Standard deviation of expectation values

[neu]

Very basic question which has confused me:

if the variance of an expectation value <A> is:

uncertainty of A=&lt;(A-&lt;A&gt;)^2&gt;^0.5

how is this equal to:

(&lt;A^2&gt;-&lt;A&gt;^2)^0.5

??

 

[Doc Al]

Expand it out:
&lt;(A-&lt;A&gt;)^2&gt; = &lt; A^2 - 2A&lt;A&gt; + &lt;A&gt;^2 &gt; = &lt; A^2 - &lt;A&gt;^2 &gt;

 

[jtbell]

Start by expanding the squared term in parentheses:

(A - &lt;A&gt;)^2 = A^2 - 2&lt;A&gt;A + &lt;A&gt;^2

Note that <A> is simply a number and can be manipulated like any other numeric constant. Simplify the resulting expectation value by taking advantage of general properties of expectation values, i.e.

&lt;A+B&gt; = &lt;A&gt; + &lt;B&gt;

&lt;cA&gt; = c&lt;A&gt;

where c is a numeric constant.