Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

CRC Press eBooks

Transformations, Conservation Laws, and Symmetries

Arjun Sharma

Published Jun 18, 2021• 0 citations
Publisher

Abstract

Transformations, conservation laws, and symmetries have played an important role in the development of modern quantum mechanics. This chapter includes translation in space and time, rotation in space and rotation operator, parity operations and space inversion, time reversal operator and its properties for both zero and non-zero spin particles, solved examples, and unsolved exercises. The chapter discusses in detail how the rotation in space leads to important relations between various components of orbital and total angular momenta, which play an important role in formulations of quantum mechanics. Parity conservation and the most striking properties of the parity operator, relevance of time reversal in classical and quantum mechanical descriptions, and the antiunitary and antilinear properties of the time reversal operator are also featured details in the chapter.

DOI
10.1201/9781003154457-4

Related papers

Symmetries and conservation laws

Gennaro Auletta, Mauro Fortunato, Giorgio Parisi A

A symmetry is an equivalence of different physical situations, and is therefore strictly associated with the concept of invariance . One of the main aims of physics is the search in nature for symmetries and, therefore, for observables that are invariant under certain classes of transformation. In this context, it becomes particularly interesting to look for the conditions under which a certain symmetry is eventually broken. We have already discussed some symmetries of quantum mechanics: time translations (Ch. 3), space translations (Ch. 2) and rotations (Ch. 6). Another example of invariance is represented by the indistinguishability of identical particles, which we discussed in the previous chapter. Symmetries are induced by physical transformations that generate some invariant properties (see Sec. 8.1). In Sec. 8.2, the continuous transformations induced by rotations and space translations are analyzed, whereas in Sec. 8.3, space reflection and time reversal ( discrete symmetries) are discussed. As we shall see in the next chapter, time reversibility is lost in the measurement process. The rigorous mathematical framework of a theory of transformations and of their invariance is called group theory (the subject of Sec. 8.4). Quantum transformations and symmetries Active and passive transformations We have seen that time evolution may be represented either in the Schrödinger picture (state vectors evolve and observables are kept constant) or in the Heisenberg picture (states vectors remain constant and observables evolve) (see Subsec. 3.6.1 and Sec. 5.3).

Cambridge University Press eBooks•Apr 16, 2009•1 citation

Rotational symmetry of space, rotation operators, conservation laws, and spin and orbital angular momenta in quantum mechanics

Masud Mansuripur

Quantization of spin and orbital angular momenta in standard quantum mechanics is rooted in the fundamental rotational symmetry of space in conjunction with the results of experiments on atomic and subatomic particles conducted with the Stern-Gerlach apparatus. We explain some of the most important features of spin and orbital angular momenta as inescapable consequences of elementary Stern-Gerlach experiments viewed in light of the rotational symmetry of space. The rotation operator is first derived for spin-½ particles, then extended to spin-1 particles by treating the case of a hydrogen atom in its ground state, where the spin of the proton combines with that of the electron to create a spin-1 triplet state. Along the way, we examine the evolution of a particle’s spin state under an applied magnetic field, where the interaction energy arises from the existence of a magnetic dipole moment for each particle in proportion to its spin angular momentum. Whereas a constant magnetic field B0 causes the particle’s magnetic moment to gyrate around the direction of B0, it will be seen that the presence of a small oscillating magnetic field perpendicular to B0 brings about the so-called Rabi oscillations, which involve periodic flippings of the particle’s spin orientation. In the case of a hydrogen atom in its ground state, the interaction between the magnetic dipole moments of the proton and the electron splits the ground state between a lower energy singlet having a net angular momentum of zero (spin-0 state), and a higher energy triplet having a total angular momentum of ħ (spin-1 state). This hyperfine splitting is the source of the well-known 21-cm line observed in the spectrum of the hydrogen atom, which arises from transitions between the atom’s singlet and triplet states. In the final section, we put forward an argument (originally made by Richard Feynman) that ties the rotational symmetry of space to the conservation of angular momentum in quantum mechanics. The argument is, in fact, quite general, enabling one to deduce other conservation laws from various inherent symmetries of quantum mechanical systems.

Spintronics•Oct 4, 2024•0 citations

Symmetry and physics

Michael El-Batanouny, F. Wooten

Introduction The application of group theory to study physical problems and their solutions provides a formal method for exploiting the simplifications made possible by the presence of symmetry. Often the symmetry that is readily apparent is the symmetry of the system/object of interest, such as the three-fold axial symmetry of an NH 3 molecule. The symmetry exploited in actual analysis is the symmetry of the Hamiltonian. When alluding to symmetry we usually include geometrical, time-reversal symmetry, and symmetry associated with the exchange of identical particles. Conservation laws of physics are rooted in the symmetries of the underlying space and time. The most common physical laws we are familiar with are actually manifestations of some universal symmetries. For example, the homogeneity and isotropy of space lead to the conservation of linear and angular momentum, respectively, while the homogeneity of time leads to the conservation of energy. Such laws have come to be known as universal conservation laws. As we will delineate in a later chapter, the relation between these classical symmetries and corresponding conserved quantities is beautifully cast in a theorem due to Emmy Noether. At the day-to-day working level of the physicist dealing with quantum mechanics, the application of symmetry restrictions leads to familiar results, such as selection rules and characteristic transformations of eigenfunctions when acted upon by symmetry operations that leave the Hamiltonian of the system invariant.

Cambridge University Press eBooks•Mar 13, 2008•0 citations

Symmetries and Invariants for Non-Hermitian Hamiltonians

Miguel Angel Otero Simón, Álvaro Buendía, Juan Gonzalo Muga

We discuss Hamiltonian symmetries and invariants for quantum systems driven by non-Hermitian Hamiltonians. For time-independent Hermitian Hamiltonians, a unitary or antiunitary transformation AHA† that leaves the Hamiltonian H unchanged represents a symmetry of the Hamiltonian, which implies the commutativity [H,A]=0 and, if A is linear and time-independent, a conservation law, namely the invariance of expectation values of A. For non-Hermitian Hamiltonians, H† comes into play as a distinct operator that complements H in generalized unitarity relations. The above description of symmetries has to be extended to include also A-pseudohermiticity relations of the form AH=H†A. A superoperator formulation of Hamiltonian symmetries is provided and exemplified for Hamiltonians of a particle moving in one-dimension considering the set of A operators that form Klein’s 4-group: parity, time-reversal, parity&time-reversal, and unity. The link between symmetry and conservation laws is discussed and shown to be richer and subtler for non-Hermitian than for Hermitian Hamiltonians.

Mathematics•Jun 27, 2018•18 citations

Symmetry and Conservation Law

Tao Xiang

Corresponding to each symmetry, there is a conservation law with a characteristic conserving quantity which is invariant under that transformation. One of the aims of physical research is to search for symmetries and the corresponding observables that are invariant under certain classes of transformation. The Stone theorem ensures the existence of a Hermitian infinitesimal generator for an Abelian group of unitary transformation. The spatial translation operator is a specific example of this theorem and the momentum operator is the generator of spatial translation. This Galilean principle of relativity must be true in quantum mechanics as well. Time-reversal symmetry is a symmetry of the system under the transformation of time reversal.

CRC Press eBooks•Mar 31, 2022•0 citations

Spin and rotation

Benjamin W. Schumacher, Michael D. Westmoreland

Spin-s systems Angular momentum is one of the fundamental quantities of Newtonian physics, and in quantum physics its importance is at least as great. In quantum mechanics we often distinguish between two types of angular momentum: orbital angular momentum , which a system of particles possesses due to particle motion through space; and spin angular momentum , which is an intrinsic property of a particle. The distinction will be important later, but for now we will ignore it. We will here refer to angular momentum of any sort as “spin” and develop general-purpose mathematical tools for its description. We have already dealt with spin systems, particularly the example of a spin-½ particle. Our approach began with the empirical observation that a measurement of any spin component of a spin-½ particle could yield only the results +ħ/2 or −ħ/2. We introduced the basis states | z ±〉 for the two-dimensional Hilbert space ℋ. We also gave other basis states such as {| x ±〉} and {| y ±〉} in terms of the | z ±〉 states. From basis states and measurement values we constructed operators for the spin components S x , S y , and S z . With the operators in hand, we could then examine the algebraic relations between them (such as the commutation relation in Exercise 3.56). Our job here is to generalize our analysis to systems of arbitrary spin. To do this, we will reverse our chain of logic.

Cambridge University Press eBooks•Mar 25, 2010•0 citations
Transformations, conservation laws, and symmetries have played an important role in the development of modern quantum mechanics. This chapter includes translation in space and time, rotation in space and rotation operator, parity operations and space inversion, time reversal operator and its properties for both zero and non-zero spin particles, solved examples, and unsolved exercises. The chapter discusses in detail how the rotation in space leads to important relations between various components of orbital and total angular momenta, which play an important role in formulations of quantum mechanics. Parity conservation and the most striking properties of the parity operator, relevance of time reversal in classical and quantum mechanical descriptions, and the antiunitary and antilinear properties of the time reversal operator are also featured details in the chapter.
A symmetry is an equivalence of different physical situations, and is therefore strictly associated with the concept of invariance . One of the main aims of physics is the search in nature for symmetries and, therefore, for observables that are invariant under certain classes of transformation. In this context, it becomes particularly interesting to look for the conditions under which a certain symmetry is eventually broken. We have already discussed some symmetries of quantum mechanics: time translations (Ch. 3), space translations (Ch. 2) and rotations (Ch. 6). Another example of invariance is represented by the indistinguishability of identical particles, which we discussed in the previous chapter. Symmetries are induced by physical transformations that generate some invariant properties (see Sec. 8.1). In Sec. 8.2, the continuous transformations induced by rotations and space translations are analyzed, whereas in Sec. 8.3, space reflection and time reversal ( discrete symmetries) are discussed. As we shall see in the next chapter, time reversibility is lost in the measurement process. The rigorous mathematical framework of a theory of transformations and of their invariance is called group theory (the subject of Sec. 8.4). Quantum transformations and symmetries Active and passive transformations We have seen that time evolution may be represented either in the Schrödinger picture (state vectors evolve and observables are kept constant) or in the Heisenberg picture (states vectors remain constant and observables evolve) (see Subsec. 3.6.1 and Sec. 5.3).
Quantization of spin and orbital angular momenta in standard quantum mechanics is rooted in the fundamental rotational symmetry of space in conjunction with the results of experiments on atomic and subatomic particles conducted with the Stern-Gerlach apparatus. We explain some of the most important features of spin and orbital angular momenta as inescapable consequences of elementary Stern-Gerlach experiments viewed in light of the rotational symmetry of space. The rotation operator is first derived for spin-½ particles, then extended to spin-1 particles by treating the case of a hydrogen atom in its ground state, where the spin of the proton combines with that of the electron to create a spin-1 triplet state. Along the way, we examine the evolution of a particle’s spin state under an applied magnetic field, where the interaction energy arises from the existence of a magnetic dipole moment for each particle in proportion to its spin angular momentum. Whereas a constant magnetic field B0 causes the particle’s magnetic moment to gyrate around the direction of B0, it will be seen that the presence of a small oscillating magnetic field perpendicular to B0 brings about the so-called Rabi oscillations, which involve periodic flippings of the particle’s spin orientation. In the case of a hydrogen atom in its ground state, the interaction between the magnetic dipole moments of the proton and the electron splits the ground state between a lower energy singlet having a net angular momentum of zero (spin-0 state), and a higher energy triplet having a total angular momentum of ħ (spin-1 state). This hyperfine splitting is the source of the well-known 21-cm line observed in the spectrum of the hydrogen atom, which arises from transitions between the atom’s singlet and triplet states. In the final section, we put forward an argument (originally made by Richard Feynman) that ties the rotational symmetry of space to the conservation of angular momentum in quantum mechanics. The argument is, in fact, quite general, enabling one to deduce other conservation laws from various inherent symmetries of quantum mechanical systems.
Introduction The application of group theory to study physical problems and their solutions provides a formal method for exploiting the simplifications made possible by the presence of symmetry. Often the symmetry that is readily apparent is the symmetry of the system/object of interest, such as the three-fold axial symmetry of an NH 3 molecule. The symmetry exploited in actual analysis is the symmetry of the Hamiltonian. When alluding to symmetry we usually include geometrical, time-reversal symmetry, and symmetry associated with the exchange of identical particles. Conservation laws of physics are rooted in the symmetries of the underlying space and time. The most common physical laws we are familiar with are actually manifestations of some universal symmetries. For example, the homogeneity and isotropy of space lead to the conservation of linear and angular momentum, respectively, while the homogeneity of time leads to the conservation of energy. Such laws have come to be known as universal conservation laws. As we will delineate in a later chapter, the relation between these classical symmetries and corresponding conserved quantities is beautifully cast in a theorem due to Emmy Noether. At the day-to-day working level of the physicist dealing with quantum mechanics, the application of symmetry restrictions leads to familiar results, such as selection rules and characteristic transformations of eigenfunctions when acted upon by symmetry operations that leave the Hamiltonian of the system invariant.
We discuss Hamiltonian symmetries and invariants for quantum systems driven by non-Hermitian Hamiltonians. For time-independent Hermitian Hamiltonians, a unitary or antiunitary transformation AHA† that leaves the Hamiltonian H unchanged represents a symmetry of the Hamiltonian, which implies the commutativity [H,A]=0 and, if A is linear and time-independent, a conservation law, namely the invariance of expectation values of A. For non-Hermitian Hamiltonians, H† comes into play as a distinct operator that complements H in generalized unitarity relations. The above description of symmetries has to be extended to include also A-pseudohermiticity relations of the form AH=H†A. A superoperator formulation of Hamiltonian symmetries is provided and exemplified for Hamiltonians of a particle moving in one-dimension considering the set of A operators that form Klein’s 4-group: parity, time-reversal, parity&time-reversal, and unity. The link between symmetry and conservation laws is discussed and shown to be richer and subtler for non-Hermitian than for Hermitian Hamiltonians.
Corresponding to each symmetry, there is a conservation law with a characteristic conserving quantity which is invariant under that transformation. One of the aims of physical research is to search for symmetries and the corresponding observables that are invariant under certain classes of transformation. The Stone theorem ensures the existence of a Hermitian infinitesimal generator for an Abelian group of unitary transformation. The spatial translation operator is a specific example of this theorem and the momentum operator is the generator of spatial translation. This Galilean principle of relativity must be true in quantum mechanics as well. Time-reversal symmetry is a symmetry of the system under the transformation of time reversal.
Spin-s systems Angular momentum is one of the fundamental quantities of Newtonian physics, and in quantum physics its importance is at least as great. In quantum mechanics we often distinguish between two types of angular momentum: orbital angular momentum , which a system of particles possesses due to particle motion through space; and spin angular momentum , which is an intrinsic property of a particle. The distinction will be important later, but for now we will ignore it. We will here refer to angular momentum of any sort as “spin” and develop general-purpose mathematical tools for its description. We have already dealt with spin systems, particularly the example of a spin-½ particle. Our approach began with the empirical observation that a measurement of any spin component of a spin-½ particle could yield only the results +ħ/2 or −ħ/2. We introduced the basis states | z ±〉 for the two-dimensional Hilbert space ℋ. We also gave other basis states such as {| x ±〉} and {| y ±〉} in terms of the | z ±〉 states. From basis states and measurement values we constructed operators for the spin components S x , S y , and S z . With the operators in hand, we could then examine the algebraic relations between them (such as the commutation relation in Exercise 3.56). Our job here is to generalize our analysis to systems of arbitrary spin. To do this, we will reverse our chain of logic.