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The Math of Quantum Mechanics

  1. Jan 27, 2012 #1
    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?
     
  2. jcsd
  3. Jan 27, 2012 #2

    phyzguy

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    The σ's are the 2x2 Pauli matrices, so what the problem means is :

    X = a0*(2x2 Identity matrix) + ax*(2x2 Pauli matrix σx) + ay*(2x2 Pauli matrix σy)+ az*(2x2 Pauli matrix σz)
     
  4. Jan 27, 2012 #3
    Ah that makes it MUCH more clear! So I basically plugged everything in and found the trace in each case and got..

    [tex]tr(X)=2a_0[/tex]
    [tex]tr(σ_{1}X)=2a_1[/tex]
    [tex]tr(σ_{2}X)=2a_2[/tex]
    [tex]tr(σ_{3}X)=2a_3[/tex]

    Which I assume is what they are looking for for part A. Part B however is making me a bit confused. Do they want me to just solve the above expressions for a?
     
  5. Jan 27, 2012 #4

    vela

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    No, you have some matrix X where
    $$X = \begin{pmatrix} X_{11} & X_{12} \\ X_{21} & X_{22} \end{pmatrix} = a_0 + \vec{a}\cdot\vec{\sigma}$$The problem wants you to solve for a0 and the ak's in terms of the Xij's. Start by writing down explicitly what ##a_0 + \vec{a}\cdot\vec{\sigma}## is equal to.
     
  6. Jan 27, 2012 #5
    Ohh okay so would this be on the right track?

    [tex]a_k=\frac{1}{2}\left[ (σ_kX)_{11}+(σ_kX)_{22} \right][/tex]
    Where k=0,1,2,3

    EDIT: I got this by looking at the expressions I posted for part A and finding a common equation that suits all of them. I now see that you said to start by writing the original expression. I will try this.
     
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