Quantum Mechanics G Aruldhas Pdf
Joseph Zyiuahndy
In physics, relativistic quantum mechanics (RQM) is any Poincar covariant formulation of quantum mechanics (QM). This theory is applicable to massive particles propagating at all velocities up to those comparable to the speed of light c, and can accommodate massless particles. The theory has application in high energy physics,[1] particle physics and accelerator physics,[2] as well as atomic physics, chemistry[3] and condensed matter physics.[4][5] Non-relativistic quantum mechanics refers to the mathematical formulation of quantum mechanics applied in the context of Galilean relativity, more specifically quantizing the equations of classical mechanics by replacing dynamical variables by operators. Relativistic quantum mechanics (RQM) is quantum mechanics applied with special relativity. Although the earlier formulations, like the Schrdinger picture and Heisenberg picture were originally formulated in a non-relativistic background, a few of them (e.g. the Dirac or path-integral formalism) also work with special relativity.
The most successful (and most widely used) RQM is relativistic quantum field theory (QFT), in which elementary particles are interpreted as field quanta. A unique consequence of QFT that has been tested against other RQMs is the failure of conservation of particle number, for example in matter creation and annihilation.[7]
Paul Dirac's work between 1927 to 1933 shaped the synthesis of special relativity and quantum mechanics.[8] His work was instrumental, as he formulated the Dirac equation and also originated quantum electrodynamics, both of which were successful in combining the two theories.[9]
There is also the problem of incorporating spin in the Hamiltonian, which isn't a prediction of the non-relativistic Schrdinger theory. Particles with spin have a corresponding spin magnetic moment quantized in units of μB, the Bohr magneton:[16][17]
where g is the (spin) g-factor for the particle, and S the spin operator, so they interact with electromagnetic fields. For a particle in an externally applied magnetic field B, the interaction term[18]
has to be added to the above non-relativistic Hamiltonian. On the contrary; a relativistic Hamiltonian introduces spin automatically as a requirement of enforcing the relativistic energy-momentum relation.[19]
Relativistic Hamiltonians are analogous to those of non-relativistic QM in the following respect; there are terms including rest mass and interaction terms with externally applied fields, similar to the classical potential energy term, as well as momentum terms like the classical kinetic energy term. A key difference is that relativistic Hamiltonians contain spin operators in the form of matrices, in which the matrix multiplication runs over the spin index σ, so in general a relativistic Hamiltonian:
and was discovered by many people because of the straightforward way of obtaining it, notably by Schrdinger in 1925 before he found the non-relativistic equation named after him, and by Klein and Gordon in 1927, who included electromagnetic interactions in the equation. This is relativistically invariant, yet this equation alone isn't a sufficient foundation for RQM for a at least two reasons: one is that negative-energy states are solutions,[2][21] another is the density (given below), and this equation as it stands is only applicable to spinless particles. This equation can be factored into the form:[22][23]
is the Dirac equation. The other factor is also the Dirac equation, but for a particle of negative mass.[22] Each factor is relativistically invariant. The reasoning can be done the other way round: propose the Hamiltonian in the above form, as Dirac did in 1928, then pre-multiply the equation by the other factor of operators E + cα p + βmc2, and comparison with the KG equation determines the constraints on α and β. The positive mass equation can continue to be used without loss of continuity. The matrices multiplying ψ suggest it isn't a scalar wavefunction as permitted in the KG equation, but must instead be a four-component entity. The Dirac equation still predicts negative energy solutions,[6][24] so Dirac postulated that negative energy states are always occupied, because according to the Pauli principle, electronic transitions from positive to negative energy levels in atoms would be forbidden. See Dirac sea for details.
Instead, what appears look at first sight a "probability density" and "probability current" has to be reinterpreted as charge density and current density when multiplied by electric charge. Then, the wavefunction ψ is not a wavefunction at all, but reinterpreted as a field.[15] The density and current of electric charge always satisfy a continuity equation:
The KG equation is applicable to spinless charged bosons in an external electromagnetic potential.[2] As such, the equation cannot be applied to the description of atoms, since the electron is a spin 1/2 particle. In the non-relativistic limit the equation reduces to the Schrdinger equation for a spinless charged particle in an electromagnetic field:[18]
the first of which is the Weyl equation, a considerable simplification applicable for massless neutrinos.[28] This time there is a 2 2 identity matrix pre-multiplying the energy operator conventionally not written. In RQM it is useful to take this as the zeroth Pauli matrix σ0 which couples to the energy operator (time derivative), just as the other three matrices couple to the momentum operator (spatial derivatives).
The Pauli and gamma matrices were introduced here, in theoretical physics, rather than pure mathematics itself. They have applications to quaternions and to the SO(2) and SO(3) Lie groups, because they satisfy the important commutator [ , ] and anticommutator [ , ]+ relations respectively:
where εabc is the three-dimensional Levi-Civita symbol. The gamma matrices form bases in Clifford algebra, and have a connection to the components of the flat spacetime Minkowski metric ηαβ in the anticommutation relation:
In 1929, the Breit equation was found to describe two or more electromagnetically interacting massive spin 1/2 fermions to first-order relativistic corrections; one of the first attempts to describe such a relativistic quantum many-particle system. This is, however, still only an approximation, and the Hamiltonian includes numerous long and complicated sums.
where p is the momentum operator, S the spin operator for a particle of spin s, E is the total energy of the particle, and m0 its rest mass. Helicity indicates the orientations of the spin and translational momentum vectors.[29] Helicity is frame-dependent because of the 3-momentum in the definition, and is quantized due to spin quantization, which has discrete positive values for parallel alignment, and negative values for antiparallel alignment.
According to the relativistic energy-momentum relation, all massless particles travel at the speed of light, so particles traveling at the speed of light are also described by two-component spinors. Historically, lie Cartan found the most general form of spinors in 1913, prior to the spinors revealed in the RWEs following the year 1927.
For equations describing higher-spin particles, the inclusion of interactions is nowhere near as simple minimal coupling, they lead to incorrect predictions and self-inconsistencies.[33] For spin greater than ħ/2, the RWE is not fixed by the particle's mass, spin, and electric charge; the electromagnetic moments (electric dipole moments and magnetic dipole moments) allowed by the spin quantum number are arbitrary. (Theoretically, magnetic charge would contribute also). For example, the spin 1/2 case only allows a magnetic dipole, but for spin 1 particles magnetic quadrupoles and electric dipoles are also possible.[28] For more on this topic, see multipole expansion and (for example) Cdric Lorc (2009).[34][35]
This is not possible for all RWEs; and is one reason the Lorentz group theoretic approach is important and appealing: fundamental invariance and symmetries in space and time can be used to derive RWEs using appropriate group representations. The Lagrangian approach with field interpretation of ψ is the subject of QFT rather than RQM: Feynman's path integral formulation uses invariant Lagrangians rather than Hamiltonian operators, since the latter can become extremely complicated, see (for example) Weinberg (1995).[38]
In non-relativistic QM, the angular momentum operator is formed from the classical pseudovector definition L = r p. In RQM, the position and momentum operators are inserted directly where they appear in the orbital relativistic angular momentum tensor defined from the four-dimensional position and momentum of the particle, equivalently a bivector in the exterior algebra formalism:[39][d]
The events which led to and established RQM, and the continuation beyond into quantum electrodynamics (QED), are summarized below [see, for example, R. Resnick and R. Eisberg (1985),[46] and P.W Atkins (1974)[47]]. More than half a century of experimental and theoretical research from the 1890s through to the 1950s in the new and mysterious quantum theory as it was up and coming revealed that a number of phenomena cannot be explained by QM alone. SR, found at the turn of the 20th century, was found to be a necessary component, leading to unification: RQM. Theoretical predictions and experiments mainly focused on the newly found atomic physics, nuclear physics, and particle physics; by considering spectroscopy, diffraction and scattering of particles, and the electrons and nuclei within atoms and molecules. Numerous results are attributed to the effects of spin.
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