Quantization of the piezoelectric constant

The well-known and celebrated first-primer classical analysis of a one-dimensional inversion-asymmetric assembly of electric point charges interconnected by mechanical springs shows that the system is piezoelectric and characterized by a parameter-dependent but constant piezoelectric coefficient <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mi>d</a:mi></a:math> defined as the ratio between the change in system length and the change in electric field. The former system is the simplest system displaying the phenomenon of piezoelectricity. We demonstrate that a quantum-mechanical analysis of the Hamiltonian for the same system of electric point charges and mechanical springs leads to a piezoelectric constant that depends not only on the system parameters but also on the eigenstate. Hence, the piezoelectric constant, determined as the ratio between the change in the expectation value of the system length and the change in the applied electric field, is quantized. It is demonstrated analytically and numerically, which is a necessary condition, that the quantized piezoelectric constant vanishes if the system Hamiltonian is inversion symmetric.