Did people try that? Or did they consider energy quantization for the translation motion of a free particle?

  • $\begingroup$ What is a non-linear oscillator, exactly? $\endgroup$ Commented Jun 11, 2017 at 9:58
  • 1
    $\begingroup$ like with a $x^4$ term in the potential $\endgroup$
    – wdlang
    Commented Jun 11, 2017 at 9:59

1 Answer 1


The Schrodinger equation is linear, so perhaps "quantization of non-linear oscillators" refers to non-linearity of the classical system that one quantizes. If this is what you mean, the answer is yes. In the very first paper on modern quantum mechanics, Heisenberg's paper of 1925, non-linear oscillators with potentials $x^2+cx^3$ and $x^2+cx^4$ play a crucial role. They are called anharmonic oscillators. They were also studied before and after Heisenberg in context of quantization.

References: B. L. van der Waerden, Sources in quantum mechanics, Dover, 1968. This contains English translations and commentaries of the most important papers on quantum mechanics.

Edit. Most people who studied this paper of Heisenberg agree that his logic is very obscure. After stating the rules of the matrix mechanics, he applies them to quantization of cubic and quartic potentials, and somehow concludes from the result that his rules must be correct.


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