What are the restrictions on the Hamiltonian in QM?

Question

In quantum mechanics, we usually write the Hamiltonian as:

$$\hat{H}=\hat{T}+\hat{V}$$

But in classical mechanics, there are several reasons why it would not have this form:

• We've chosen some particular coordinate system where there is no conservation of energy (we're neglecting the source of external forces).

• We can modify the Hamiltonian without changing the equations of motion:

  The Lagrangian can be written as: $L=\frac{1}{3}T^2+2TV-V^2$. The Hamiltonian would be:

  $\hat{H}=p\left[\frac{\left(\sqrt{9p^2m^8 +32m^9V^3}+3pm^4\right)^{1/3}}{\sqrt[3]{2}m^2}-\frac{2\sqrt[3]{2}mV}{\left(\sqrt{9p^2m^8 +32m^9V^3}+3pm^4\right)}\right]-L=(T+V)^2$

  The eigenvalues of this operator are $E^2$, not the energy.

Do we consider this possibilities in quantum mechanics? In the negative case, why do we discard them?

And (besides the necessity to be self-adjoint and bounded from below) does the Hamiltonian have some restriction on the operators it includes (position, angular momentum,...)?

Answer 1

Myth 1: We Need Unitary Time Evolution.

Problem a: Self-Adjointness. This would require the Hamiltonian to be self-adjoint. But there are interesting situations when this won't be true, consider the following papers:

1. F. Bagarello, A. Inoue, C. Trapani, "Non-self-adjoint hamiltonians defined by Riesz bases". Eprint arXiv:1402.6199
2. Fabio Bagarello, Miloslav Znojil, "The dynamical problem for a non self-adjoint Hamiltonian". Operator Theory: Advances and Applications 221 (2012) 109--119, arXiv:1105.4716

Problem b: Constrained Systems. For topological field theory, parametrized fields, or even the relativistic particle (taking the action to be its proper time), the Hamiltonian operator no longer controls time-evolution.

The Hamiltonian instead becomes a constraint for these systems. It vanishes when acting on the physical states. Asking the time evolution operator to be "unitary" is meaningless for physical states.

For more on constrained systems, see, e.g.,

1. Henneaux and Teitelboim's Quantization of Gauge Systems
2. Andreas W. Wipf, "Hamilton's Formalism for Systems with Constraints". Eprint arXiv:hep-th/9312078

Myth 2: We Can Quantize Anything.

Not quite true. For example, as I noted in my comment, if we consider the simple harmonic oscillator (dropping constants for simplicity) $$ T = v^{2},\quad\mbox{and}\quad V=x^{2}$$ then $$ H = (p^{2}+x^{2})^{2}. $$ But look, when we try to quantize this, we can't put hats on everything and hope it all works for the best. Why? Well, how do we quantize $x^{2}p^{2}$? Observe $$ (\hat{x}\hat{p})^{2}\neq \hat{x}^{2}\hat{p}^{2}\neq\left(\frac{\hat{p}\hat{x}+\hat{x}\hat{p}}{2}\right)^{2}$$ They give inequivalent commutators.

This ambiguity is well-known in the literature. See, for example:

1. S. Twareque Ali, Miroslav Engliš, "Quantization Methods: A Guide for Physicists and Analysts". Rev.Math.Phys. 17 (2005) 391-490, arXiv:math-ph/0405065

So...what DO we do?

"We do what works" is the motto! It'd be nice to have some algorithm that always works no matter what...but instead we restrict our focus to the physically relevant situations.

And even then, we only use the quantized systems that make sense.

In a sense, this is the proper route: instead of asking for a procedure to turn something classical into something quantum, we should think that nature already is quantized and we need to seek out the quantum model that emulates the natural phenomena we're interested in.