1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Problem with vectors and matrices.

  1. Feb 6, 2016 #1
    1. The problem statement, all variables and given/known data
    Calculate ##(\vec a \cdot \vec \sigma)^2##, ##(\vec a \cdot \vec \sigma)^3##, and ##(\vec a \cdot \vec \sigma)^4##, where ##\vec a## is a 3D-vector and ##\vec \sigma## is a 3D-vector formed from the ##\sigma_i## vectors.

    2. Relevant equations
    $$\sigma_1 = \begin{bmatrix}
    0 & 1\\
    1 & 0
    \end{bmatrix}$$
    $$\sigma_2 = \begin{bmatrix}
    0 & -i\\
    i & 0
    \end{bmatrix}$$
    $$\sigma_3 = \begin{bmatrix}
    1 & 0\\
    0 & -1
    \end{bmatrix}$$
    3. The attempt at a solution
    This makes very little sense to me, since there are no ##\sigma_i## vectors, just matrices. My main thought was to take the determinant of each matrix and set it as a component, such that
    $$\vec \sigma_i = <-1,-1,-1>$$

    Then ##\vec a \cdot \vec \sigma_i## is ##-(a_1 + a_2 + a_3)##.
    ##(\vec a \cdot \vec \sigma_i)^2## is ##(a_1 + a_2 + a_3)^2##,
    ##(\vec a \cdot \vec \sigma_i)^3## is ##-(a_1 + a_2 + a_3)^3## and
    ##(\vec a \cdot \vec \sigma_i)^4## is ##(a_1 + a_2 + a_3)^4##.

    I have no idea, however, whether this is the right approach. Some guidance on this would be nice, thank you!!
     
  2. jcsd
  3. Feb 7, 2016 #2

    blue_leaf77

    User Avatar
    Science Advisor
    Homework Helper

    A notation like ##\mathbf{a}\cdot\mathbf{\sigma}## is just a way of saying ##a_1\sigma_1 + a_2\sigma_2 + a_3\sigma_3##. ##\mathbf{a}## is known to be a 3D vector, therefore its components are just numbers.
     
  4. Feb 7, 2016 #3
    Yes, I know that. Thank you though. I'm confused though as to how to form a 3D vector from three 2x2 matrices. Is there a prescribed method or do I just need to do something along the lines of what I did?
     
  5. Feb 7, 2016 #4

    blue_leaf77

    User Avatar
    Science Advisor
    Homework Helper

    The sigma vector ##\mathbf{\sigma}## is just one example of vector operators, another example would be the orbital angular momentum operator ##\mathbf{L}##. Operators in quantum mechanics (as well as in linear algebra in general) need not always be represented by a matrix. The matrix representation is especially helpful when you are working in the basis formed by the eigenstates of the operator being represented as a matrix. If, on the other hand, you are given a problem in which you have to operate ##L_z## on ##Y_{lm}(\theta,\phi)## (##|l,m\rangle## in position basis), you will then resort to the position form of ##L_z##, which is equal to ##-i\hbar\frac{\partial}{\partial\phi}##, instead of its matrix form.
    Especially for spin operators, there is no position representation for them. Therefore, the most commonly used representation for these operators are the matrix form.
     
  6. Feb 8, 2016 #5

    haruspex

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    I think what's throwing you is the last word in the problem statement. If you change it from 'vectors' to 'matrices' does it make sense for you?
    ##a_i\sigma_i## is just a scalar multiplied by a 2x2 matrix, giving another 2x2 matrix. The dot product ##\vec a.\vec \sigma## is a sum of these, so is another 2x2 matrix. This can be raised to integer powers.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Problem with vectors and matrices.
  1. Matrice problem (Replies: 5)

  2. Matrices problem (Replies: 1)

  3. Matrices problem (Replies: 2)

Loading...