Energy of free particle not quantized?

[Aziza]

what does it mean that the energy of a free particle is not quantized, but continuous just like in classical physics? I thought energy is always quantized??

 

[Khashishi]

Energy is not quantized for a free particle. The particle can have whatever kinetic energy.

 

[HallsofIvy]

A particle constrained to a finite interval has quantized energy. A "free particle", that can move any where in space, has continuous energy. Mathematically, that is because the eigenvalues on a finite interval (where you can use a Fourier series) are discrete while the eigenvalues on an infinite interval (where you can use a Fourier integral) are continuous.

 

[tannerbk]

Energy is not quantized in this case because the free particle does not represent a possible physical state. Rather, it is a useful description in the study of one dimensional scattering. None of the eigenfunctions of the moment operator live in Hilbert Space, thus they do not represent a physically realizable state. However, you can recover Dirac Orthonormality and the eigenfunctions are complete, so the free particle is very useful when applied to other problems.

 

[haruspex]

What's always quantised is action; energy * time, momentum * position,...
Energy becomes quantised in consequence when time is constrained to discrete values, such as the period of a photon, or of an electron in orbit. Free electrons have no such time constraint.