Quantum pendulum

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.

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 \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 = \frac{1}{2} m l^2 \dot{\phi}^2, U = mgl (1 - \cos\phi). This results in the Hamiltonian \hat{H} = \frac{\hat{p}^2}{2 m l^2} + mgl (1 - \cos\phi). The time-dependent Schrödinger equation for the system is i \hbar \frac{d\Psi}{dt} = -\frac{\hbar^2}{2 m l^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: \eta = \phi + \pi, \Psi = \psi e^{-iEt/\hbar}, E \psi = -\frac{\hbar^2}{2 m l^2} \frac{d^2 \psi}{d \eta^2} + mgl (1 + \cos\eta) \psi. This is simply Mathieu's differential equation \frac{d^2 \psi}{d \eta^2} + \left(\frac{2 m E l^2}{\hbar^2} - \frac{2 m^2 g l^3}{\hbar^2} - \frac{2 m^2 g l^3}{\hbar^2} \cos\eta\right) \psi = 0, whose solutions are Mathieu functions. ==Solutions== ===Energies=== Given q , for countably many special values of a , called characteristic values, the Mathieu equation admits solutions that are periodic with period 2\pi . The characteristic values of the Mathieu cosine, sine functions respectively are written a_n(q), b_n(q) , where n is a natural number. The periodic special cases of the Mathieu cosine and sine functions are often written CE(n,q,x), SE(n,q,x) respectively, although they are traditionally given a different normalization (namely, that their L^2 norm equals \pi ). The boundary conditions in the quantum pendulum imply that a_n(q), b_n(q) are as follows for a given q : \frac{d^2 \psi}{d \eta^2} + \left(\frac{2 m E l^2}{\hbar^2} - \frac{2 m^2 g l^3}{\hbar^2} - \frac{2 m^2 g l^3}{\hbar^2} \cos\eta\right) \psi = 0, a_n(q), b_n(q) = \frac{2 m E l^2}{\hbar^2} - \frac{2 m^2 g l^3}{\hbar^2}. The energies of the system, E = m g l + \frac{\hbar^2 a_n(q), b_n(q)}{2 m l^2} for even/odd solutions respectively, are quantized based on the characteristic values found by solving the Mathieu equation. The effective potential depth can be defined as q = \frac{m^2 g l^3}{\hbar^2}. A deep potential yields the dynamics of a particle in an independent potential. In contrast, in a shallow potential, Bloch waves, as well as quantum tunneling, become of importance. ===General solution=== The general solution of the above differential equation for a given value of a and q is a set of linearly independent Mathieu cosines and Mathieu sines, which are even and odd solutions respectively. In general, the Mathieu functions are aperiodic; however, for characteristic values of a_n(q), b_n(q) , the Mathieu cosine and sine become periodic with a period of 2\pi . ===Eigenstates=== For positive values of q, the following is true: C(a_n(q), q, x) = \frac{CE(n, q, x)}{CE(n, q, 0)}, S(b_n(q), q, x) = \frac{SE(n, q, x)}{SE'(n, q, 0)}. Here are the first few periodic Mathieu cosine functions for q = 1 . Note that, for example, CE(1, 1, x) (green) resembles a cosine function, but with flatter hills and shallower valleys. ==References==

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