Free Hamiltonian problem for relativistic mechanics

In summary, the equation is a result of trying to make the second-order Klein-Gordon Equation look like the first-order Dirac Equation. The Hamiltonian is expressed in terms of Pauli matrices for convenience, but it is not Hermitian. To fix this, the definition of the adjoint is modified, leading to a conserved quantity.
  • #1
forhad_jnu
2
0
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 ?
 
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  • #2
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.
 

What is the Free Hamiltonian problem for relativistic mechanics?

The Free Hamiltonian problem for relativistic mechanics is a fundamental problem in physics that involves finding the equations of motion for a system of particles in a relativistic framework. It is based on the Hamiltonian formulation of classical mechanics, which uses the concept of energy to describe the dynamics of a system.

What is the difference between the Free Hamiltonian problem and the Classical Hamiltonian problem?

The main difference between the Free Hamiltonian problem and the Classical Hamiltonian problem is that the former takes into account the effects of relativity, while the latter is based on classical mechanics. In the Free Hamiltonian problem, the equations of motion are derived using the relativistic energy and momentum equations, while in the Classical Hamiltonian problem, the equations are derived using the classical energy and momentum equations.

What are the challenges in solving the Free Hamiltonian problem for relativistic mechanics?

One of the main challenges in solving the Free Hamiltonian problem for relativistic mechanics is the complexity of the equations of motion. Unlike in classical mechanics, where the equations are relatively simple and can be solved analytically, the equations in relativistic mechanics are much more complex and often require numerical methods to solve. Another challenge is the need to take into account the effects of special relativity, such as time dilation and length contraction, which can complicate the calculations.

What are some applications of the Free Hamiltonian problem in physics?

The Free Hamiltonian problem has many applications in various fields of physics, such as particle physics, astrophysics, and cosmology. It is used to study the dynamics of particles in high-energy collisions, the behavior of matter in extreme environments, and the evolution of the universe. It also plays a crucial role in the development of theories such as quantum field theory and general relativity.

Is the Free Hamiltonian problem applicable to all systems in relativistic mechanics?

No, the Free Hamiltonian problem is not applicable to all systems in relativistic mechanics. It is primarily used for systems that can be described by a set of point particles, such as particles in a particle accelerator or stars in a galaxy. It is not suitable for systems that involve continuous matter, such as fluids or solids, as the equations of motion for these systems are more complex and require different approaches.

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