This article concerns the rotation operator, as it appears in quantum mechanics.
, we postulate a quantum mechanical rotation operator
that is the rule that assigns to each vector in the space
We will show that, in terms of the generators of rotation,
indicating the rotation axis and the second
for infinitesimal rotations as explained below.
This is why, it is first shown how the translation operator is acting on a particle at position x (the particle is then in the state
Translation of the particle at position
Because a translation of 0 does not change the position of the particle, we have (with 1 meaning the identity operator, which does nothing):
Additionally, suppose a Hamiltonian
Because the translation operator can be written in terms of
This result means that linear momentum for the system is conserved.
Classically we have for the angular momentum
Classically, an infinitesimal rotation
unchanged can be expressed by the following infinitesimal translations (using Taylor approximation):
-component of the angular momentum according to the classical cross product.
, we construct the following differential equation using the condition
Similar to the translation operator, if we are given a Hamiltonian
This result means that angular momentum is conserved.
For the spin angular momentum about for example the
is the Pauli Y matrix) and we get the spin rotation operator
Operators can be represented by matrices.
From linear algebra one knows that a certain matrix
can be represented in another basis through the transformation
is the basis transformation matrix.
are the z-axis in one basis respectively another, they are perpendicular to the y-axis with a certain angle
in the first basis can then be transformed into the spin operator
From standard quantum mechanics we have the known results
are the top spins in their corresponding bases.
, a result that can be generalized to arbitrary axes.