We study the one-dimensional Schrödinger equation for the Hamiltonian involving the bipartite hard-core potential, [Formula: see text]. We select the suitable unitary transformation and employ the entangled state representation |η⟩, η = η1 + iη2, to find the quantization condition for energy E, [Formula: see text] which seems new.
By virtue of the entangled state representation, we show that the quantum teleportated state in Bob, after Alice makes an Einstein–Podolsky–Rosen measurement and Bob makes an appropriate unitary transformation on hearing Alice's measurement result via a classical channel, is described by [Formula: see text] where F(η)≡23〈η|ρ23|η〉23, ρ23 stands for a quantum channel in 2–3 modes, |η〉23 is the bipartite entangled state, and [Formula: see text] is a displacement transform performed by Bob.
The Schrödinger equation is one of the cores in quantum mechanics, but bears a contradiction. In the process to obtain the energy and momentum operators, the relation [Formula: see text] is used for [Formula: see text]. However, when they are applied to the Hamiltonian equation, the kinetic energy is set as [Formula: see text]. Based on the Energy Circulation Theory, we examine in this paper the quantization of motions of a particle. We clarify in which situation and for what energy we can use the relation [Formula: see text]. We derive a wave equation de novo to provide wave functions representing a concrete motion and energy distribution of a particle. The Schrödinger equation has a similar form by chance but the mass in our new equation is that of energy quantum expressed by [Formula: see text], which is common for any energies of any particles and decided only by the moving speed. A solution shows an energy location in the 3D real space even if it is expressed in complex. When a motion of a particle gets in circle, its circular frequency becomes quantized. In an atom, an electron circulates around the hidden dimensional axis, and the circulation can further rotate. We propose the quantization conditions for the electron orbiting, and derive the wave functions in concrete for S and P orbitals, which are different from current perceptions. We also demonstrate that the uncertainty principle is not valid for a motion of a particle.