## Spin Angular Momentum

The angular momentum of an electromagnetic wave involves dual constituents: (1) a spin angular momentum and (2) an orbital angular momentum. The spin angular momentum L s = lℏ is associated with wave polarization, and the orbital angular momentum L 0 = nℏ is correlated to reliance of a wave phase from a direction where l and n are quantum.

Broadly speaking, a classical extended object (e.g., the Earth) can possess two different types of angular momentum. The first type is due to the rotation of the object’s center of mass about some fixed external point (e.g., the Sun)—this is generally known as *orbital angular momentum*. The second type is due to the object’s internal motion—this is generally known as *spin angular momentum* (because, for a rigid object, the internal motion consists of spinning about an axis passing through the center of mass). By analogy, quantum particles can possess both orbital angular momentum due to their motion through space (see Chapter [sorb]), and spin angular momentum due to their internal motion. Actually, the analogy with classical extended objects is not entirely accurate, because electrons, for instance, are structureless point particles. In fact, in quantum mechanics, it is best to think of spin angular momentum as a kind of intrinsic angular momentum possessed by particles. It turns out that each type of elementary particle has a characteristic spin angular momentum, just as each type has a characteristic charge and mass.

Addition of angular momentum April 21, 2015 Often we need to combine diﬀerent sources of angular momentum to characterize the total angular momentum of a system, or to divide the total angular momentum into parts to evaluate the eﬀect of a. An example of conservation of angular momentum is seen in an ice skater executing a spin, as shown in. The net torque on her is very close to zero, because 1) there is relatively little friction between her skates and the ice, and 2) the friction is exerted very close to the pivot point. Spin is sometimes called angular momentum, which is defined as: (mass) x (velocity) x (radius), where radius is the distance from the spinning object to the axis. Conservation of angular momentum, one of the fundamental laws of physics, observes that the angular rotation of a spinning object remains constant unless acted upon by external torque.

- 9.1: Spin Operators
- Because spin is a type of angular momentum, it is reasonable to suppose that it possesses similar properties to orbital angular momentum. Thus, by analogy, we would expect to be able to define three operators that represent the three Cartesian components of spin angular momentum. Moreover, it is plausible that these operators possess analogous commutation relations to the three corresponding orbital angular momentum operators.

- 9.2: Spin Space
- Unlike regular wavefunctions, spin wavefunctions do not exist in real space. Likewise, the spin angular momentum operators cannot be represented as differential operators in real space. Instead, we need to think of spin wavefunctions as existing in an abstract (complex) vector space. The different members of this space correspond to the different internal configurations of the particle under investigation. Note that only the directions of our vectors have any physical significance.

- 9.3: Eigenstates of Sz and S²
- Because the operators Sz and S² commute, they must possess simultaneous eigenstates.

- 9.4: Pauli Representation
- Up to now, we have discussed spin space in rather abstract terms. In the following, we shall describe a particular representation of electron spin space due to Pauli . This so-called Pauli representation allows us to visualize spin space, and also facilitates calculations involving spin.

- 9.5: Spin Precession
- The expectation value of the spin angular momentum vector subtends a constant angle α with the z -axis, and precesses about this axis. This behavior is actually equivalent to that predicted by classical physics.

- 9.E: Spin Angular Momentum (Exercises)

## Contributors and Attributions

Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)

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## The Angular Momentum Eigenfunctions

The angular momentum eigenstates are **eigenstates of two operators**.

**raising and lowering operators**raise or lower , leaving unchanged.

The differential operators take some work to derive.

Its easy to find functions that give the eigenvalue of .

To find the dependence, we will use the fact that there are limits on . The state with maximum must give zero when raised.

This gives us a differential equation for that state.

The solution is

Check the solution.

Its correct.

Here we should note that only the integer value of work for these solutions. If we were to use half-integers, the wave functions would not be single valued, for example at and . Even though the probability may be single valued, discontinuities in the amplitude would lead to infinities in the Schrödinger equation. We will find later that the **half-integer angular momentum states are used** for internal angular momentum (spin), for which no or coordinates exist.

Therefore, **the eigenstate is**.

We can continue to lower to get all of the eigenfunctions.

We call these eigenstates the **Spherical Harmonics**. The spherical harmonics are **normalized**.

Since they are eigenfunctions of Hermitian operators, they are

**orthogonal**.

We will use the **actual function** in some problems.

## Spin Angular Momentum Quantum Number

The spherical harmonics with negative can be easily compute from those with positive .Any function of and can be **expanded in the spherical harmonics**.

The spherical harmonics form a **complete set**.

When using bra-ket notation, is sufficient to identify the state.

The spherical harmonics are **related to the Legendre polynomials** which are functions of .

## Spin Angular Momentum Quantum Mechanics

**Subsections**