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- Thread starter pivoxa15
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In what context, careful?

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Are you sure about that? I mean, the wavevector of a particle inside a barrier is complex so that we get exponentially decaying wavefunctions instead of oscillating ones, right? This means that the kinetic energy [tex]\hbar^2 k^2 / 2 m < 0[/tex] since [tex]k[/tex] is complex.

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Kinetic energy is defined by the positive hermitian operatorAre you sure about that? I mean, the wavevector of a particle inside a barrier is complex so that we get exponentially decaying wavefunctions instead of oscillating ones, right? This means that the kinetic energy [tex]\hbar^2 k^2 / 2 m < 0[/tex] since [tex]k[/tex] is complex.

[tex]\hat{p}^2 / 2 m[/tex]

The operator [tex]\hat{p}[/tex] is hermitian so its eigenvalues cannot be imaginary, but only real.

A wave function [tex]e^{kx}[/tex], which is formally an eigenstate of the momentum operator with an imaginary eigenvalue, is not square integrable even as a functional (a plane wave is not square integrable either, but it is square integrable at least as a functional), so it is not a physical state. In a barrier, the wave function has the form [tex]e^{kx}[/tex] only on a small portion of space, not everywhere, so such a wave function is not really an eigenstate of the momentum operator. The momentum is a property of the whole wave function, not of a wave function on a small portion of space.

Although the local momentum does not make sense in the usual formulation of QM, it does make sense in the Bohmian formulation. In this case, the local momentum associated with a wave function proportional to [tex]e^{kx}[/tex] is - zero. A local particle does not move and does not have a kinetic energy in this region.

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Cesiumfrog, write out the Hamiltonian constraint, the terms quadratic in the momentum are indefinite : 1/2 (P_c^c)^2 - P_ab P^ab. The ``potential'' term is the densitized Ricci curvature of the spacelike metric and can be positive and negative as well.

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I'm reminded of quantum tunneling. When, say, an electron is tunneling through a potential barrier then during the time the particle is inside the barrier the particle's kinetic energy is zero. This, of course, assumes that the potential energy of the barrier is zero at a certain point so we can meaningfully say that the particle is inside the potential and the potential is zero otherwise (or something of that manner.

Pete

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