HOME

TheInfoList



OR:

This article concerns the
rotation Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
operator, as it appears in
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
.


Quantum mechanical rotations

With every physical rotation R, we postulate a quantum mechanical rotation operator D(R) which rotates quantum mechanical states. , \alpha \rangle_R = D(R) , \alpha \rangle In terms of the generators of rotation, D (\mathbf,\phi) = \exp \left( -i \phi \frac \right), where \mathbf is rotation axis, \mathbf is
angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
, and \hbar is the
reduced Planck constant The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivalen ...
.


The translation operator

The
rotation Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
operator \operatorname(z, \theta), with the first argument z indicating the rotation axis and the second \theta the rotation angle, can operate through the translation operator \operatorname(a) 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 State may refer to: Arts, entertainment, and media Literature * ''State Magazine'', a monthly magazine published by the U.S. Department of State * ''The State'' (newspaper), a daily newspaper in Columbia, South Carolina, United States * ''Our S ...
, x\rangle according to
Quantum Mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
). Translation of the particle at position x to position x + a: \operatorname(a), x\rangle = , x + a\rangle Because a translation of 0 does not change the position of the particle, we have (with 1 meaning the
identity operator Identity may refer to: * Identity document * Identity (philosophy) * Identity (social science) * Identity (mathematics) Arts and entertainment Film and television * ''Identity'' (1987 film), an Iranian film * ''Identity'' (2003 film) ...
, which does nothing): \operatorname(0) = 1 \operatorname(a) \operatorname(da), x\rangle = \operatorname(a), x + da\rangle = , x + a + da\rangle = \operatorname(a + da), x\rangle \Rightarrow \operatorname(a) \operatorname(da) = \operatorname(a + da) Taylor development gives: \operatorname(da) = \operatorname(0) + \frac da + \cdots = 1 - \frac p_x da with p_x = i \hbar \frac From that follows: \operatorname(a + da) = \operatorname(a) \operatorname(da) = \operatorname(a)\left(1 - \frac p_x da\right) \Rightarrow \frac = \frac = - \frac p_x \operatorname(a) This is a
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, ...
with the solution \operatorname(a) = \exp\left(- \frac p_x a\right). Additionally, suppose a Hamiltonian H is independent of the x position. Because the translation operator can be written in terms of p_x, and _x,H= 0, we know that , \operatorname(a)0. This result means that linear
momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass ...
for the system is conserved.


In relation to the orbital angular momentum

Classically we have for the
angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
\mathbf L = \mathbf r \times \mathbf p. This is the same in
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
considering \mathbf r and \mathbf p as operators. Classically, an infinitesimal rotation dt of the vector \mathbf r = (x,y,z) about the z-axis to \mathbf r' = (x',y',z) leaving z unchanged can be expressed by the following infinitesimal translations (using
Taylor approximation In calculus, Taylor's theorem gives an approximation of a ''k''-times differentiable function around a given point by a polynomial of degree ''k'', called the ''k''th-order Taylor polynomial. For a smooth function, the Taylor polynomial is the t ...
): \begin x' &= r \cos(t + dt) = x - y \, dt + \cdots \\ y' &= r \sin(t + dt) = y + x \, dt + \cdots \end From that follows for states: \operatorname(z, dt), r\rangle = \operatorname(z, dt), x, y, z\rangle = , x - y \, dt, y + x \, dt, z\rangle = \operatorname_x(-y \, dt) \operatorname_y(x \, dt), x, y, z\rangle = \operatorname_x(-y \, dt) \operatorname_y(x \, dt) , r\rangle And consequently: \operatorname(z, dt) = \operatorname_x (-y \, dt) \operatorname_y(x \, dt) Using T_k(a) = \exp\left(- \frac p_k a\right) from above with k = x,y and Taylor expansion we get: \operatorname(z,dt)=\exp\left \frac \left(x p_y - y p_x\right) dt\right= \exp\left(-\frac L_z dt\right) = 1-\fracL_z dt + \cdots with L_z = x p_y - y p_x the z-component of the angular momentum according to the classical
cross product In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and ...
. To get a rotation for the angle t, we construct the following differential equation using the condition \operatorname(z, 0) = 1 : \begin &\operatorname(z, t + dt) = \operatorname(z, t) \operatorname(z, dt) \\ .1ex\Rightarrow & \frac = \frac = \operatorname(z, t) \frac = - \frac L_z \operatorname(z, t) \\ .1ex\Rightarrow & \operatorname(z, t) = \exp\left(- \frac\, t \, L_z\right) \end Similar to the translation operator, if we are given a Hamiltonian H which rotationally symmetric about the z-axis, _z,H0 implies operatorname(z,t),H0. This result means that angular momentum is conserved. For the spin angular momentum about for example the y-axis we just replace L_z with S_y = \frac \sigma_y (where \sigma_y is the Pauli Y matrix) and we get the
spin Spin or spinning most often refers to: * Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning * Spin, the rotation of an object around a central axis * Spin (propaganda), an intentionally ...
rotation operator \operatorname(y, t) = \exp\left(- i \frac \sigma_y\right).


Effect on the spin operator and quantum states

Operators can be represented by
matrices Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
. From
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrice ...
one knows that a certain matrix A can be represented in another basis through the transformation A' = P A P^ where P is the basis transformation matrix. If the vectors b respectively c are the z-axis in one basis respectively another, they are perpendicular to the y-axis with a certain angle t between them. The spin operator S_b in the first basis can then be transformed into the spin operator S_c of the other basis through the following transformation: S_c = \operatorname(y, t) S_b \operatorname^(y, t) From standard quantum mechanics we have the known results S_b , b+\rangle = \frac , b+\rangle and S_c , c+\rangle = \frac , c+\rangle where , b+\rangle and , c+\rangle are the top spins in their corresponding bases. So we have: \frac , c+\rangle = S_c , c+\rangle = \operatorname(y, t) S_b \operatorname^(y, t) , c+\rangle \Rightarrow S_b \operatorname^(y, t) , c+\rangle = \frac \operatorname^(y, t) , c+\rangle Comparison with S_b , b+\rangle = \frac , b+\rangle yields , b+\rangle = D^(y, t) , c+\rangle. This means that if the state , c+\rangle is rotated about the y-axis by an angle t, it becomes the state , b+\rangle, a result that can be generalized to arbitrary axes.


See also

* Symmetry in quantum mechanics *
Spherical basis In pure and applied mathematics, particularly quantum mechanics and computer graphics and their applications, a spherical basis is the basis used to express spherical tensors. The spherical basis closely relates to the description of angular m ...
*
Optical phase space In quantum optics, an optical phase space is a phase space in which all quantum states of an optical system are described. Each point in the optical phase space corresponds to a unique state of an ''optical system''. For any such system, a plot o ...


References

*L.D. Landau and E.M. Lifshitz: ''Quantum Mechanics: Non-Relativistic Theory'', Pergamon Press, 1985 *P.A.M. Dirac: ''The Principles of Quantum Mechanics'', Oxford University Press, 1958 *R.P. Feynman, R.B. Leighton and M. Sands: ''The Feynman Lectures on Physics'', Addison-Wesley, 1965 {{DEFAULTSORT:Rotation Operator (Quantum Mechanics) Rotational symmetry Quantum mechanics