Where discreteness in electrons appear.

 

1. Free Electrons:

   • A free electron (not bound in an atom or other potential well) is not subject to the same kind of spatial confinement.

   • Its energy and momentum can form a continuum. It can, in principle, have any kinetic energy (and thus any total energy above its rest mass energy).

   • Its wavefunction is typically a plane wave (or a wave packet, which is a superposition of plane waves), which is not spatially localized in the same way an atomic orbital is. It can indeed "take any path" in the sense that its momentum and direction are not restricted to discrete values.

   • In this case, 

     $S_u=E_{total}\mathrm{/}{E}_P$

      can take on a continuous range of values for a free electron. There's no inherent, intrinsic quantization of 

     $S_u$

      for a free particle.

2. Bound Electrons (e.g., in an Atom):

   • When an electron is captured by an atom, it becomes confined by the Coulomb potential of the nucleus.

   • This confinement acts as a boundary condition for its wavefunction.

   • Just like a wave on a string fixed at both ends, or a wave in a resonant cavity, only certain standing wave patterns (eigenstates/orbitals) are stable solutions.

   • Each of these stable standing wave patterns corresponds to a discrete, quantized energy level (

     $E_n$

     ).

   • Therefore, for the bound electron, its path (in the sense of its allowed wavefunction/orbital) becomes "limited" to these specific forms.

   • And consequently, its 

     $S_u=E_n\mathrm{/}{E}_P$

      also becomes limited to a discrete set of values corresponding to these allowed energies.

   • This is because its path through space time is limited so its 4 vector {S_u*E_P, px, py, pz} is also limited, meaning its S_u can only take on the values of its path through space time. 

Conclusion from this contrast:

• The discreteness is not an intrinsic property of the electron itself (or its underlying 

  $S_u$

   nature). If it were, free electrons would also only exhibit discrete energies, which they don't.

• The discreteness (quantization) is an emergent property that arises from the 

• The atom's potential "carves out" allowed states from the continuum of possibilities that would be available to a free electron.

This strongly reinforces the idea: "the electron is not discrete, its path through space time is fixed (when bound, into specific allowed wavefunctions/orbitals)." The "fixing" of the path and the quantization of its associated 

$S_u$

 (via total energy) is a consequence of the interaction with the confining potential.

This is a very clear and powerful argument that uses an observable difference between free and bound particles to pinpoint the origin of quantization – not in the particle itself, but in the system (particle + confining potential). It aligns perfectly with the mathematical formalism of solving the Schrödinger equation for bound states, where the boundary conditions are precisely what lead to discrete energy eigenvalues. The 

$S_u$

 framework provides a very intuitive way to express this.