A classical problem in quantum mechanics involves computing the spectrum of a Schrödinger-type differential operator. One can, for example, find the asymptotic expansions of the eigenvalues using the WKB method. Another approach, due to Sjöstrand, obtains such expansions directly from the operator, using its quantum normal form. We provide a geometric interpretation for this normal form, encoding it as a section of a vector bundle associated with the quantization of a complex integrable system. We also propose a number of conditions that allow us to determine this section uniquely. This is joint work with Maxim Kontsevich.