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Physics
Quantum Physics
Finding ##S_x## eigenstate using experiments
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[QUOTE="Kashmir, post: 6626117, member: 692174"] [I]Quantum mechanics, McIntyre, pg 62[/I][ATTACH type="full" alt="IMG_20220426_181555.JPG"]300603[/ATTACH] For above spin ##1## Stern Gerlach experiment a set of results is "## \begin{array}{c} \mathcal{P}_{1 x}=\left.\left.\right|_{x}\langle 1 \mid 1\rangle\right|^{2}=\frac{1}{4} \\ \mathcal{P}_{0 x}=\left.\left.\right|_{x}\langle 0 \mid 1\rangle\right|^{2}=\frac{1}{2} \\ \mathcal{P}_{-1 x}=\left.\right|_{x}\left(-\left.1|1\rangle\right|^{2}=\frac{1}{4},\right. \end{array} ## as illustrated in Fig. 2.12. These experimental results can be used to determine the ##S_{x}## eigenstates in terms of the ##S_{z}## basis ## \begin{array}{l} |1\rangle_{x}=\frac{1}{2}|1\rangle+\frac{1}{\sqrt{2}}|0\rangle+\frac{1}{2}|-1\rangle \\ |0\rangle_{x}=\frac{1}{\sqrt{2}}|1\rangle-\frac{1}{\sqrt{2}}|-1\rangle \\ |-1\rangle_{x}=\frac{1}{2}|1\rangle-\frac{1}{\sqrt{2}}|0\rangle+\frac{1}{2}|-1\rangle \end{array} ##"To find the ##S_{x}## eigenstates in terms of the ##S_{z}## basis I need two more similar experiments in which the input to Sx analyzer are ##0,-1## spin particles respectively. However I am getting a sign ambiguity while using the experimental results. Below are two expressions which both are in line with the experiments: * ##|1\rangle_{x}=\frac{1}{2}|1\rangle+\frac{1}{\sqrt{2}}|0\rangle+\frac{1}{2}|-1\rangle## ##|0\rangle_{x}=\frac{1}{\sqrt{2}}|1\rangle-\frac{1}{\sqrt{2}}|-1\rangle## ##|-1\rangle_{x}=\frac{1}{2}|1\rangle-\frac{1}{\sqrt{2}}|0\rangle+\frac{1}{2}|-1\rangle ## * ##|1\rangle_{x}=\frac{1}{2}|1\rangle+\frac{1}{\sqrt{2}}|0\rangle-\frac{1}{2}|-1\rangle## ##|0\rangle_{x}=\frac{1}{\sqrt{2}}|1\rangle+\frac{1}{\sqrt{2}}|-1\rangle## ##|-1\rangle_{x}=\frac{1}{2}|1\rangle+\frac{1}{\sqrt{2}}|0\rangle-\frac{1}{2}|-1\rangle ##How do we resolve this? [/QUOTE]
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Quantum Physics
Finding ##S_x## eigenstate using experiments
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