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Equation 2.2 can be rewritten as:
C d2
dt 2 V (t) + ∂
∂V
( 1
2L V 2 )
= 0 (2.4)
Here, the “force” acting on the oscillator has been expressed as the partial
derivative of the system’s potential energy as a parabolic function of the “position”
V . Solving the time independent Schrödinger equation (see Section 3.1) yields the
familiar quantum eigenstates of the potential:
〈 x | ψ n 〉 =
√ (√ )
1 4 C ω
√ C ω C ω
2n n! π e− 2 x2 H n
x for n = 0, 1, 2, . . . (2.5)
Here, H n are the Hermite polynomials and the angular frequency ω is defined as
1/ √ LC. The energy levels corresponding to the eigenstates are:
E n = ω
(
n + 1 )
2
(2.6)
It is possible to operate the LC oscillator in a regime that allows the observation
of quantum mechanical behavior, provided three conditions are met:
• The wiring metal needs to be chosen such that it can be operated below
its superconducting transition temperature. Otherwise the resistance in the
leads will cause energy decay rates that are much too high to observe meaningful
quantum effects. In the analogy, the damping term in the pendulum
has to be small enough to allow for quantum excitations to exist for a meaningful
time.
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