Angular momentum operator

Angular momentum operator is a concept in quantum mechanics. It is a Hermitian vector operator whose components satisfy the following Kommutatorrelation ( the epsilon tensor ):

The angular momentum operator plays a central role in atoms and other quantum mechanical problems with rotational symmetry. The orbital angular momentum operator is the quantum mechanical analogue of the classical angular momentum. Also, there are the spin operator, which is also an angular momentum operator, but does not have a classical analog.

• 2.1 Local representation of the orbital angular momentum in Cartesian coordinates
• 2.2 Local representation of the orbital angular momentum in spherical coordinates
• 2.3 Generating a rotation

• 4.1 Orientation and direction quantization
• 4.2 A concrete behavior under rotations and mirroring
• 4.3 states, in contrast to intuition

• 5.1 Spin -orbit coupling
• 5.2 Spin -spin coupling

Properties

As a result, the Einstein summation convention is used, that is, over repeated indices are summed.

From Kommutatorrelation automatically follows:

Since the components of the angular momentum operator do not exchange, by definition, it often goes over to the common eigenvectors of and an arbitrary angular momentum component (usually ). denote the eigenvectors of the common base of and and with the following eigenvalue equations:

The quantum number, the quantum number and the values ​​have the values ​​. Thus, equal - fold degenerate.

The indices and correspond to the orbital angular momentum of the quantum number (integer ) or the magnetic quantum number of the orbital angular momentum and the spin analog to the two spin quantum numbers ( half-integer ) and.

We define ladder operators with which the spectrum can be traversed at given:

• Upgrade operator:

• Lowering operator:

Details on the quantum numbers j and m

Are the eigenvalues ​​and determined from means and ladder operators. First, various commutators are determined with ladder operators, which can be attributed to:

Now, the operation of the operators head is to be examined on the state:

When using a ladder operator does not change, but is increased or decreased by 1 ( therefore the terms Open and lowering operator justified ). In the next step, the constant is determined. These initially be extended to products of ladder operators and returned:

The commutator leads to as well. Inserting provides:

Since all eigenstates are to be normalized, the length of the vector and can be determined with the standard square; it is exploited that are up and lowering operator adjoint to each other:

Since the norm of a vector is non-negative, must be true. It follows:

Applying the upgrade operator on the highest state at will; applies to the lowering operator to be at. In either case, abort the ladder and one obtains the zero vector:

By -fold Apply () of the operator to upgrade the state gets you to:

Therefore, the possible quantum numbers must be non-negative integer or half-integer.

Orbital angular momentum operator

A specific implementation of an angular momentum, the angular momentum operator dar. This is defined as follows:

It is the position operator, the momentum operator and the - th unit vector. The quantum number is usually not with, but with designated. It applies ( " the orbital angular momentum is perpendicular to the position vector and the momentum vector "). Consequently, the quantum numbers are integers:

The eigenvectors can be identified in real space with the spherical harmonics ( see below).

Coordinate representation of the orbital angular momentum in Cartesian coordinates

For the momentum operator and position operator apply in coordinate representation, or. This used to, results with:

Coordinate representation of the orbital angular momentum in spherical coordinates

With the nabla operator and the gradient in spherical coordinates is obtained after performing the cross products firstly

The Cartesian components now can be connected to the Cartesian components of the unit vectors and read:

Of the last line can be seen that the component of the angular momentum of the generator rotation ( with angles) to the axis.

The operator corresponds in position representation just the angular part of the Laplace operator (except the constant). The eigenfunctions of the angular part and thus by and are the spherical harmonics:

The quantum numbers and are restricted to integer values:

The spherical harmonics form a complete orthonormal system of square integrable functions normals on the unit sphere:

Generating a rotation

The operator turn the location coordinates by the angle around the z -axis:

For infinitesimally small rotation angle of the trigonometric functions can be developed up to first order in to (see also: Infinitesimal rotations ) and also the wave function:

In the last step, the definition of the z- component of the angular momentum operator was used. Since Hermitian is the infinitesimal rotation operator is unitary.

In spherical coordinates, an infinitesimal rotation about the Z axis is analogous to the above:

To produce a finite rotation of such infinitesimal rotation, consider the following border crossing:

Since the rotation operator is composed of unitary operators, it is itself unitary.

Rotation about an arbitrary axis ( with ) by the angle can be generally written as:

Spin operator

(see also spin operator for the base states and spin 1/2 in the article spin)

The spin is another degree of freedom of a quantum particle and describes the angular momentum in its rest frame ( intrinsic angular momentum ). For point-like particles is there not classical analogue ( hence no local representation). The spin operator commutes with all other degrees of freedom of the particle, eg, the momentum operator and the orbital angular momentum operator. Unlike the orbital angular momentum it must not be perpendicular to the momentum operator. The quantum numbers and can be full-or half-integer. In the most common case, they have the values ​​:

All quarks and leptons are spin 1/2-Teilchen, as many composite particles such as protons and neutrons. However, there are particles with different spin, such as the photon and other exchange bosons with spin 1, the baryonic Delta = 3/2, etc.

In the spin 1/2 is called the two eigenstates are often called " spin up" and " spin down".

These states satisfy the eigenvalue equations

The ladder operators on the eigenstates have the effect of:

The spin components can be expressed via the ladder operators:

Often the matrix representation of the operators is used, where the eigenstates following column vectors ( spinors ) are assigned:

Finally, on the relationship

The spin components associated with the Pauli matrices.

Angular momentum operator and angular momentum vector

Orientation and direction quantization

The eigenstates are called aligned to the z -axis. The vector of the three expected values ​​is here parallel to the z -axis. The magnitude of this vector and therefore does not reach at maximum orientation ( ) is the length of the angular momentum vector, which is given by. Accordingly applies to the squares of the operators for the x -and y- component, that the expected value can not be considered to be small. Therefore, the quantum mechanical angular momentum of an illustrative accessible vector differs in three-dimensional space: it can no axis are parallel in the sense that its component along this axis is the same size as its magnitude or length. Still in the maximum possible physical text alignment is often referred to as " parallel orientation " simplified.

For a vector in three-dimensional space, the angular distance resulting from the z-axis, wherein the length of the vector. This transferred to the quantum mechanical angular momentum, leading to

The discrete eigenvalues ​​of the z-component one can therefore illustrate such a way that the angular momentum vector in these conditions may take only certain angle to the z- axis. This is known as space quantization. The smallest possible angle is given by. For large values ​​of the angular momentum tends to zero. However, what is the vivid description contradicts a "parallel position" for the smallest ( non-zero ) quantum mechanical angular momentum. The perpendicular to the z -axis component of the angular momentum in any state having a well-defined self- value, as is apparent from. Only their direction in the xy plane is completely undetermined, since the expectation values ​​of both the x - component and y- component of angular momentum is are zero.

A concrete behavior under rotations and mirroring

The angular momentum operator is equivalent in some aspects of the ideological image of the classical angular momentum. In particular, it behaves upon rotation of the coordinate system just like any other vector, that is, its three components along the new coordinate axes are linear combinations of the three operators along the old axes according to the same formulas as (for example ) the classical angular momentum vector. This also applies to (in any state of the system under consideration ) for the three expected values ​​, which together form the vector of the expected value. Therefore, the length of the expected value of the angular momentum vector with rotations of the coordinate system ( or condition ) is the same.

In reflection of the coordinate system, the angular momentum operator and its expected value also behave the same way as the mechanical angular momentum vector. Stay in fact the same, as well as all other axial vectors (eg, angular velocity, magnetic field, magnetic dipole moment ), unlike polar vectors ( such as position vector, velocity vector, momentum vector ), the change mirroring its sign. Axial vectors are also called pseudo vectors.

States, in contrast to intuition

Although the amount of the expected value vector is the same for all rotations and reflections of the system, there are for quantum numbers states at the same quantum number in which the vector has a different length, and which can not be converted by rotation and mirroring one another accordingly. For example, in a state of the expected value and its magnitude. This results depending on the value of different values ​​, except in the cases and. For the length is zero. A length of zero is obtained for the expectation value of the angular momentum vector also in conditions such as long and will continue to apply to more than distinguish and thus for the expected values ​​. In such conditions, the system is a so-called "alignment ", in German "Orientation" (using the German word but is often generally used in the case that the system be viewed based on its own state relating to a previously selected z-axis should ).

In case ( see section " spin 1 /2 and three-dimensional vector " in the article spin) is that in every possible given state the expectation value of the angular momentum operator has the length and a direction in space can specify, according to the associate this state, the quantum number is.

Addition of angular momenta

It is of two angular momentum operators and from, each of which has the quantum numbers and or and. Each of these has its own angular momentum eigenspace spanned by the eigenvectors to or to. The angular momentum exchange with each other.

Now the individual angular momenta couple to a total angular momentum:

Thus automatically applies. The states of the whole system the product space ( tensorial product) of the states of the individual systems. In it the products of the basis states of the individual systems form a basis:

However, these are (mostly), so that it has no diagonal form no eigenvectors of the total angular momentum in this base. Therefore, one goes over the complete set of commuting operators with the eigenstates for the complete set of commuting operators with the eigenstates. In the new basis of the total angular momentum has again a simple diagonal form:

The quantum numbers for the total angular momentum and can have the following values ​​:

The transition from the product base in the eigenbasis is done through the following development ( exploiting the completeness of the product base):

Here are the Clebsch -Gordan coefficients.

Spin -orbit coupling

It is coupled to an orbital angular momentum of a 1/2-Spin.

The spin quantum numbers are limited and the orbital angular momentum quantum numbers and. Thus, the total angular momentum quantum number only take the following values:

• For:
• For:.

Each state of total angular momentum basis is composed of exactly two product basis states. Given to can be.

From the requirement of Ortho Normalized awareness of the states, the coefficients are defined:

As an example, the angular momentum is to be coupled with a spin. The following abbreviation and write for the product base.

For there is a quartet:

For there is a doublet:

Spin -spin coupling

In the following two 1/2-Spins are coupled.

The spin quantum numbers are limited to and. Thus, the total spin quantum numbers and only the following values ​​can accept:

• Then
• Then

The following abbreviation and write for the product base

For there is a triplet:

For there is a singlet: