# Quantum energy spectrum of coupled LC harmonic oscillators

• For a LC harmonic oscillator, the energy spectrum is evenly spaced by

$$\Delta E = \hbar \omega \quad \omega = {1\over \sqrt{LC} }$$

• For two inductively coupled LC harmonic oscillators with mutual inductance $$M$$, starting from:

$$\begin{cases} \quad (i\omega L_1 + {1\over i\omega C_1})I_1 + i\omega M I_2 &= U \\ \quad (i\omega L_2 + {1\over i\omega C_2})I_2 + i\omega M I_1 &= 0 \end{cases}$$

• The eigen-energies are (calculated by Mathematica):

$$\left[ \left[W = \sqrt{\frac{1}{2}} \sqrt{-\frac{L_{1} C_{1} + L_{2} C_{2} - \sqrt{L_{1}^{2} C_{1}^{2} + 4 \, C_{1} M^{2} C_{2} - 2 \, L_{1} C_{1} L_{2} C_{2} + L_{2}^{2} C_{2}^{2}}}{C_{1} M^{2} C_{2} - L_{1} C_{1} L_{2} C_{2}}} \right], \left[W = \sqrt{\frac{1}{2}} \sqrt{-\frac{L_{1} C_{1} + L_{2} C_{2} + \sqrt{L_{1}^{2} C_{1}^{2} + 4 \, C_{1} M^{2} C_{2} - 2 \, L_{1} C_{1} L_{2} C_{2} + L_{2}^{2} C_{2}^{2}}}{C_{1} M^{2} C_{2} - L_{1} C_{1} L_{2} C_{2}}} \right] \right]$$

• In the case where $$L_1 = L_2, C_1 = C_2$$, the above becomes:

$$\left[ \left[W = \frac{1}{\sqrt{L_{1} C_{1} + C_{1} M}}\right], \left[W = \frac{1}{\sqrt{L_{1} C_{1} - C_{1} M}}\right] \right]$$

• The question is: Now that there are two different frequencies (close to each other when $$M$$ is small), how does the energy spectrum look like?

• I have found this: https://en.wikipedia.org/wiki/Quantum_LC_circuit, but the spectrum is not discussed.

When the two (identical) oscillators are decoupled, the energy spectrum is identical to that of a harmonic oscillator albeit each energy level is doubly degenerate. In the case of the coupled oscillator, it can be viewed as this degeneracy being lifted. As the coupling $$M$$ increases, the splitting is higher. 