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State of Harmonic Oscillator with spin half

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Homework Statement


The state of Harmonic Oscillator with spin half is

[tex]|\psi>=\frac{1}{\sqrt{2}}( |n=0,\uparrow> + |n=1, \downarrow>)[/tex]

a, say which one is the possible outcome for a measure of [tex]^{^}S_{x}[/tex] and find the probability of measuring each possible outcome.

b, find the state of the system after [tex]^{^}S_{x}[/tex] has been measured and each possible outcome obtained.


Homework Equations



[tex]|S_{z} \uparrow> = a|S_{x} \uparrow> or b|S[tex]_{x} [tex]\downarrow[/tex]>
[tex]|S_{z} = c|S_{x} \uparrow> or d|S_{x} \downarrow> where a,b,c,d, are your probabilities (or related to it..)

Therefore |PSI> = 1/sqrt(2) ( a|n=0,Sx up> + c|n=1,Sx up> + b|n=0,Sx down> + d|n=1,Sx down> )


The Attempt at a Solution



probable eigen values correspond to yr |Sx up> and |Sx down> with respective probabilities 1/2(a^2 + c^2) and 1/2(b^2 + d^2).
 
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