Physics:Quantum Symmetry in quantum mechanics - Revision history

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	{{Short description\|Quantum Collection topic on Quantum Symmetry in quantum mechanics}}~~Book I~~		{{Short description\|Quantum Collection topic on Quantum Symmetry in quantum mechanics}}

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	'''Symmetry in quantum mechanics''' ~~is a Book I topic~~ in ~~the Quantum Collection. In~~ quantum mechanics, a symmetry is a transformation that leaves the physical properties of a system unchanged. In general, symmetry in physics, invariance, and conservation laws, are fundamentally important constraints for formulating physical theories and models. In practice, they are powerful methods for solving problems and predicting what can happen. Symmetries play a central role in determining conservation laws and the structure of quantum systems. In quantum mechanics, a symmetry is a transformation that leaves the physical properties of a system unchanged. In general, symmetry in physics, invariance, and conservation laws, are fundamentally important constraints for formulating physical theories and models. In practice, they are powerful methods for solving problems and predicting what can happen.		'''Symmetry in quantum mechanics''' in quantum mechanics, a symmetry is a transformation that leaves the physical properties of a system unchanged. In general, symmetry in physics, invariance, and conservation laws, are fundamentally important constraints for formulating physical theories and models. In practice, they are powerful methods for solving problems and predicting what can happen. Symmetries play a central role in determining conservation laws and the structure of quantum systems. In quantum mechanics, a symmetry is a transformation that leaves the physical properties of a system unchanged. In general, symmetry in physics, invariance, and conservation laws, are fundamentally important constraints for formulating physical theories and models. In practice, they are powerful methods for solving problems and predicting what can happen. A transformation is a symmetry if it preserves transition probabilities:
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	then the corresponding observable is conserved in time.<ref>{{cite book \|last=Ballentine \|first=Leslie E. \|title=Quantum Mechanics: A Modern Development \|publisher=World Scientific \|year=1998}}</ref>		then the corresponding observable is conserved in time.<ref>{{cite book \|last=Ballentine \|first=Leslie E. \|title=Quantum Mechanics: A Modern Development \|publisher=World Scientific \|year=1998}}</ref>

	This is the quantum analogue of ~~[[Wikipedia:Noether's theorem\|~~Noether’s theorem]].		This is the quantum analogue of Noether’s theorem.

	=== Physical significance ===		=== Physical significance ===
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	=== Spherical harmonics ===		=== Spherical harmonics ===

	In position space, the eigenfunctions of angular momentum are the ~~[[Wikipedia:Spherical harmonics\|~~spherical harmonics]] <math>Y_{lm}(\theta,\phi)</math>, which form an orthonormal basis on the sphere.		In position space, the eigenfunctions of angular momentum are the spherical harmonics <math>Y_{lm}(\theta,\phi)</math>, which form an orthonormal basis on the sphere.

	They arise naturally when solving the Schrödinger equation for systems with spherical symmetry, such as the hydrogen atom.		They arise naturally when solving the Schrödinger equation for systems with spherical symmetry, such as the hydrogen atom.
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	=== Spin-½ systems ===		=== Spin-½ systems ===

	For spin-½ particles, the spin operators can be written in terms of the ~~[[Wikipedia:~~Pauli matrices~~\|Pauli matrices]]~~:		For spin-½ particles, the spin operators can be written in terms of the Pauli matrices:

	<math>		<math>
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	'''Noether’s theorem''' establishes a fundamental connection between symmetries and conservation laws. In quantum mechanics, it states that every continuous symmetry of a system corresponds to a conserved observable.<ref>{{cite book \|last=Ballentine \|first=Leslie E. \|title=Quantum Mechanics: A Modern Development \|publisher=World Scientific \|year=1998}}</ref>		'''Noether’s theorem''' establishes a fundamental connection between symmetries and conservation laws. In quantum mechanics, it states that every continuous symmetry of a system corresponds to a conserved observable.<ref>{{cite book \|last=Ballentine \|first=Leslie E. \|title=Quantum Mechanics: A Modern Development \|publisher=World Scientific \|year=1998}}</ref>

	Originally formulated in classical mechanics by ~~[[Wikipedia:~~Emmy Noether~~\|Emmy Noether]]~~, the theorem extends naturally to quantum systems through operator methods.		Originally formulated in classical mechanics by Emmy Noether, the theorem extends naturally to quantum systems through operator methods.

	=== Symmetry and generators ===		=== Symmetry and generators ===

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	In quantum mechanics, a ~~'''~~symmetry~~'''~~ is a transformation that leaves the physical properties of a system unchanged. In general, symmetry in physics, ~~[[Wikipedia:invariant (physics)\|~~invariance]], and ~~[[Wikipedia:Conservation law (physics)\|~~conservation ~~law]]s~~, are fundamentally important constraints for formulating ~~[[Wikipedia:Theoretical physics\|~~physical theories]] and models. In practice, they are powerful methods for solving problems and predicting what can happen. Symmetries play a central role in determining conservation laws and the structure of quantum systems.~~<ref>{{cite book \|last=Sakurai \|first=J~~. J. ~~\|title=Modern Quantum Mechanics \|publisher=Addison-Wesley \|year=1994}}</ref>~~		'''Symmetry in quantum mechanics''' is a Book I topic in the Quantum Collection. In quantum mechanics, a symmetry is a transformation that leaves the physical properties of a system unchanged. In general, symmetry in physics, invariance, and conservation laws, are fundamentally important constraints for formulating physical theories and models. In practice, they are powerful methods for solving problems and predicting what can happen. Symmetries play a central role in determining conservation laws and the structure of quantum systems. In quantum mechanics, a symmetry is a transformation that leaves the physical properties of a system unchanged. In general, symmetry in physics, invariance, and conservation laws, are fundamentally important constraints for formulating physical theories and models. In practice, they are powerful methods for solving problems and predicting what can happen.
	~~{{Lie groups \|Physics}}~~
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	{{Author\|Harold Foppele}}		{{Author\|Harold Foppele}}

	{{Sourceattribution\|Quantum Symmetry in quantum mechanics\|1}}		{{Sourceattribution\|Physics:Quantum Symmetry in quantum mechanics\|1}}

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