Main Answer
In the context of a scattering collision between a relativistic electron and a hydrogen atom in the ground state, the interaction time can be estimated as Δt ≈ μDvwith a coefficient μ in the range of 0.333 < μ < 3.
Step-by-step explanation
In quantum mechanics, the motion of particles is described by wavefunctions, which are mathematical functions that represent the probability of finding a particle at a given position and time.
In the case of a scattering collision between a relativistic electron and a hydrogen atom, the wavefunctions of both particles overlap during the collision, leading to an exchange of energy and momentum.
To estimate the interaction time, we can use the principles of wavefunction overlap integrals. This involves calculating the overlap integral between the wavefunctions of the electron and hydrogen atom during the collision.
The overlap integral provides a measure of how much the wavefunctions overlap during the collision, which is related to the probability of finding both particles in close proximity to each other.
The diameter D of a ball within which the probability to find the hydrogenic-electron (in the ground state) is at least 0.99 can be used to estimate the size of the region where both wavefunctions overlap significantly during the collision.
The velocity v is a relativistic velocity that takes into account the fact that both particles are moving at high speeds close to the speed of light.
The coefficient μ is introduced to account for various factors that affect the interaction time, such as the strength of the interaction potential and the angular momentum of both particles.
In general, μ is expected to be in the range of 0.333 < μ < 3, based on considerations from classical mechanics and intuition about how long it takes for two objects to interact significantly when they are moving at high speeds.
While classical mechanics cannot be directly applied to this problem because both particles are described by quantum wavefunctions, some justified argument must exist to estimate and justify this estimate without invoking the full formalism of relativistic quantum scattering, which is extremely heavy.
The estimate provided here seems true based on principles of wavefunction overlap integrals and intuition about how long it takes for two objects to interact significantly when they are moving at high speeds.