# Quantum mechanics space and spin functions Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. It only spinning chair gif a minute to sign up. Connect and share knowledge within a single location that is structured and easy to search. This question comes to mind while reading the quote in different books which is "Hilbert space is infinite dimensional". While the electron spin is 2 Dimensional problem. Secondly, we assume that spin wave function is square integrable which is a property of a function in Hilbert space. THen how do we know if spin wavefunctions belongs to Hilbert Space or not?
• quantum mechanics - Do spin wavefunctions live on a Hilbert space? - Physics Stack Exchange
• Phase Space Quantum Mechanics [The Physics Travel Guide]
• Spin (physics) - Wikipedia
• The total angular momentum conserved in function processes is then the sum of the orbital angular momentum and the spin. In quantum mechanics all particles are either bosons or fermions. In some speculative relativistic quantum field theories " supersymmetric " particles also exist, where linear combinations of bosonic and fermionic components appear.

The above permutation postulate for N -particle state functions has most important consequences in daily life, e. As described above, quantum mechanics states that components of angular momentum measured along any direction can only take a number of discrete values. The most convenient quantum-mechanical description of particle's spin is therefore with a set of complex numbers corresponding to amplitudes of finding a given value of projection of its intrinsic angular momentum on a given axis.

Since these numbers depend on the choice of the axis, they transform into each other non-trivially when this axis is rotated. It is clear that the transformation law must be linear, so we can represent it by associating a matrix with each rotation, and the product of two transformation matrices corresponding to rotations A and B must be equal up to phase to the matrix representing rotation AB.

Further, rotations preserve the quantum-mechanical inner product, and so should our transformation matrices:. Mathematically speaking, these matrices furnish a unitary projective representation of the rotation group SO 3. Each such representation corresponds to a representation of the covering group of SO 3which is SU 2.

Starting with S x. Using the spin operator commutation relationswe see functiohs the commutators evaluate to i S y for the odd terms in the series, and to S x for all of the even terms. Note that since we only relied on the spin operator commutation relations, this proof holds for any dimension i.

A generic mecjanics in 3-dimensional space can be built by compounding operators of this type using Euler angles :. An irreducible representation of this group of operators is furnished by the Wigner D-matrix :.

### quantum mechanics - Do spin wavefunctions live on a Hilbert space? - Physics Stack Exchange

Recalling that a generic spin state can be written as a superposition of states with definite mwe see that if s is an integer, the values of m are all integers, and this matrix corresponds quanthm the identity operator. This fact is a crucial element of the proof of the spin—statistics theorem.

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

These spinors transform under Lorentz transformations according to the law. It can be shown that the scalar product.

### Phase Space Quantum Mechanics [The Physics Travel Guide]

The corresponding normalized eigenvectors are. Because any eigenvector multiplied by a constant is still an eigenvector, there is ambiguity about the overall sign. Space 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.

By the postulates of quantum mechanicsan experiment designed to measure the electron spin on the xyor z axis can only yield an eigenvalue of the and spin operator S xS y or S z on that axis, i. The quantum state of a particle with respect to spincan be represented by a two-component spinor :.

Following the measurement, the spin state of the particle collapses into the corresponding eigenstate. The operator to measure spin along an arbitrary axis direction is easily obtained from the Pauli spin matrices. Then the operator for spin in this direction is simply. This method of finding the operator for spin in an arbitrary direction generalizes to higher spin states, mechanics takes the dot product of the direction with a vector of the three operators for the three x - y - z -axis directions.

In quantum mechanics, vectors are termed "normalized" spin multiplied by a normalizing factor, which results in the vector having a length of unity. Since the Pauli matrices do not commutemeasurements of spin along the different axes are incompatible. This means that if, for example, we know the spin along the x axis, and we then measure the spin along the y axis, we have invalidated our previous knowledge of the x axis spin.

This can be seen functions the property of the eigenvectors i. This implies quantum the original measurement of the spin along the x axis is no longer valid, since the spin along the x axis will now be measured to have either eigenvalue with equal probability. By taking Kronecker products of this representation with itself repeatedly, one may construct all higher irreducible representations. That is, the resulting spin operators for higher-spin systems in three spatial dimensions can be calculated for arbitrarily large s using this functlons operator and ladder operators. The resulting irreducible representations yield the following spin matrices and eigenvalues in the z-basis:.

Also useful in the quantum mechanics of multiparticle systems, the general Pauli group G n is defined to consist of all n -fold tensor products of Mschanics matrices. The analog formula of Euler's formula in terms of the Pauli matrices. For example, see the isotopes of bismuthdpace which the list of isotopes includes the column nuclear spin and parity.

Spin has important theoretical implications and practical applications. Well-established direct applications of spin mechaniics. Electron spin plays an important role in magnetismwith 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.

Spin—orbit coupling leads to the fine structure of atomic spectra, which is used in atomic clocks and in the modern definition of the second. Precise measurements of the g -factor of the electron have played an important role in the development and verification of quantum electrodynamics. Photon spin is associated with the polarization of light photon polarization.

An emerging application of spin is as a binary information carrier in spin transistors.

The original concept, proposed inis known as Datta—Das spin transistor. The manipulation of spin in dilute magnetic semiconductor materialssuch as metal-doped ZnO or TiO 2 imparts a further degree of freedom and has the potential to facilitate the fabrication of more efficient electronics.

There are many indirect applications and manifestations of spin and the associated Pauli exclusion principlestarting with the periodic table of chemistry. Spin was first discovered in the context of the emission functtions of alkali metals. InWolfgang Pauli introduced what he called a "two-valuedness not describable classically"  associated with the electron in the outermost shell.

This allowed him to formulate the Pauli exclusion principlestating that no two electrons can have the same quantum state in the same quantum system. The physical interpretation of Pauli's "degree of freedom" was initially unknown. When Pauli heard about the idea, he criticized it severely, noting that the electron's hypothetical surface would have to be moving fuctions than the speed of light in order for it to rotate quickly enough to and the necessary angular momentum.

This would violate the theory of relativity. Largely due to Space criticism, Kronig decided not to publish his idea. Under the advice of Paul Ehrenfestthey published their results. This discrepancy was due to the orientation of the electron's tangent frame, in addition to its position.

Mathematically speaking, a fiber bundle description is quantum. The tangent bundle effect is additive and relativistic; that is, it vanishes if c goes to infinity. It is one half of the value obtained without regard for the tangent-space mechanics, but with opposite sign.

Thus the combined effect differs from the latter by a factor two Thomas precessionknown to Ludwik Silberstein in He pioneered the use of Pauli matrices as a representation of the spin operators and introduced a two-component spinor wave-function. Uhlenbeck and Goudsmit treated spin as arising from classical rotation, while Pauli emphasized, that mechaniccs is non-classical and intrinsic property.

Pauli's theory of spin was non-relativistic. However, inPaul Dirac published the Dirac equationwhich described the relativistic electron. In the Dirac equation, a four-component spinor known as a spin Dirac spinor " was used for the electron wave-function. Relativistic spin explained gyromagnetic anomaly, which was in retrospect first observed by Samuel Jackson Barnett in see Einstein—de Haas effect.

InPauli proved the spin—statistics theoremwhich states that fermions have half-integer spin, and bosons have integer spin. In retrospect, the first direct experimental evidence of the electron spin was the Stern—Gerlach experiment of However, the correct explanation of this experiment functions only given in s;ace From Wikipedia, the free encyclopedia.

(5) Runge-Lenz vector in quantum mechanics (6) Hydrogenlike Atoms (7) Radial Wave functions of Hydrogen Atom Lecture 8- Symmetry and Spin (1)Spin and Rotations (2)Generators of rotation in the spinor space (3) Spin wave functions (4) Total angular momentum (5)Addition of Angular Momenta (6) Clebsch-Gordan Coefficients. Oct 11,  · In the phase space formulation, the transition from classical mechanics to quantum mechanics is known as deformation quantization. The map between function in the phase space formulation of quantum mechanics and the operators in the corresponding Hilbert space is known as Wigner–Weyl transform. Sep 16,  · The space of states of position (or momentum) wavefunctions of objects without spin in quantum mechanics is the infinite-dimensional Hilbert space of square-integrable functions L 2 (R n). The space of (spin) states for a particle with spin s is the finite-dimensional Hilbert space C 2 s + casinocanli.cos: 1.

Intrinsic form of angular momentum as a property of quantum particles. This article is about spin in quantum mechanics. For rotation in classical mechanics, see Angular momentum. Main article: Spin quantum number. Main article: spin—statistics theorem.

Main article: Spin magnetic moment. Further information: Angular momentum operator. This sspace does not cite any sources. Please help improve this section by adding citations to reliable sources. Unsourced material may be challenged and removed. May Learn how and when to remove this template message.

Main article: Pauli matrices. See also: Symmetry in quantum mechanics. Click "show" at right to see a proof or "hide" to hide it. This section needs additional citations for verification. Please help improve this article by adding citations to reliable sources. September Learn how and when to remove this template message.

### Spin (physics) - Wikipedia

Quantum Mechanics 3rd ed. Introduction to Quantum Mechanics 2nd ed. Niels Bohr's Times. Oxford: Clarendon Press. ISBN February Retrieved 25 July Bibcode : PhRv Quantum field theoryCh. The Advanced Book Program. Bransden, C. Retrieved Physics Letters. B67 3 : — Bibcode : PhLB Physical Review.

Mechanixs 5 : — Bibcode : PhRvD.

S2CID Fujikawa, R. Shrock Physical Review Letters. Bibcode : PhRvL. S;in Hall Quantum Theory for Mathematicians. SakuraiModern Quantum Mechanicsp. S and Functions. Das And Physics Letters. Bibcode : ApPhL. Journal of Applied Physics. Bibcode quantum JAP Nobel Lecture. In this paper, we shall attempt to interpret quantum mechanics as a form of funtcions a general statistical dynamics.

Quantum Mechanics as a statistical theory by J. The time evolution in the phase space formulation of quantum mechanics is described by the vonNeumann equation. The difference between the von Neumann equation and the Liouville equation is that in the former the density of points in phase space is not conserved. Formulated differently, the probability fluid is diffusive and compressible.

In the phase space spin, the space from classical mechanics to quantum mechanics is known as deformation quantization. The map between function in the phase space formulation of quantum mechanics and the operators in the corresponding Hilbert space is known as Wigner—Weyl transform.

The chief advantage of the phase space formulation is that it makes quantum mechanics appear as similar to Hamiltonian mechanics as possible by avoiding the operator formalism, thereby "'freeing' the quantization of the 'burden' of the Hilbert space". User Tools Register Log Mecahnics.

Site Tools Search. Sidebar Formalisms. Quantum Mechanics in Phase space by Curtright and Zachos Classical statistical mechanics is a ' crypto-deterministic' theory, where each element of the probability distribution of the dynamical variables specifying a given system evolves with time according to deterministic laws of motion; the whole uncertainty is contained in the form of the initial distributions.

Reading Recommendations A great introduction is The algebraic way by Hiley. See also the corresponding chapter in Ballentine's Quantum Mechanics mechanics, and also. Canonical transformation in quantum mechanics Y.