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Free Hamiltonian problem for relativistic mechanics

  1. Oct 9, 2012 #1
    I need to elaborate the equation ,and need to know what is the physical significance and how matrices will manipulate in the equation $$
    \hat{H} = (\hat{\tau_3}+i\hat{\tau_2})\frac{\hat{p}^2}{2m_0}+ \hat{\tau_3}m_0 c^2 = \frac{\hat{p}^2}{2m_0}
    \left| \begin{array}{ccc}
    1 & 1 \\
    -1 & -1 \\
    \end{array}\right| \frac{\hat{p}^2}{2m_0} + \left| \begin{array}{ccc}
    1 & 0 \\
    0 & -1 \\
    \end{array}\right| m_0 c^2
    $$

    Where $$\tau_1 , \tau_2,\tau_3
    $$ are Pauli matrices and Hamiltonian comes from "Schrodinger form of the free Klein_Gordon equation
    And also why did we added Pauli matrices in the free Hamiltonian ?
     
  2. jcsd
  3. Oct 10, 2012 #2

    Bill_K

    User Avatar
    Science Advisor

    This equation arises from trying to make the second-order Klein-Gordon Equation look like the first-order Dirac Equation. Define a matrix Ψ = (u v) where u = φ - iħ/mc2 ∂φ/∂t and v = φ + iħ/mc2 ∂φ/∂t. Then Ψ obeys a matrix Schrodinger Equation, iħ∂Ψ/∂t = HΨ where the matrix Hamiltonian H is what you have written. Expressing it in terms of Pauli matrices is just for convenience.

    The problem is that H is not Hermitian due to the iτ2 term. What Feshbach and Villars did to fix this was again by analogy with the Dirac Equation. In the Dirac Equation we modified the definition of the adjoint, and instead of ψ† we use ψ ≡ ψ†γ0. Here we use φ ≡ φ†τ3. In terms of this "metric" the norm of ψ is (ψ, ψ) ≡ ∫φ†τ3φ d3x = ∫(u*u - v*v) d3x = iħ/mc2 ∫ (φ* ∂φ/∂t - ∂φ*/∂t φ) d3x, which is the familiar conserved quantity.
     
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