# Quantizing a hamiltonian

1. Jan 8, 2009

### the1ceman

1. The problem statement, all variables and given/known data
The Lagrangian of a non-relativistic particle propagating on a unit circle is
$$L=\frac{1}{2}\dot{\phi}^{2}$$

where ϕ is an angle 0 ≤ ϕ < 2π.
(i) Give the Hamiltonian of the theory, and the Poisson brackets of the ca-
nonical variables. Quantize the theory by promoting the Poisson brackets into
commutators, and write the angular momentum operator, L, which is the con-
jugate (momentum) variable of ϕ, in the position representation. Note that in
the position representation
$$\hat{\phi}|\phi\rangle=\phi|\phi\rangle\;,\;\langle\phi'|\phi\rangle=\delta(\phi'-\phi)$$
2. Relevant equations

3. The attempt
i am stuck on the part where i have to write down L, how do i find its form in the $\phi$ representation? Please help
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Jan 9, 2009

### the1ceman

Anyone?

3. Jan 11, 2009