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Quaternions multiplication

  1. Jun 27, 2012 #1
    1. The problem statement, all variables and given/known data
    Its not really homework problem, and you may find it silly because its only multiplication problem, but I cannot get the right answer by multiplying quaternions.

    Basically this is what i want to show:

    exp(iψ/2)exp(kθ/2)exp(iф/2) = cos(θ/2)exp(i[ψ+ф]/2) + ksin(θ/2)exp(i[ψ-ф]/2)


    2. Relevant equations



    3. The attempt at a solution

    I begin writing:
    exp(kθ/2) = cos(θ/2) + ksin(θ/2)

    Then multiplying:
    exp(iψ/2)*[cos(θ/2) + ksin(θ/2)] = exp(iψ/2)cos(θ/2) + exp(iψ/2)*ksin(θ/2) =
    = cos(θ/2)exp(iψ/2) + sin(θ/2) exp(iψ/2)*k

    Since scalar terms can are commutative in quaternions algebra.

    Finally multiplying answer above with the final exp(iф/2)

    [cos(θ/2)exp(iψ/2) + sin(θ/2) exp(iψ/2)*k] * exp(iф/2) =
    = cos(θ/2)exp(iψ/2)exp(iф/2) + sin(θ/2)exp(iψ/2)*k*exp(iф/2) =
    = cos(θ/2)exp(i(ψ+ф)/2) + sin(θ/2)*k*exp(-iψ/2)*exp(iф/2) =
    = cos(θ/2)exp(i(ψ+ф)/2) + k*sin(θ/2)*exp(i(ф-ψ)/2)

    Here I used that exp(iψ/2)*k = k*exp(-iψ/2)

    And no matter how I do it I always get the same answer, with the last exponential term having ф-ψ, and the paper says it should be ψ-ф
     
  2. jcsd
  3. Jun 27, 2012 #2

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    hi amiras! :smile:

    yours looks correct to me …

    exp(iψ/2)exp(kθ/2)exp(iф/2)

    = exp(iψ/2)cos(θ/2)exp(iф/2) + exp(iψ/2)ksin(θ/2)exp(iф/2)

    = cos(θ/2)exp(i(ψ+ф)/2) + kexp(-iψ/2)sin(θ/2)exp(iф/2)
     
  4. Jun 27, 2012 #3
    Thanks for confirming me, from now on I'l stop blindly following and start thinking by myself. :)
     
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