State of Harmonic Oscillator with spin half

[JayKo]

Homework Statement

The state of Harmonic Oscillator with spin half is

$$|\psi>=\frac{1}{\sqrt{2}}( |n=0,\uparrow> + |n=1, \downarrow>)$$

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

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

Homework Equations

[tex]|S_{z} \uparrow> = a|S_{x} \uparrow> or b|S[tex]_{x} $$\downarrow$$>
[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).

 

[NateD]

After measuring Sx, the state of the system will depend upon the outcome.if it is |Sx up>, then the state is |n=0,Sx up> + c|n=1,Sx up>.if it is |Sx down>, then the state is b|n=0,Sx down> + d|n=1,Sx down>.