Rotation operator
Encyclopedia
This article concerns the rotation
operator
, as it appears in quantum mechanics
.
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 
according to Quantum Mechanics
).
Translation of the particle at position x to position x+a:
Because a translation of 0 does not change the position of the particle, we have (with 1 meaning the identity operator
, which does nothing):
Taylor
development gives:
with
From that follows:
This is a differential equation
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
for the system is conserved.
. This is the same in quantum mechanics
considering
and 
as operators. Classically, an infinitesimal rotation 
of the vector r=(x,y,z) about the z-axis to r'=(x',y',z) leaving z unchanged can be expressed by the following infinitesimal translations (using Taylor approximation
):
From that follows for states:
And consequently:
Using
from above with 
and Taylor development we get:
with lz = x py - y px the z-component of the angular momentum according to the classical cross product
.
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 z axis, 
implies 
. This result means that angular momentum is conserved.
For the spin angular momentum about the y-axis we just replace
with 
and we get the spin
rotation operator
.
. From linear algebra
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 y-axis by an angle 
, it becomes the state 
, a result that can be generalized to arbitrary axes. It is important, for instance, in Sakurai%27s Bell inequality.
Rotation
A rotation is a circular movement of an object around a center of rotation. A three-dimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...
operator
Operator (physics)
In physics, an operator is a function acting on the space of physical states. As a resultof its application on a physical state, another physical state is obtained, very often along withsome extra relevant information....
, as it appears in quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
.
The translation operator
The rotationRotation
A rotation is a circular movement of an object around a center of rotation. A three-dimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...
operator
Operator (physics)
In physics, an operator is a function acting on the space of physical states. As a resultof its application on a physical state, another physical state is obtained, very often along withsome extra relevant information....
, with the first argument indicating the rotation axisCartesian coordinate system
A Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length...
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 according to Quantum MechanicsQuantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
).
Translation of the particle at position x to position x+a:
Because a translation of 0 does not change the position of the particle, we have (with 1 meaning the identity operator
Identity function
In mathematics, an identity function, also called identity map or identity transformation, is a function that always returns the same value that was used as its argument...
, which does nothing):
Taylor
Taylor series
In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....
development gives:
with
From that follows:
This is a differential equation
Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
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 momentumMomentum
In classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...
for the system is conserved.
In relation to the orbital angular momentum
Classically we have for the angular momentumAngular momentum
In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...
. This is the same in quantum mechanicsQuantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
considering
and as operators. Classically, an infinitesimal rotation of the vector r=(x,y,z) about the z-axis to r'=(x',y',z) leaving z unchanged can be expressed by the following infinitesimal translations (using Taylor approximationTaylor series
In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....
):
From that follows for states:
And consequently:
Using
from above with and Taylor development we get:with lz = x py - y px the z-component of the angular momentum according to the classical cross product
Cross product
In mathematics, the cross product, vector product, or Gibbs vector product is a binary operation on two vectors in three-dimensional space. It results in a vector which is perpendicular to both of the vectors being multiplied and normal to the plane containing them...
.
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 z axis, implies . This result means that angular momentum is conserved.For the spin angular momentum about the y-axis we just replace
with and we get the spinSpin (physics)
In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...
rotation operator
.Effect upon the spin operator and upon states
Operators can be represented by matricesMatrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
. From linear algebra
Linear algebra
Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
one knows that a certain matrix
can be represented in another basisBasis (linear algebra)
In linear algebra, a basis is a set of linearly independent vectors that, in a linear combination, can represent every vector in a given vector space or free module, or, more simply put, which define a "coordinate system"...
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 y-axis by an angle , it becomes the state , a result that can be generalized to arbitrary axes. It is important, for instance, in Sakurai%27s Bell inequality.