3 Spin 1 2 Particles

[Mandy Geise]

Theearliest models for electron spin imagined a rotating charged mass, but this model fails when examined in detail: the required space distribution does not match limits on the electron radius: the required rotation speed exceeds the speed of light.[4] In the Standard Model, the fundamental particles are all considered "point-like": they have their effects through the field that surrounds them.[5] Any model for spin based on mass rotation would need to be consistent with that model.

Wolfgang Pauli, a central figure in the history of quantum spin, initially rejected any idea that the "degree of freedom" he introduced to explain experimental observations was related to rotation. He called it "classically non-describable two-valuedness". Later he allowed that it is related to angular momentum, but insisted on considering spin an abstract property.[6] This approach allowed Pauli to develop a proof of his fundamental Pauli exclusion principle, a proof now called the spin-statistics theorem.[7] In retrospect this insistence and the style of his proof initiated the modern particle physics era, where abstract quantum properties derived from symmetry properties dominate. Concrete interpretation became secondary and optional.[6]

Unlike classical wavefield circulation which allows continuous values of angular momentum, quantum wavefields allow only discrete values.[10] Consequently energy transfer to or from spins states always occurs in fixed quantum steps. Only a few steps are allowed: for many qualitative purposes the complexity of the spin quantum wavefields can be ignored and the system properties can be discussed in terms of "integer" or "half-integer" spin models as discussed in quantum numbers below.

Since elementary particles are point-like, self-rotation is not well-defined for them. However, spin implies that the phase of the particle depends on the angle as e i S θ \displaystyle e^iS\theta , for rotation of angle θ around the axis parallel to the spin S. This is equivalent to the quantum-mechanical interpretation of momentum as phase dependence in the position, and of orbital angular momentum as phase dependence in the angular position.

For fermions, the picture is less clear. Angular velocity is equal by the Ehrenfest theorem to the derivative of the Hamiltonian to its conjugate momentum, which is the total angular momentum operator J = L + S. Therefore, if the Hamiltonian H is dependent upon the spin S, dH/dS is non-zero, and the spin causes angular velocity, and hence actual rotation, i.e. a change in the phase-angle relation over time. However, whether this holds for free electron is ambiguous, since for an electron, S2 is constant, and therefore it is a matter of interpretation whether the Hamiltonian includes such a term. Nevertheless, spin appears in the Dirac equation, and thus the relativistic Hamiltonian of the electron, treated as a Dirac field, can be interpreted as including a dependence in the spin S.[9]

Composite particles also possess magnetic moments associated with their spin. In particular, the neutron possesses a non-zero magnetic moment despite being electrically neutral. This fact was an early indication that the neutron is not an elementary particle. In fact, it is made up of quarks, which are electrically charged particles. The magnetic moment of the neutron comes from the spins of the individual quarks and their orbital motions.

In ordinary materials, the magnetic dipole moments of individual atoms produce magnetic fields that cancel one another, because each dipole points in a random direction, with the overall average being very near zero. Ferromagnetic materials below their Curie temperature, however, exhibit magnetic domains in which the atomic dipole moments spontaneously align locally, producing a macroscopic, non-zero magnetic field from the domain. These are the ordinary "magnets" with which we are all familiar.

In paramagnetic materials, the magnetic dipole moments of individual atoms will partially align with an externally applied magnetic field. In diamagnetic materials, on the other hand, the magnetic dipole moments of individual atoms align oppositely to any externally applied magnetic field, even if it requires energy to do so.

The study of the behavior of such "spin models" is a thriving area of research in condensed matter physics. For instance, the Ising model describes spins (dipoles) that have only two possible states, up and down, whereas in the Heisenberg model the spin vector is allowed to point in any direction. These models have many interesting properties, which have led to interesting results in the theory of phase transitions.

In classical mechanics, the angular momentum of a particle possesses not only a magnitude (how fast the body is rotating), but also a direction (either up or down on the axis of rotation of the particle). Quantum-mechanical spin also contains information about direction, but in a more subtle form. Quantum mechanics states that the component of angular momentum for a spin-s particle measured along any direction can only take on the values[22]

where Si is the spin component along the i-th axis (either x, y, or z), si is the spin projection quantum number along the i-th axis, and s is the principal spin quantum number (discussed in the previous section). Conventionally the direction chosen is the z axis:

We could try the same approach to determine the behavior of spin under general Lorentz transformations, but we would immediately discover a major obstacle. Unlike SO(3), the group of Lorentz transformations SO(3,1) is non-compact and therefore does not have any faithful, unitary, finite-dimensional representations.

(Because any eigenvector multiplied by a constant is still an eigenvector, there is ambiguity about the overall sign. In this article, the convention is chosen to make the first element imaginary and negative if there is a sign ambiguity. The present convention is used by software such as SymPy; while many physics textbooks, such as Sakurai and Griffiths, prefer to make it real and positive.)

The above spinor is obtained in the usual way by diagonalizing the σu matrix and finding the eigenstates corresponding to the eigenvalues. In quantum mechanics, vectors are termed "normalized" when multiplied by a normalizing factor, which results in the vector having a length of unity.

The nuclear spin of atoms can be determined by sophisticated improvements to the original Stern-Gerlach experiment.[27] A single-energy (monochromatic) molecular beam of atoms in an inhomogeneous magnetic field will split into beams representing each possible spin quantum state. For an atom with electronic spin S and nuclear spin I, there are (2S + 1)(2I + 1) spin states. For example, neutral Na atoms, which have S = 1/2, were passed through a series of inhomogeneous magnetic fields that selected one of the two electronic spin states and separated the nuclear spin states, from which four beams were observed. Thus, the nuclear spin for 23Na atoms was found to be I = 3/2.[28][29]

Electron spin plays an important role in magnetism, with applications for instance in computer memories. The manipulation of nuclear spin by radio-frequency waves (nuclear magnetic resonance) is important in chemical spectroscopy and medical imaging.

The physical interpretation of Pauli's "degree of freedom" was initially unknown. Ralph Kronig, one of Alfred Land's assistants, suggested in early 1925 that it was produced by the self-rotation of the electron. When Pauli heard about the idea, he criticized it severely, noting that the electron's hypothetical surface would have to be moving faster than the speed of light in order for it to rotate quickly enough to produce the necessary angular momentum. This would violate the theory of relativity. Largely due to Pauli's criticism, Kronig decided not to publish his idea.[35]

In the autumn of 1925, the same thought came to Dutch physicists George Uhlenbeck and Samuel Goudsmit at Leiden University. Under the advice of Paul Ehrenfest, they published their results.[36] The young physicists immediately regretted the publication: Hendrik Lorentz and Werner Heisenberg both pointed out problems with the concept of a spinning electron.[37]

Pauli was especially unconvinced and continued to pursue his two-valued degree of freedom. This allowed him to formulate the Pauli exclusion principle, stating that no two electrons can have the same quantum state in the same quantum system.

In 1927 Pauli formalized the theory of spin using the theory of quantum mechanics invented by Erwin Schrdinger and Werner Heisenberg. He pioneered the use of Pauli matrices as a representation of the spin operators and introduced a two-component spinor wave-function.

What you are looking for is Angular Velocity in the Animated Velocity section (the location of some properties changed in Godot 4.2). Check out my video about creating a snow effect with snowflake particles.

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Composites formed from charged particles and vortices in (2+1)-dimensional models, or flux tubes in three-dimensional models, can have any (fractional) angular momentum. The statistics of these objects, like their spin, interpolates continuously between the usual boson and fermion cases. How this works for two-particle quantum mechanics is discussed here.

I only did one QFT course so there may be something obvious I'm missing.Studying quantum fields, some of the easiest examples are bosonic scalar $\phi$ and vector $A_\mu$ fields, and fermionic spinor $\psi$ field. On the book from Peskin and Schroeder, it also mentions the (probable) bosonic tensor $g_\mu\nu$ for gravity. Are there theories regarding higher-order fermionic fields, for example with spin 3/2? If not, why not? If yes, how would one build such a theory?

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