# Quantising Classical Lagrangian

Suppose you have a system described by the following Lagrangian: $$L=(1-gq²)\dot{q}^2/2.$$

How would you quantize this theory? Do you need to symmetrize the Hamiltonian before promoting the coordinate and momentum to operators?

• What is usually understood as "canonical quantization" depends very much on whether the system is constrained. Can you check this? Dec 15, 2019 at 16:43

1. The 1D particle $$L=\frac{1}{2}m(q)\dot{q}^2$$ with a position-dependent mass $$m(q)$$ has classical momentum $$p=m(q)\dot{q}$$ and Hamiltonian $$H=\frac{p^2}{2m(q)}$$.
3. Following the same quantization strategy as my Phys.SE answer here where the mass $$m(q)$$ is viewed as the lone component $$g_{qq}$$ of a 1D metric, in the Schrödinger representation the momentum operator becomes $$\hat{p}~=~ \frac{\hbar}{i} m(q)^{-1/4}\frac{\partial }{\partial q} m(q)^{1/4},$$ and the Hamiltonian operator becomes $$\hat{H}~=~ -\frac{\hbar^2}{2} m(q)^{-1/4}\frac{\partial }{\partial q} m(q)^{-1/2} \frac{\partial }{\partial q}m(q)^{-1/4}.$$