Rotations and Angular Momentum
نویسنده
چکیده
2 Exercise. Prove it. ◦ In fact, as sophisticates can show you, on a complex space a linear operator which preserves all norms will also preserve all inner products. But the key thing is to remember is that a unitary operator preserves all inner products (and therefore all norms). • This is analogous to the fact that translations and rotations in ordinary three-dimensional space preserve distances between points, and also angles between intersecting lines. • Note that while a simple application of the unitary U is needed to transform a ket, the corresponding transformation on an operator, A = UAU , (3)
منابع مشابه
Rotation and Angular Momentum
There is a fundamental connection between the angular momentum (a mechanical quantity) and the rotation operators (which is a symmetry group). The angular momentum is the generator of rotations. Because of this connection, many properties of angular momentum operators can be derived from the properties of the rotation group. We will derive these properties (mainly the commutation relations) sta...
متن کاملCoupling of Ket Spaces and the Addition of Angular Momenta
In these notes we discuss the coupling of ket spaces of systems to form the ket spaces of larger systems. The constituent systems may be independent of each other in some physical circumstances or in some models, while in other circumstances or more realistic models they may become coupled. We will see many examples of this process in this course, including the coupling of spin and orbital degr...
متن کاملNotes 17 : Coupling of Angular Momenta 3
In these notes we discuss the coupling or addition of angular momenta, a problem that arises whenever different subsets of the degrees of freedom of a system, each possessing its own angular momentum operator, are combined to form the whole system. In such cases, the whole system also possesses an angular momentum operator. Alternatively, we may say that there is an action of the rotations on e...
متن کاملAmplification of Angular Rotations Using Weak Measurements
We present a weak measurement protocol that permits a sensitive estimation of angular rotations based on the concept of weak-value amplification. The shift in the state of a pointer, in both angular position and the conjugate orbital angular momentum bases, is used to estimate angular rotations. This is done by an amplification of both the real and imaginary parts of the weak-value of a polariz...
متن کاملOn angular momentum operator in quantum field theory
Relations between two definitions of (total) angular momentum operator, as a generator of rotations and in the Lagrangian formalism, are explored in quantum field theory. Generally, these definitions result in different angular momentum operators, which are suitable for different purposes in the theory. From the spin and orbital angular momentum operators (in the Lagrangian formalism) are extra...
متن کاملOrbital Angular Momentum and Spherical Harmonics
In Notes 13, we worked out the general theory of the representations of the angular momentum operators J and the corresponding rotation operators. That theory did not make any assumptions about the kind of system upon which the angular momentum and rotation operators act, for example, whether it is a spin system or has spatial degrees of freedom, whether single particle or multiparticle, whethe...
متن کامل