Abstract
An investigation of origins of the quantum mechanical momentum operator has shown that it corresponds to the nonrelativistic momentum of classical special relativity theory rather than the relativistic one, as has been unconditionally believed in traditional relativistic quantum mechanics until now. Taking this correspondence into account, relativistic momentum and energy operators are defined. Schrödinger equations with relativistic kinematics are introduced and investigated for a free particle and a particle trapped in the deep potential well.
Highlights
The known attempts to apply the ideas of special relativity theory (SRT) in quantum mechanics, formulated in the third decade of 20-th century and present in numerous textbooks, are based on using the quantum mechanical momentum operator p =−i ∇ in the nonrelativistic Schrödinger equation for the free particle with the Hamiltonian corresponding to the classical SRT expression for energy: E= m2c4 + p2c2 = mc2 + p2 2m − p4 8m3c2 (1)The first term of this expansion mc2 is constant in an arbitrary reference frame, it can be considered as part of the potential, defined with an accuracy up to a constant
An investigation of origins of the quantum mechanical momentum operator has shown that it corresponds to the nonrelativistic momentum of classical special relativity theory rather than the relativistic one, as has been unconditionally believed in traditional relativistic quantum mechanics until now
Schrödinger equations with relativistic kinematics are introduced and investigated for a free particle and a particle trapped in the deep potential well
Summary
The known attempts to apply the ideas of special relativity theory (SRT) in quantum mechanics, formulated in the third decade of 20-th century and present in numerous textbooks, are based on using the quantum mechanical momentum operator p =−i ∇ in the nonrelativistic Schrödinger equation for the free particle with the Hamiltonian corresponding to the classical SRT expression for energy: E=. ∂ ∂t is that it is associated with the total relativistic energy, i.e This Schrödinger equation with relativistic kinematics does not correspond to the requirement that the operator of relativistic equation has to be invariant in respect of Lorentz transformations. The first one gives the Klein-Gordon equation, following directly from square of total energy of free particle expression applying defined quantum mechanical operators pand E :. The other method is introduced by Dirac He postulated the possibility of quantum operator Elinearization, i.e. presentation in form. Where β j is fourth order matrices The conditions for these matrices follow from square of relativistic energy expression, present in operator form:. The new definition of operators is further inspected solving the well-known problems for a free particle and particle trapped in the deep potential well
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