Question
Asked Nov 4, 2019
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It's a quantum mechanics problem.

Consider the 2 x 2 matrix defined by
aoi a
U
.
(5)
ао — iд .a
where ao is a real number, and a is a three-dimensional vector with real components.
a. Prove that U is unitary and unimodular
b. In general, a 2 x 2 unitary unimodular matrix represents a rotation in three dimensions. Find the
axis and angle of rotation appropriate for U in terms of ao, ai, a2, a3
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Consider the 2 x 2 matrix defined by aoi a U . (5) ао — iд .a where ao is a real number, and a is a three-dimensional vector with real components. a. Prove that U is unitary and unimodular b. In general, a 2 x 2 unitary unimodular matrix represents a rotation in three dimensions. Find the axis and angle of rotation appropriate for U in terms of ao, ai, a2, a3

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Expert Answer

Step 1

(a)Consider.

 

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А-а, +io A in matrix form a+ia А- а, tia, a, -ia, -а, tia,

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Step 2

The determinant of A is.

 

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la, +iaj det Ala+ia = az +la\ =det A 1 det A

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Step 3

Consider the given 2x2 ma...

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U-4 AA AA AA det A 1 (det A) 1 -AA det A =UU =1 thus the matrix U is unitary 1 -det A2 (det .A) det U =1 U is unimodular

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