Collision of Two Neutrons in the de Broglie–Bohm Approach
Neutron scattering is a spectroscopic method of measuring the motions of magnetic particles. This can be used to probe a wide variety of different physical phenomena: interference effects, diffusional or hopping motions of atoms, rotational modes of molecules, acoustic modes and molecular vibrations, recoil in quantum fluids and magnetic quantum excitations .[more]
In the causal interpretation of nonrelativistic quantum mechanics, a particle such as a neutron possesses a definite position and a momentum at all times. The possible trajectories are determined by the gradient of real phase function (or quantum potential) in the total wavefunction (pilot wave) . We show a very simple model of a time-dependent collision of two neutrons, without spin or magnetic moment, neglecting the influence of gravitation and quark structure and with low momentum, displayed in a three-dimensional configuration space.
Two neutrons represented by two three-dimensional Gaussian wave profiles with different initial positions and momenta are used as projectiles. These interact in some regions, if the initial momenta are chosen appropriately . In an elastic neutron scattering event, the momentum is transferred from one neutron to the another.
For a symmetric initial momentum distribution of the two waves (monochromatic beam), but with opposite signs in one direction, for example, in the direction, the model can be interpreted as a neutron interferometer (beam-splitter). After splitting by amplitude division (Mach–Zehnder type) via Bragg reflection, but with only one neutron present in the device at a time, the interference effect still appears. In this case, the neutron wave packet is split into two coherent waves (sub-beams) and the possible trajectories of one particle are affected by the wave that is not carrying the other particle (empty wave) .
In the graphics you see the wave density (if enabled), the initial momentum (large red arrows), the velocity vector field, the initial starting points of the eight trajectories (red points, shown as small red spheres), the actual position (colored points, shown as small spheres) and eight possible trajectories of the two neutrons.[less]
The Schrödinger equation for the motion of a free particle is given by:
It is readily shown that the general normalized solution of the two sub-beams is of the form
where is the considered point of space and is the midpoint of the wave. The dispersion relationship holds with , here (atomic units). The solution as stated is defined in the infinite space . Obviously, the initial condition at determines (inverse Fourier transform). It follows
For the neutron beam, the is chosen by:
with initial density , , and with the wavenumber vector for the particle. The total wavefunction becomes
From the total wavefunction for , the equation for the phase function could be calculated, and therefore the components of the velocity. This is a standard procedure [5–7] in Bohmian mechanics, which will not be described here.
When PlotPoints, AccuracyGoal, PrecisionGoal and MaxSteps are increased (if enabled), the results will be more accurate. The initial distance between the starting trajectories is determined by the factor .
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