each of the remaining two analyzers. d) Compute the fraction of particles expected at each of the four counters. Is the simulation result shown consistent with theory? Try it for 100,000 particles. Is the result better?$

Added by Melinda W.

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