The state of a spin ½ particle can be represented using the eigenstate
![\mid\uparrow\rangle](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_tFnY5U4KZocRFArjs1ruYeqKz_TtUJJ8ds0G3s_wsbtcsdascf5NwV4P6P3i5g5_KHsPy69XsLZGVihe25zDMAGGUbdV2D12DN7xcoptd0hbJH74J8NY4fBYsJPgtl3nU=s0-d)
and
![\mid\downarrow\rangle](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vunCIZqosVNIgEatBMRmAGuN1BGtz2-BTAyeFm6YfCcbIY46yhXplMwEizOYKEQDzRo5E1iE-YfOoQ1tbEjzWyEf4gjSxuFdkc0SrFrvoyxQ-bzIv15Q6aqTtgJKchCgIZkg=s0-d)
of the
![S_z](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_uR3GLP4NMgGurY4p_XeqrPnoH4rnaQpKrRUUdy7XPhrftJnSL1epCapOL3iHWp55qr5PdVkZBniG5oWWm9vvqiBYIzJeMtB-itsA=s0-d)
operator.
Given the Pauli matrix
![\sigma_x=\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vmtF1mBQJn_OwdS9QnvCjONpDHa0tjVpz8GE214CTdwM1mBBTByPWUYCvUXqU-XYkHPanvd5aLKDgQqTXx6-i_5xGUe9FJtAJRKXrCoQrr2zUY43H8o0GaG-lR97sO19QVUKP8oemxMFE-TRkjLyXCfyMlDBWmIdJPgVW3jjHxLvgHm46LhhZ6Zi4sBefcjeB0d3pOAhFa681gBJDWD-rh9B6HN-SbKbS7IEbnVLYXRVMh52Wvmnbgd0zLw2C5Ig=s0-d)
which of the following is an eigenstate of
![S_x](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_usehfxFgrFO1R3wRU5NEKZlE-1AA4KPoE3Sagk5ovq7kWNWc7Vp8qtA_cMPtgy3kbFcdCBNz_VafBKnKOwwGU3yLE2eptNzQYtmA=s0-d)
with eigenvalue
![-\frac{1}{2} \hbar](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vOYzXkYGzH_lywvBl-LhkOdIjcm_ThROM5drTSlc62RSLSxs0SAASwvUSNQuHiHG9sDrHNo9Zhc96zqIOCCihDGcUHsfYWWf17jhKz1q3iOoyH341uUgcw56lS91jocZ-2ZfAKHO_dzLnIcw=s0-d)
?
(GR0177 #83)
Solution:
Spin Operator:
Eigenvalue:
Eigenfunction:
Normalized
![\chi](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_sgwJXSSaThR0HxmXOwpt1gkCMkLxbtFPGsl6iPIJF1h18AMpCCV_g17AN6jaCQxLRrk3b3hkXlCxH79NAOMEaAp7KtnKls6HK7x_oa=s0-d)
:
Answer: C
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