Translation operator on a sphere

[kashokjayaram]

I'm considering a system where an electron is subjected to magnetic field which is produced by dirac monopole. Here I'm interested in looking for a translation operator. Now how can I get a translation operator in presence of field and in absence of field.?? I need both the operators. Can anybody help me..? Thank you...

 

[tom.stoer]

The translation operator is

$$\hat{T}(a) = e^{-ia\hat{p}}$$

In position-space this is

$$\hat{p} = -i\nabla$$

Are you thinking about using the covariant derivative instead?

$$\nabla \to \nabla - ieA$$

 

[kashokjayaram]

is this applicable to sphere also..?? Why both are same...? Since this operator is used for translation in plane, how come it is same in sphere?

 

[tom.stoer]

You haven't said anything about a particle on a sphere. Is your particle subject to a constraint r²=const.? Or equivalently, are you using spherical coordinates with r being a constant?

Then the flat-space translation operator is no longer valid. On the sphere only operators respecting the symmetry of the sphere (and the constraint) are allowed. A simple approach is to use spherical coordinates, to fix r=const. and to drop all ∂_r terms. I am not absolutely sure whether this approach is valid for all operators. Perhaps an approach like constraint quantization is required.

But in principle it's obvious that instead of momentum p as a generator of translation you have to consider angular momentum L as a generator of rotation.

 

[DrDu]

You haven't said anything about a particle on a sphere.

Have a look at the title.

The problem is that with a constant field, the hamiltonian is not directly translation invariant. But the translated hamiltonian can be brought back to its original form applying an additional gauge transformation. Effectively, the translation operator becomes the covariant derivative. Physically, this leads to the appearance of Landau levels.
On a sphere, the translation operators are the same as that for a flat surface tangential to the point you want to rotate. So it should be possible to use a covariant version of the angular momentum operators.

 

[tom.stoer]

Ah, I see, it's in the title of the thread ;-)

What you have to do is to use

$$\hat{D}(n,\phi) = e^{-i\,n^a\,\hat{L}^a\,\phi}$$

which generates rotations with angle phi and axis n; n is the unit vector in direction n.

Note that formally L is an angular momentum operator, but not necessarily an orbital angular momentum operator. That means it satisfies the algebra

$$[\hat{L}^a,\hat{L}^b] = i\epsilon^{abc}\,\hat{L}^c$$

but it need not be represented using r, theta and phi. Of course it has the usual eigenstates $|lm\rangle$.

This allows you treat the particle on a sphere as a rigid rotor.

But I am not sure whether this helps for your problem with a external field.