How to quantize a particle confined to the surface of a sphere?

[wdlang]

how to go step by step from the classical lagrangian to the Schrödinger equation?

i would like to work with the two angles.

whether the quantization is right or not is a matter of experiment, is not it? I mean, you might have many schemes of quantization, but which one is the right one is up to experiment.

 

[DrDu]

Do you know the Podolsky trick?

B. Podolsky, Phys. Rev., 32, 812 (1928).

 

[wdlang]

Do you know the Podolsky trick?

B. Podolsky, Phys. Rev., 32, 812 (1928).

Thanks a lot. It looks interesting.

 

[WannabeNewton]

Is there a specific reason you want to start from the Lagrangian (there is no need for the qualifier "classical" by the way)? The quantization for the system in the OP is much easier than that. All you have to do is take the total energy of a free particle confined to a sphere, which will involve the total angular momentum and the moment of inertia, and then promote the angular momentum to an operator. From there you can easily write down the Schrödinger equation and immediately get the general solution (which will be in terms of the spherical harmonics-the eigenstates of the angular momentum).

 

[wdlang]

Is there a specific reason you want to start from the Lagrangian (there is no need for the qualifier "classical" by the way)? The quantization for the system in the OP is much easier than that. All you have to do is take the total energy of a free particle confined to a sphere, which will involve the total angular momentum and the moment of inertia, and then promote the angular momentum to an operator. From there you can easily write down the Schrödinger equation and immediately get the general solution (which will be in terms of the spherical harmonics-the eigenstates of the angular momentum).

I would feel safe if i start from the classical lagrangian or the classical hamiltonian.

It is at first a psychological issue.

I do not understand why you just mention energy, not the hamiltonian. Of course, the hamiltonian is time independent and energy is conserved.

Another fundamental issue is, what justifies a quantum hamiltonian? Only experiment can do it or we have some theoretical criterion?

 

[WannabeNewton]

I do not understand why you just mention energy, not the hamiltonian. Of course, the hamiltonian is time independent and energy is conserved.

There are no external influences so the total energy is the Hamiltonian.

Another fundamental issue is, what justifies a quantum hamiltonian? Only experiment can do it or we have some theoretical criterion?

We promote the classical Hamiltonian to an operator because state vectors are mapped into other state vectors by operators acting on state space and the Hamiltonian generates time-translations which is what we need in order to propagate the state vector in state space.