Wave function, equivalent inductance and magnetic energy stored in the mesoscopic two-loop L–C circuit derived in entangled state representation

Abstract

We analyse the mesoscopic two-loop LC circuit in which a capacitance C is sharing and mutual inductance M exists between two inductances L1, L2. Instead of using the usual Kirhoff's voltage law to examine the equivalent inductance, we solve the Schrödinger equation for this quantized two-loop LC circuit to find the wave function in entangled state representation (ESR). In so doing, we derive the equivalent inductance L1L2−M2L1+2M+L2, and the quantized energy level formula En=L1I1+MI1+I2+L2I222L1+2M+L2+n+12L1+2M+L2L1L2−M2C, (n = 0, 1, …) where the term L1I1+MI1+I2+L2I222L1+2M+L2, as comparing the common energy stored in an inductance LI22, is a newly found magnetic energy stored in the two-loop circuit while quantum radiation with the characteristic frequency L1+2M+L2L1L2−M2C taking place. We do this by virtue of the ESR method, since quantum entanglement arises from both the mutual inductance coupling and the capacitance in sharing, and using ESR is convenient.