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Mathematics

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EastWind [94]2 years ago

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An ensemble of 100 identical particles is sent through a Stern-Gerlach apparatus and the z-component of spin is measured. 46 yie

Elodia [21]

Answer:

The standard deviation is 0.4984 $\hbar$

Step-by-step explanation:

In order to find standard deviation, The equation is given as

$\sigma=\sqrt{\frac{1}{n} \sum_{i=1}^{100} (\mu-x_i)^2$

Here μ is the mean which is calculated as follows

$\mu=\frac{\sum_{i=1}^{100} x_i}{n}\\\mu=\frac{46\times \frac{\hbar}{2}+54\times \frac{-\hbar}{2}}{100}\\\mu=\frac{-4 \hbar}{100}\\\mu=-0.04 \hbar$

Now the standard deviation is given as

$\sigma=\sqrt{\frac{1}{100} \sum_{i=1}^{100} (-0.04 \hbar-x_i)^2}\\\sigma=\sqrt{\frac{1}{100} [[46 \times(-0.04 \hbar-0.5 \hbar)^2]+[54 \times(-0.04 \hbar+0.5 \hbar)^2]}]\\\sigma=\sqrt{\frac{1}{100} [[46 \times(-0.54 \hbar)^2]+[54 \times(0.46 \hbar)^2]}]\\\sigma=\sqrt{\frac{1}{100} [[46 \times(0.2916 \hbar)]+[54 \times(0.2116 \hbar)]}]\\\sigma=\sqrt{\frac{1}{100} [13.4136 \hbar+11.4264 \hbar}]\\\sigma=\sqrt{\frac{24.84 \hbar}{100}}\\\sigma =0.4984 \hbar$