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
, we postulate a quantum mechanical rotation operator
which rotates quantum mechanical states.
In terms of the generators of rotation,
where
is rotation axis,
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
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 , with the first argument
indicating the rotation
axis and the second
the rotation angle, can operate through the
translation operator 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 ...
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
to position
:
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):
Taylor development gives:
with
From that follows:
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
Additionally, suppose a
Hamiltonian is independent of the
position. Because the translation operator can be written in terms of
, and
, we know that
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 ...
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
and
as operators. Classically, an infinitesimal rotation
of the vector
about the
-axis to
leaving
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 ...
):
From that follows for states:
And consequently:
Using
from above with
and Taylor expansion we get:
with
the
-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
, we construct the following differential equation using the condition
:
Similar to the translation operator, if we are given a Hamiltonian
which rotationally symmetric about the
-axis,
implies
. This result means that angular momentum is conserved.
For the spin angular momentum about for example the
-axis we just replace
with
(where
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
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
can be represented in another
basis through the transformation
where
is the basis transformation matrix. If the vectors
respectively
are the z-axis in one basis respectively another, they are perpendicular to the y-axis with a certain angle
between them. The spin operator
in the first basis can then be transformed into the spin operator
of the other basis through the following transformation:
From standard quantum mechanics we have the known results
and
where
and
are the top spins in their corresponding bases. So we have:
Comparison with
yields
.
This means that if the state
is rotated about the
-axis by an angle
, it becomes the state
, 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