3 votes

"Constrain then quantise" vs. "quantise then constrain"

The phrase "quantization commutes with constraints" usually refers to Guillemin-Sternberg conjecture. It has only been proved for a limited class of (gauge) theories. It should stressed that ...
  • 182k
2 votes

"Constrain then quantise" vs. "quantise then constrain"

Take your first example of a particle in $\mathbb R^d$ constrained to lie on $x_1=0$. In Dirac Quantization the constraint on the wavefunctions is not $x_1 \psi(\vec x)=0$, but rather $$\frac{\partial}...
  • 259
2 votes

Example of a transformation that is not canonical

One way is to spoil a canonical transformation. But in reality, canonical transformations are a very restricted subset of all possible transformations. Therefore, if you just 'randomly' pick a ...
1 vote
Accepted

Need help finding Hamiltonian for equations of motion

Hints: The ansatz (2) is too optimistic: The generalized velocity $\dot{\theta}$ is not necessarily the canonical momentum. Instead use the method outlined in my Phys.SE answer here to find a ...
  • 182k
1 vote

Using the EoM in the canonical quantization of EM field

I think you can use the equation of motion for the simple harmonic oscillator too. The Euler Lagrange equation for the position operator $X$ is $$\frac{d^2X}{dt^2}+\omega ^2 X=0$$ so we get: $$X=ae^{-...
  • 5,695
1 vote
Accepted

Derivation of Hamilton-Jacobi (HJ) Equation

Hamilton's principal function$^1$ $S\equiv F$ is the sole unknown variable in HJ equation. E.g. in contrast the Hamiltonian $H$ is assumed to be known. That's how separation of variables (SOV) work. ...
  • 182k
1 vote
Accepted

What are good books/chapters of books or articles to study canonical transformations in quantum mechanics at a graduate level?

I am unaware of many detailed resources in the area targeted towards typical physics graduates. However, there is a CRM monograph dealing with the Function Theory on Symplectic Manifolds which is ...

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