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FIG. 1. Two-slit interference experiment for electrons.

Figure 1 Two-slit interference experiment for electrons.

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The term Hy, will be treated as a perturbation. The Hamiltonian without perturbation If we consider that a photon can be emitted, these are characterized by the potential vector A(r) and then the Hamiltonian of the total electron + photon system is
The term Hy, will be treated as a perturbation. The Hamiltonian without perturbation If we consider that a photon can be emitted, these are characterized by the potential vector A(r) and then the Hamiltonian of the total electron + photon system is
the destruction and creation operators, and Q = volume of the box where the electromagnetic field is quantized
the destruction and creation operators, and Q = volume of the box where the electromagnetic field is quantized
and substituting the expansion in normal modes for A
and substituting the expansion in normal modes for A
FIG. 2. Polarization of photons for the sum in Eq. (19). Ukilur2 and both perpendicular to the direction of propa- gation of the photon k
FIG. 2. Polarization of photons for the sum in Eq. (19). Ukilur2 and both perpendicular to the direction of propa- gation of the photon k
FIG. 3. Trajectories of the electrons in the two-slit inter- ference experiment predicted by Bohm-de Broglie mechanics (Extracted from [5]). 6 plane of the slits: we see a series of ”‘potential wells” (or valleys) corresponding to the troughs (Fig. 5). The electron ”‘fall”’ in the potential well where is first accel- erated with a strong force je and then decelerated (the quantum potential for this experiment depends only on y, see [5], [6]). From the quantum theoretical point of view we can say that there exist a certain probability for the accelerated electron emit a photon.
FIG. 3. Trajectories of the electrons in the two-slit inter- ference experiment predicted by Bohm-de Broglie mechanics (Extracted from [5]). 6 plane of the slits: we see a series of ”‘potential wells” (or valleys) corresponding to the troughs (Fig. 5). The electron ”‘fall”’ in the potential well where is first accel- erated with a strong force je and then decelerated (the quantum potential for this experiment depends only on y, see [5], [6]). From the quantum theoretical point of view we can say that there exist a certain probability for the accelerated electron emit a photon.
FIG. 4. The quantum potential for the two (gaussian) slits viewed from the screen S2 (Extracted from [5]).
FIG. 4. The quantum potential for the two (gaussian) slits viewed from the screen S2 (Extracted from [5]).
FIG. 5. A section of the quantum potential at 18cm from the slits S2 (Extracted from [5]). that the electron follows trajectories. To do that ° we are going to re-write the bracket:
FIG. 5. A section of the quantum potential at 18cm from the slits S2 (Extracted from [5]). that the electron follows trajectories. To do that ° we are going to re-write the bracket:
We see that 7# = ae) is the eigenvalue of acceleratior operator. Note that operators r and r commute. It is easy to show that the local expectation value (LEV o1 the beable) associated with the operator acceleration, ir the BdB theory, coincides with the eigenvalue of operatot acceleration® #. Substituting (39) in (33) we obtain:
We see that 7# = ae) is the eigenvalue of acceleratior operator. Note that operators r and r commute. It is easy to show that the local expectation value (LEV o1 the beable) associated with the operator acceleration, ir the BdB theory, coincides with the eigenvalue of operatot acceleration® #. Substituting (39) in (33) we obtain:
Accepting that during this time a photon is emitted, by using the mean emission power given by Eq.(56) we have for the energy of the photon: FIG. 6. Approximating the section of the quantum potential given in Fig. 5.
Accepting that during this time a photon is emitted, by using the mean emission power given by Eq.(56) we have for the energy of the photon: FIG. 6. Approximating the section of the quantum potential given in Fig. 5.
FIG. 7. Qualitative frequency spectrum for the emitted pho- tons when the electron cross each well of the quantum poten- tial. Here, I(0) is given by Eq.(72).
FIG. 7. Qualitative frequency spectrum for the emitted pho- tons when the electron cross each well of the quantum poten- tial. Here, I(0) is given by Eq.(72).
Related topics:
PhysicsQuantum Mechanicsde Broglie-Bohm theoryInterpretations of Quantum MechanicsDouble Slit Experiment
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