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WavePacket demo example: Simple pendulum

The problem of a physical pendulum has played a great role in the development of analytic classical mechanics. Hence, soon after Schrödinger's fundamental work appeared, a first paper on the physical pendulum in quantum mechanics was published by E. U. Condon in 1928. He recognized that the eigenfunctions of the time-independent Schrödinger equation with a simple cosine potential can be expressed in terms of Mathieu' (co-)sine elliptic functions for which no analytic expressions are available. Since that time, however, the problem has rarely been touched. Only very recently, there has been a renewed interest in the quantum dynamics of a simple pendulum, see, e. g., our work on quantum dynamics of a plane pendulum.

Note that some analytic eigenfunctions of the time-independent Schrödinger equation are available for a linear combinations of a cosine potential and a cosine-squared potential, see our Wiki page on generalized pendula.

Stationary states of the simple pendulum

The (stationary) eigenstates of a plane quantum pendulum beautifully demonstrate the transition from harmonic oscillator states for the strongly hindered case to free rotor states in the essentially unhindered case. The corresponding wavefunctions are Mathieu functions for which no closed expressions exist. However, their Fourier representations are easily generated by diagonalizing a tri-diagonal matrix. Learn more ...

Pendular analogues of squeezed states

In a harmonic oscillator, a "squeezed" state is obtained by changing the width of the ground state wavefunction. As a characteristic property, it does not change its Gaussian shape upon time evolution but its width is oscillating periodically in time. For the Mathieu states of the quantum pendulum, this is not the case because of the anharmonicity of the trigonometric potential energy curve, but in the free rotor limit intriguing (full and fractional) revival phenomena are found for long times. Learn more ...

Pendular analogues of coherent states

In a harmonic oscillator, coherent (or Glauber) states are described by stationary wavefunctions, shifted along the independent coordinate. As a characteristic property, they do not change their shape upon time evolution with their centers following classical trajectories. For the Mathieu states of the quantum pendulum, this is not the case because of the anharmonicity of the trigonometric potential energy curve, but intriguing interference phenomena are found for long times. Learn more ...

Pendular analogues of double well potentials

Novel features arise when passing from the periodic single well potentials to pendular analogues of double well potentials. In addition to the formation of (nearly degenerate) tunneling doublets in deep potential wells, higher states of different parity exhibit drastically different time evolution. In particular, different (full and fractional) revival times are found for even and odd states, or for combinations thereof. Learn more ...