The quantum pendulum is a theoretical model and experimental system that studies how a pendulum behaves under quantum mechanics. It is fundamental in understanding hindered internal rotations in chemistry, quantum features of scattering atoms, as well as numerous other quantum phenomena. Though a pendulum not subject to the small-angle approximation has an inherent nonlinearity, the Schrödinger equation for the quantized system can be solved relatively easily.
Schrödinger equation Using Lagrangian mechanics, one can develop a Hamiltonian for the system. A simple pendulum has one generalized coordinate (the angular displacement ϕ {\displaystyle \phi } ) and two constraints (the length of the string and the plane of motion). The kinetic and potential energies of the system can be found to be
T = 1 2 m l 2 ϕ ˙ 2 , {\displaystyle T={\frac {1}{2}}ml^{2}{\dot {\phi }}^{2},}
U = m g l ( 1 − cos ϕ ) . {\displaystyle U=mgl(1-\cos \phi ).}
This results in the Hamiltonian
H ^ = p ^ 2 2 m l 2 + m g l ( 1 − cos ϕ ) . {\displaystyle {\hat {H}}={\frac {{\hat {p}}^{2}}{2ml^{2}}}+mgl(1-\cos \phi ).}
The time-dependent Schrödinger equation for the system is
i ℏ d Ψ d t = − ℏ 2 2 m l 2 d 2 Ψ d ϕ 2 + m g l ( 1 − cos ϕ ) Ψ . {\displaystyle i\hbar {\frac {d\Psi }{dt}}=-{\frac {\hbar ^{2}}{2ml^{2}}}{\frac {d^{2}\Psi }{d\phi ^{2}}}+mgl(1-\cos \phi )\Psi .}
One must solve the time-independent Schrödinger equation to find the energy levels and corresponding eigenstates. This is best accomplished by changing the independent variable as follows:
η = ϕ + π , {\displaystyle \eta =\phi +\pi ,}
Ψ = ψ e − i E t / ℏ , {\displaystyle \Psi =\psi e^{-iEt/\hbar },}
E ψ = − ℏ 2 2 m l 2 d 2 ψ d η 2 + m g l ( 1 + cos η ) ψ . {\displaystyle E\psi =-{\frac {\hbar ^{2}}{2ml^{2}}}{\frac {d^{2}\psi }{d\eta ^{2}}}+mgl(1+\cos \eta )\psi .}
This is simply Mathieu's differential equation
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