Rotation Operator

4 Key excerpts on "Rotation Operator"

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  eBook - PDF

  Advanced Modern Physics

  Theoretical Foundations

  • John Dirk Walecka(Author)
  • 2010(Publication Date)
  • WSPC

    (Publisher)

  Chapter 3 Angular Momentum The theory of angular momentum in quantum mechanics is of great utility, as it provides the basis for the calculation of matrix elements in any finite system. 1 Furthermore, it provides an example of the role of a continuous symmetry in quantum mechanics, in this case the very deep symmetry of the isotropy of space and invariance under rotations. 3.1 Translations The translation operator in quantum mechanics is analyzed in Probs. 2.6-2.7. Let us arrive at this operator in a more systematic fashion. Consider a classical coordinate translation to a new primed coordinate system trans-lated by a distance a along the x -axis from the unprimed one, so that x prime = x -a ; translation p prime = p (3.1) The second relation follows since dx prime /dt = dx/dt . In quantum mechan-ics (ˆ p, ˆ x ) are operators . What do we mean by a translation in quantum mechanics? Let us ask the following question: Is there an operator which induces a transformation on our opera-tors and state vectors such that the new quantities can be put into a one-to-one correspondence with the translated coordinate system? We start with an infinitesimal transformation where a ≡ ε → 0; the finite transformation will then be built up by repeated application of in-1 See [Edmonds (1974)]; see also appendix B in [Fetter and Walecka (2003a)]. 35 36 Advanced Modern Physics finitesimals. We look for an operator of the form ˆ U = 1 -i planckover2pi1 ε ˆ K ; infinitesimal a ≡ ε → 0 ˆ U -1 = 1 + i planckover2pi1 ε ˆ K ˆ U ˆ U -1 = 1 + O ( ε 2 ) (3.2) ˆ K is referred to as the generator of the transformation.

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  eBook - ePub

  A Textbook on Modern Quantum Mechanics

  • A C Sharma(Author)
  • 2021(Publication Date)
  • CRC Press

    (Publisher)

  point x + (δ θ z) y, y − (δ θ z) x. 4.4 Quantum Generalization of the Rotation Operator It is well known that in quantum mechanics orbital angular momentum is not the total angular momentum. It is found experimentally that particles like the electron exhibit an internal angular momentum, called spin angular momentum. The total angular momentum, called J, is the sum of the orbital angular momentum L and spin angular momentum S J = L + S The operator J is defined as the generator of rotations on any wave function, including possible spin components. Therefore, Eqn. (4.16b) is generalized to: R (δ θ) ψ (r) = e − i δ θ. J / ℏ ψ (r) (4.20) which is similar to the equation for L, whose components are written as differentials. Up to this point, we considered ψ (r) a complex valued function of position. But the wave function at a point can have several components in some vector space. The Rotation Operator will operate in that space, and it is a differential operator too with respect to position. Then, the state vector ψ is a vector at each point, and the rotation of the system rotates this vector as well as moving it to a different value of r. The ψ in general, has n -components at each point in space; R (δ θ) is then a n × n matrix in the component space, and Eqn. (4.20) is the definition of J in that component space. This definition is used to study the properties of J. For an infinitesimal angle ε, we can write: R x (ε) ≃ 1 − i ℏ ε J x ; R y (ε) ≃ 1 − i ℏ ε J y and R z (ε 2) ≃ 1 − i ℏ ε 2 J x, which on substituting into Eqn. (4.15b) yield: 1 − i ℏ ε J x 1 − i ℏ ε J y − 1 − i ℏ ε 2 J z 1 − i ℏ ε J y 1 − i ℏ ε J x ψ = 0 (4.21) All the zeroth and first-order terms in ε cancel, and the second-order term gives: J x, J y = i ℏ J z (4.22a) which is generalized to: J i, J j = i ℏ ε i j k J k (4.22b) where the. symbol ε i j k is equal to +1, if i j k take values in cyclic order (123, 231, 312) and it is equal to −1 when i j k take values that are not in cyclic order

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  Unitary Symmetry And Combinatorics

  • James D Louck(Author)
  • 2008(Publication Date)
  • World Scientific

    (Publisher)

  1.1. BACKGROUND AND VIEWPOINT 9 at position x with linear momentum p relative to the origin of a coor-dinate frame ( e 1 , e 2 , e 3 ) , gives L / = − i R = − i x × ∇ . Classical linear momentum and angular momentum of a point particle are identified in their quantum mechanical interpretation as the generators of translations and rotations, as described above. The founders of quantum mechanics rediscovered relations (1.15) and (1.16) in the guise of the unitary oper-ators exp ( i a · p / ) and exp ( − iφ n · L ) / acting in the Hilbert space of states of a physical system. From the definition of the quantum angu-lar momentum, L / = − i x × ∇ , the operations of translation, inversion, and rotation effect the following transformations of the quantal angular momentum: translation : L → L − i a × ∇ , inversion : L → L , (1.19) rotation : L → L . 1.1.2 Newtonian physics A physical system is said to be isolated if it has no influence on its surroundings, and conversely. Such ideal systems do not exist in nature, but much of the progress in physics can be attributed to the approximate validity of the concept of an isolated physical system. In Newtonian physics, the Euclidean 3-space E 3 is taken as the back-ground against which the changes in an isolated physical system take place: the space is considered to be void of (isolated from) all other physical objects, homogeneous (sameness at every point), and isotropic (sameness in every direction). The physical system occupies a collection (subset) of points in E 3 , which may be a single point, but this collection of points can change relative to the fixed Euclidean background and the fixed reference frames used to assign coordinates to the points of E 3 . The measure of this change requires a new concept, that of time, which itself is assumed to have any value on the real-line (ignoring units).

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  Lectures On Quantum Mechanics

  • Gordon Baym(Author)
  • 2018(Publication Date)
  • CRC Press

    (Publisher)

  (17-4) . We can similarly study the transformation properties of operators under rotations. By the transformation of an operator A under a rotation ω we mean the unitary transformation:

  A   →   R ω AR ω – 1

  (17-48)

  (Cf. Eq. (6-15) , for a similar transformation by the inverse rotation.) The expectation value of an operator A in a state |Φ〉 is the same as the expectation value of the “rotated” operator

  R ω

  AR ω

  – 1

  in the rotated state Rω |Φ〉.

  Certain operators are scalars under rotation; this means that they commute with the total angular momentum J and are unchanged by the transformation (17-48) . Then there are operators, such as the three components of the position operator x, y, z, that transform among themselves under rotation like the components of a vector. Other operators, such as px 5 z + y, for example, have no nice transformation properties under rotation. The types of operators having simple transformation properties under rotation are known generally as tensor operators. By an irreducible tensor operator T(k) of order k we shall mean a set of 2k + 1 operators

  T q

  ( k )

  , q = –k, –k + 1,…, k – 1, k, that transform among themselves under rotation according to the transformation law

  R ω T q ( k )   R ω – 1   =   ∑ q ′   =   -k k T q ′ ( k ) d q ′ q ( k ) ( ω ) ;

  (17-49)

  This transformation law is the equivalent for operators of Eq. (17-4) . An irreducible tensor T(0) is clearly a scalar, while one of older 1 is a vector; the components Tq (1) are linear combinations (which we shall construct shortly) of the x, y, and z components of the vector. The

  d

  q ′

  q

  ( k )

  ( ω )

  coefficients are the matrix elements of the irreducible representation of the rotation group of dimension 2k + 1. It follows from the irreducibility of the d(k) matrices that there is no subset of components,

  T q

  ( k )

  , or linear combination of components, that transforms among itself privately under rotations; this is why a tensor operator transforming according to (17-49) is called irreducible. [Later we shall meet reducible tensor operators.]

  If we consider an infinitesimal rotation ε, then

  R ε   =   e – i J   ⋅   ε   ≈   1   –   i J   ⋅   ε ,

  (17-50)

  and the transformation law (17-49) becomes, to first order in ε

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