Linked Questions

14
votes
3answers
2k views

Are the path integral formalism and the operator formalism inequivalent?

Abstract The definition of the propagator $\Delta(x)$ in the path integral formalism (PI) is different from the definition in the operator formalism (OF). In general the definitions agree, but it is ...
6
votes
4answers
4k views

Equations of motion for a free particle on a sphere

I derived the equations of motion for a particle constrained on the surface of a sphere Parametrizing the trajectory as a function of time through the usual $\theta$ and $\phi$ angles, these equations ...
10
votes
2answers
2k views

Derivative interaction: $\mathcal{H}_\mathrm{int}\neq - \mathcal{L}_\mathrm{int}$. Question about Feynman Rules

As we known, if there is time derivative interaction in $\mathcal L_\mathrm{int}$, then $\mathcal{H}_\mathrm{int}\neq -\mathcal{L}_\mathrm{int}$. For example, Scalar QED, $$ \begin{aligned} \mathcal{...
3
votes
1answer
710 views

Can I Weyl-order the following Hamiltonian?

I am trying to perform a path integral but I am having trouble with the Weyl ordering of my Hamiltonian. The Lagrangian of the system in question is $$L~=~\frac{1}{2}f(q)\dot{q}^2,$$ where $f(q)$ ...
4
votes
1answer
720 views

Quantum mechanics with non-cartesian coordinates

Let say we have the classical hamiltonian of a harmonic oscillator: $$H=\frac{p_x^2+p_y^2+p_z^2}{2m}+\frac{k_1x^2+k_2y^2+k_3z^2}{2}$$ and we want to find the hamiltonian operator in quantum mechanics, ...
1
vote
3answers
404 views

Classical Hamiltonian involving product of factors whose quantum analogues don't commute

Dirac remarked in his quantum mechanics book: One can usually assume that the Hamiltonian is the same function of the canonical coordinates and momenta in the quantum theory as in the ...
1
vote
0answers
729 views

Hamiltonian for particle moving in a sphere

Context: I know that if we have a particle (say, with unit mass) moving in the plane $\Bbb R^2$ subject to a spherically symmetric potential $V\colon \Bbb R^2 \to \Bbb R$, it will move along the ...
0
votes
0answers
174 views

General geometric interpretation for the Hamiltonian and of the cases when it is not the total energy of the system

What is the geometric interpretation for the Hamiltonian? Also, is there geometric interpretation of when and why it is not equal to the total energy of the system? Lastly, what is the most general ...
2
votes
1answer
116 views

Rewriting the Laplacian on a curved manifold

I guess there is a sense in which the following is true: "The Laplacian written on a Riemannian manifold $(M,g)$ can be seen as adding a coordinate dependent mass field to the Laplacian on ...
2
votes
1answer
89 views

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 ...
1
vote
1answer
65 views

Classic analog of quantum mechanics when dealing with Hamiltonian operator

I am reading The Principles of Quantum Mechanics by Dirac, in chapter 28 Heisenberg's form for the equations of motion, there is ...
2
votes
1answer
45 views

What is the general prescription for constructing the quantum mechanical momentum operator conjugate to a given coordinate?

Obviously, if $x$ is a Cartesian coordinate, then the corresponding momentum operator is $-i \hbar \partial_x$. But what if $x$ is something more complicated, like some sort of curvilinear coordinate ...
0
votes
1answer
52 views

Inverse of canonical quantization: Classical kinetic energy corresponding to a Laplacian on a sphere

In canonical quantization, we replace the canonical conjugate of position($x$), $p$ by $-i \hbar\frac{d}{dx}$. In general we replace $\mathbf{p}$ by $-i\hbar \nabla$. My question is, what if I wanted ...
0
votes
0answers
63 views

Momentum Operator on a Riemannian Manifold

Consider a non-linear sigma model on a Riemannian manifold with metric $g_{ij}$ with the action $$S= \frac{1}{2} \int dt g_{ij}(X) \frac{dX^i}{dt} \frac{dX^j}{dt}.$$ The momentum operator is $$P_i= \...
0
votes
0answers
63 views

Generalized momenta and quantum mechanics

In my introductory quantum mechanics book it was stated that the operator $-i\hbar\vec{\nabla}$ represents the momentum $\vec{p}=m\vec{v}$ of a particle. In the book "Physics of atoms and molecules" ...

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