# Why don't we use Hamilton-Jacobi method in QM?

In classical mechanics, we usually try to find a set of coordinates by Hamilton-Jacobi method to transform the Hamiltonian to zero such that the coordinates are conservations.

However, we never try by similar steps to transform the Hamiltonian to zero because what we can get is nothing from the zero-Hamiltonian.

Why don't we use Hamilton-Jacobi method in QM?

• The short answer is: Because canonical transformations generally do not correspond to unitary transformations. Commented Mar 18, 2014 at 14:18

I) Well semiclassically, the Hamilton-Jacobi equation famously appears to lowest order in $\hbar$ in a WKB expansion of the Schrödinger equation. See also Ref. 1.

II) The quantum concept of a canonical transformation (CT)

$$z^I~=~(\hat{q}^i; \hat{p}_i) \quad\longrightarrow\quad Z^J~=~(\hat{Q}^j; \hat{P}_j) \tag{1}$$

(where the old and new canonical variables both satisfy the CCR) typically becomes very difficult to implement to all quantum orders unless we are talking about an affine transformation

$$z^I \quad\longrightarrow\quad Z^J ~=~ A^J{}_I z^I +b^J. \tag{2}$$

Thus the great flexibility of classical CT is (at least from a practice computational point of view, although not from a theoretical point of view) replaced by rigidity at the full quantum level.

III) In practice, when quantizing a theory, one would first seek out the simplest classical formulation of the problem (which yields the same classical equations of motions, and which is most susceptible to quantization), and then try to quantize that.

E.g. if the classical Lagrangian contains an overall square root, one would typically try to find an equivalent classical Lagrangian that is quadratic in the fundamental variables before attempting to quantize the system.

References:

1. B.S. DeWitt, The Global Approach to QFT, Vol 1, 2003; eq. (13.12).
• I presume the restricted form of your (2) can be traced back to avoiding ordering issues? Commented Apr 29, 2018 at 18:41
• $\uparrow$ Right. Commented Apr 29, 2018 at 19:07
• I can probably find some literature on this but if you have something handy please share. (see v.g. jstor.org/stable/pdf/…) Commented Apr 29, 2018 at 19:09

You can ... and it leads directly to Bohm's formulation of Quantum Mechanics; as that's the essence of what it is: the quantum version of the Hamilton-Jacobi formulation of mechanics. The Quantum Hamilton-Jacobi Equation (under The Quantum Potential); and The Quantum Potential (under Stanford Encyclopedia Of Philosophy archive's Bohmian Mechanics).

Bohm's $$S$$ is the same $$S$$ as in the Hamilton-Jacobi formulation, but it's got an extra "Quantum Potential" $$Q$$ contributing to it, now.

But, I don't know where you're getting the idea from that the Hamilton-Jacobi formulation is about "setting the Hamiltonian to zero". The formulation is not defined or characterized that way, and it's not even true in general, as you can see here: Hamilton-Jacobi Equation.

The Hamiltonian does in fact go to zero though, except it is under a canonical transformation, which comes from solving the Hamilton-Jacobi equation. The hamiltonian is not just set to 0. See Hand and Finch.