Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

I Need help with Dirac Equation

  1. Dec 23, 2016 #1
    I just started learning this so I am a bit lost. This is where I am lost http://www.nyu.edu/classes/tuckerman/quant.mech/lectures/lecture_7/node1.html .

    Why when E>0, we use $$\phi_p=
    \begin{pmatrix}
    1 \\
    0 \\
    \end{pmatrix}
    $$ or $$
    \begin{pmatrix}
    0 \\
    1 \\
    \end{pmatrix}
    $$

    while when E<0, we use this instead
    $$x_p=
    \begin{pmatrix}
    1 \\
    0 \\
    \end{pmatrix}
    $$ or $$
    \begin{pmatrix}
    0 \\
    1 \\
    \end{pmatrix}
    $$
    where ∅p is the upper component while xp is the lower component of the bispinor in Dirac equation.
    Can we do it the other way round or
    $$\phi_p=
    \begin{pmatrix}
    1 \\
    0 \\
    0 \\
    0 \\
    \end{pmatrix}
    ....$$ instead?


    Secondly, how did the author convert $$\phi_p = \frac{c \sigma .p}{E_p -mc^2}x_p=?=\frac{-c \sigma .p}{|E_p| +mc^2}x_p$$? Does the mod sign means anything?

    Can someone help me or point me in the right direction cause this is my first time learning this. Thanks a lot!
     
  2. jcsd
  3. Dec 25, 2016 #2
    It's because of the ##\beta## matrix (as it's called in the notation you're using, from Dirac; see lesson 6). The top two rows are +1 (on the diagonal), the bottom two -1. The corresponding eigenvalues are pos and neg, obviously, when setting the momentum to zero, as shown in lesson 7. If you wrote the ##\beta## matrix "the other way round" then the E>0 and E<0 cases would also be switched, that is, ##\phi_p## and ##\chi_p## would play opposite roles. There are many other valid ways to write ##\beta## and ##\alpha## matrix (called "representations") all physically equivalent. BTW that second component is "chi" not "x".

    In my answer above I assumed you meant, can we switch the roles of ##\phi_p## (E>0) and ##\chi_p## (E<0), and ignored this. For one thing ##\phi_p## is a 2-vector not 4 but even if you meant ##u_p## it still doesn't make sense, AFAIK.

    The term ##|E_p|## is not a mod but an absolute value, since ##E_p## is just a real number. So you get the RHS simply by multiplying LHS numerator and denominator each by -1, remembering that ##E_p## is negative in this case.
     
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: Need help with Dirac Equation
  1. Dirac equation (Replies: 37)

  2. Dirac equation (Replies: 2)

Loading...