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Exponential of the Pauli Matrix

Anonymous Wed 09 Oct 16:36:16 2019 No.11044875 View Reply Original Report

Quoted By: >>11044901 >>11045714

Can any if you guys help me to prove 2.35?

Anonymous

Anonymous Wed 09 Oct 2019 16:46:54 No.11044892 Report

Quoted By:

>Can any if you guys help me to prove 2.35?
What have you tried?

Anonymous

Anonymous Wed 09 Oct 2019 16:49:22 No.11044901 Report

Quoted By: >>11045239

>>11044875
Probably just expand out the Taylor series on both sides.

Anonymous

Anonymous Wed 09 Oct 2019 19:00:30 No.11045239 Report

Quoted By:

>>11044901
write the matrix M = v dot sigma
compute its characteristic polynomial
You find from the Cayley Hamilton Theorem that M^2 = |v|^2
= I since v is unit

e^(i*t*M) = I + itM - (t^2)(M^2)/2 - i(t^3)(M^3)/6 + ...
=I(1-(t^2)/2 + (t^4)/4 - ...) + iM(t/1 - (t^3)/6 + (t^5)/120 - ...)
= Icos(t) +iMsin(t)

Anonymous

Anonymous Wed 09 Oct 2019 21:37:09 No.11045628 Report

Quoted By: >>11045634

Left side is scalar right side is a matrix so this problem is wrong

Anonymous

Anonymous Wed 09 Oct 2019 21:39:38 No.11045634 Report

Quoted By:

>>11045628
The dot product used is an abuse of notation.
It means a linear combination of the pauli matrices.

Anonymous

Anonymous Wed 09 Oct 2019 22:30:14 No.11045714 Report

Quoted By:

>>11044875
Use the Taylor expansion of the exponential function to get a sum of powers of matrices that you can add together. After doing the first few you should recognise the pattern that is the thing you're trying to prove

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