(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Suppose a 2x2 matrix X (not necessarily Hermitian, nor Unitary) is written as..

[tex]X=a_0+σ \cdot a[/tex]

(In the book σ and a are both bold and are being dotted.)

Where [itex]a_0[/itex] and [itex]a_{1,2,3}[/itex] are numbers.

a.)How are [itex]a_0[/itex] and [itex]a_k, (k=1,2,3)[/itex] related to [itex]tr(X)[/itex] and [itex]tr(σ_kX)[/itex]?

b.)Obtain [itex]a_0[/itex] and [itex]a_k[/itex] in terms of the matrix elements [itex]X_{ij}[/itex].

2. Relevant equations

[itex]tr(X)[/itex]= The trace of X, meaning the sum of its diagonal components.

[itex]tr(X)=\sum_{a'}\left\langle a'|X|a' \right\rangle[/itex]

Where the name a' represents base kets.

3. The attempt at a solution

I do not know where to start to be honest. My first question is how can a 2x2 matrix operator equal a number [itex]a_0[/itex] plus the dot product of two vectors? I know I must be misinterpreting this. Can anyone help?

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# The Math of Quantum Mechanics

**Physics Forums | Science Articles, Homework Help, Discussion**