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# Commutation relations and the uncertainty principle: normalized wave function

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Consider two hermitian operators A and B which satisfy the following commutation relation:

[A, B] = AB-BA=iC, where C is also a hermitian operator in general. Let us introduce a new operator Q defined by: Q=A+ i&#955;B, with &#955; being a real number, and consider the following scalar product:

Where is any normalized wave function?

(a) Show that Eq. (1) leads to the following result:
(b) Define the uncertainties as follows: With U&#8801;A - <A> and V&#8801;B - < B>, respectively. Show that [U, V] = [A, B] = iC
(c) Thus, replacing A, B in (a) by U,V, we obtain: Regarding I(&#955;) as a quadratic function of&#955;, show that I(&#955;) is minimum when
(d) Let A=X, B= px, show that the quantum condition [x, px]= i leads to the uncertainty principle:

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