Is the time derivative of the Hamiltonian integral over phase space (the gamma distribution) the equivalent of an action which conserves the probability phase/volume.
Probability Wave Dispersion
The function $$ S_0(l) $$ is called the action of the path, and to each path we define an action,
$$ S_0(l) = \int_{t_l} E dt $$
It is weighted against Planck’s constant, which also has units of action per radian.
The presence of an electromagnetic field adds the interaction term, (superscript $$ I$$),
$$ {S_0}^I(l) = q \int_{t_1}^{t_2} A^\alpha g_{\alpha\beta} \partial x^{\beta} $$
which is the action of the electromagnetic Lorentz force. This is the quantum mechanical Bohm-Aharonov Effect, where $$ q $$ is the charge and $$ A^\alpha $$ is the electromagnetic gauge potential. In the absence of a magnetic field this becomes the prepared state of a quantum computation.
Derive the action that shifts the phase angle $$ \theta_I $$ for a static EM field...?
This is the integral.
$$ {d \over dt} \int_{\Gamma} f(x,p) dxdp = \int_{\Gamma}({\partial f \over \partial x} {dx \over dt} \,+\, {\partial f \over \partial p } {dp \over dt})dxdp $$
$$ \;\;\;\;\;\;\;\;\;\; = \int_{\Gamma}({\partial f \over \partial x} {\partial \hat H \over \partial p} \,+\, {\partial f \over \partial p } {\partial \hat H \over \partial x})dxdp $$
Probability Wave Dispersion
The function $$ S_0(l) $$ is called the action of the path, and to each path we define an action,
$$ S_0(l) = \int_{t_l} E dt $$
It is weighted against Planck’s constant, which also has units of action per radian.
The presence of an electromagnetic field adds the interaction term, (superscript $$ I$$),
$$ {S_0}^I(l) = q \int_{t_1}^{t_2} A^\alpha g_{\alpha\beta} \partial x^{\beta} $$
which is the action of the electromagnetic Lorentz force. This is the quantum mechanical Bohm-Aharonov Effect, where $$ q $$ is the charge and $$ A^\alpha $$ is the electromagnetic gauge potential. In the absence of a magnetic field this becomes the prepared state of a quantum computation.
Derive the action that shifts the phase angle $$ \theta_I $$ for a static EM field...?
This is the integral.
$$ {d \over dt} \int_{\Gamma} f(x,p) dxdp = \int_{\Gamma}({\partial f \over \partial x} {dx \over dt} \,+\, {\partial f \over \partial p } {dp \over dt})dxdp $$
$$ \;\;\;\;\;\;\;\;\;\; = \int_{\Gamma}({\partial f \over \partial x} {\partial \hat H \over \partial p} \,+\, {\partial f \over \partial p } {\partial \hat H \over \partial x})dxdp $$
Last edited: