A very important aspect of the Hamiltonian operator is that it only acts at vertices ( a consequence of this is that Thiemann's Hamiltonian operator, like Ashtekar's operator, annihilates non-intersecting loops except now it is not just formal and has rigorous mathematical meaning ).
32.
Where " E " = total energy, " H " = hamiltonian, " T " = kinetic energy and " V " = potential energy of the particle, substituting the energy and Hamiltonian operators and multiplying by the wavefunction obtains the Schr�dinger equation
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In which \ psi is the wavefunction of the system, H is the Hamiltonian operator, and T and V are the operators for the kinetic energy and potential energy, respectively . ( Common forms of these operators appear in the square brackets . ) The quantity " t " is the time.
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There are however severe difficulties with this particular approach, for example the Hamiltonian operator is not self-adjoint, in fact it is not even a normal operator ( i . e . the operator does not commute with its adjoint ) and so the spectral theorem cannot be used to define the exponential in general.
35.
Where \ hat H is the Hamiltonian operator, T is elapsed time, E is the energy change due to the disturbance, W =-E T is the change in action due to the disturbance, \ varphi is the field of the virtual particle, the integral is over all paths, and the classical action is given by
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Sen's initiation and completion by Ashtekar, finally, for the first time, in a setting where the Wheeler DeWitt equation could be written in terms of a well-defined Hamiltonian operator on a well-defined Hilbert space, and led to construction of the first known exact solution, the so-called Chern Simons form or Kodama state.
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The name comes from the Green's functions used to solve inhomogeneous differential equations, to which they are loosely related . ( Specifically, only two-point'Green's functions'in the case of a non-interacting system are Green's functions in the mathematical sense; the linear operator that they invert is the Hamiltonian operator, which in the non-interacting case is quadratic in the fields .)
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