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Sakurai Problems - strange notation

  1. Sep 25, 2006 #1
    Hello!

    I'm just doing the Problems of Chapter 1 of Sakurai: Modern Quantum Mechanics. On page 60, problem 2 he writes:

    "Suppose a 2x2 matrix X, (not necessary Hermitian, nor unitary) is written as

    [itex] X = a_0 + \mathbf{\sigma \cdot a} [/itex],

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

    which confuses me a bit, because you can't add a number and a matrix, and what is [itex] \sigma [/itex] anyway? Would be nice if someone knows what is meant by this (and tells me :biggrin: ).
     
  2. jcsd
  3. Sep 25, 2006 #2

    dextercioby

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    I'm sure he means that [itex] \vec{\sigma}[/itex] is a short name for the Pauli matrices and a_{0} is multiplied by the 2*2 unit matrix.

    Daniel.
     
  4. Sep 25, 2006 #3
    dexter is correct. I was confused with this problem too and actually asked the same question about it here some time ago.
     
  5. Sep 25, 2006 #4
    Thanks.:smile:

    That [itex] \mathbf{\sigma} [/itex] is one of the Pauli-Matrices seems strange to me from the context of the book, because he never defined this during the first chapter. Nevertheless, I'll try to figure it out this way.
     
  6. Sep 25, 2006 #5
    I think you're just supposed to take the sigmas abstractly as some 2x2 (complex-valued) matrices such that any 2x2 (complex-valued) matrix can be written as

    [itex] X = a_0 I + \mathbf{\sigma \cdot a} [/itex]

    with no assumptions about the form of the [itex]\mathbf{\sigma}[/itex]s. In math speak, the [itex]\mathbf{\sigma}[/itex]s together with [itex] I [/itex] are some basis for the vector space of 2x2 complex-valued matrices.
     
    Last edited: Sep 25, 2006
  7. Sep 26, 2006 #6
    Eh, now that I try to actually do the problem, I think he (or the editor) is just assuming you know the Pauli matrices from undergrad QM.
     
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