Physics:Quantum Angular momentum operator - Revision history

Maintenance script: Normalize quantum page header order

2026-05-22T11:31:39Z

Normalize quantum page header order

← Older revision		Revision as of 11:31, 22 May 2026
Line 2:		Line 2:
	{{Quantum book backlink\|Mathematical structure and systems}}		{{Quantum book backlink\|Mathematical structure and systems}}
	{{Quantum article nav\|previous=Physics:Quantum Noether theorem\|previous label=Noether theorem\|next=Physics:Quantum Runge–Lenz vector\|next label=Runge–Lenz vector}}		{{Quantum article nav\|previous=Physics:Quantum Noether theorem\|previous label=Noether theorem\|next=Physics:Quantum Runge–Lenz vector\|next label=Runge–Lenz vector}}

	<div style="display:flex; gap:24px; align-items:flex-start; max-width:1200px;">		<div style="display:flex; gap:24px; align-items:flex-start; max-width:1200px;">

Maintenance script: Clean Book label remnants and backlink spacing

2026-05-22T11:09:57Z

Clean Book label remnants and backlink spacing

← Older revision		Revision as of 11:09, 22 May 2026
Line 1:		Line 1:
	{{Short description\|Quantum-mechanical operator associated with rotational symmetry}}		{{Short description\|Quantum-mechanical operator associated with rotational symmetry}}

	{{Quantum book backlink\|Mathematical structure and systems}}		{{Quantum book backlink\|Mathematical structure and systems}}
	{{Quantum article nav\|previous=Physics:Quantum Noether theorem\|previous label=Noether theorem\|next=Physics:Quantum Runge–Lenz vector\|next label=Runge–Lenz vector}}		{{Quantum article nav\|previous=Physics:Quantum Noether theorem\|previous label=Noether theorem\|next=Physics:Quantum Runge–Lenz vector\|next label=Runge–Lenz vector}}

Maintenance script: Remove hidden BOM characters and direct Book label after Short description

2026-05-22T11:02:09Z

Remove hidden BOM characters and direct Book label after Short description

← Older revision		Revision as of 11:02, 22 May 2026
Line 1:		Line 1:
	{{Short description\|Quantum-mechanical operator associated with rotational symmetry}}~~Book I~~		{{Short description\|Quantum-mechanical operator associated with rotational symmetry}}

	{{Quantum book backlink\|Mathematical structure and systems}}		{{Quantum book backlink\|Mathematical structure and systems}}

Maintenance script: Clean Short description prefix and add Book I label

2026-05-22T10:47:12Z

Clean Short description prefix and add Book I label

← Older revision		Revision as of 10:47, 22 May 2026
Line 1:		Line 1:
	~~~~{{Short description\|Quantum-mechanical operator associated with rotational symmetry}}		{{Short description\|Quantum-mechanical operator associated with rotational symmetry}}Book I

	{{Quantum book backlink\|Mathematical structure and systems}}		{{Quantum book backlink\|Mathematical structure and systems}}

Maintenance script: Apply continuous Quantum previous-next navigation

2026-05-20T21:58:44Z

Apply continuous Quantum previous-next navigation

← Older revision		Revision as of 21:58, 20 May 2026
Line 1:		Line 1:
	~~~~{{Short description\|Quantum-mechanical operator associated with rotational symmetry}}		{{Short description\|Quantum-mechanical operator associated with rotational symmetry}}

	{{Quantum book backlink\|Mathematical structure and systems}}		{{Quantum book backlink\|Mathematical structure and systems}}
	{{Quantum article nav\|previous=Physics:Quantum ~~Symmetry in quantum mechanics~~\|previous label=~~Symmetry in quantum mechanics~~\|next=Physics:Quantum Runge–Lenz vector\|next label=Runge–Lenz vector}}		{{Quantum article nav\|previous=Physics:Quantum Noether theorem\|previous label=Noether theorem\|next=Physics:Quantum Runge–Lenz vector\|next label=Runge–Lenz vector}}

	<div style="display:flex; gap:24px; align-items:flex-start; max-width:1200px;">		<div style="display:flex; gap:24px; align-items:flex-start; max-width:1200px;">

Maintenance script: Apply Quantum previous-next navigation

2026-05-20T12:23:28Z

Apply Quantum previous-next navigation

← Older revision		Revision as of 12:23, 20 May 2026
Line 1:		Line 1:
	~~~~{{Short description\|Quantum-mechanical operator associated with rotational symmetry}}		{{Short description\|Quantum-mechanical operator associated with rotational symmetry}}

	{{Quantum book backlink\|Mathematical structure and systems}}		{{Quantum book backlink\|Mathematical structure and systems}}
			{{Quantum article nav\|previous=Physics:Quantum Symmetry in quantum mechanics\|previous label=Symmetry in quantum mechanics\|next=Physics:Quantum Runge–Lenz vector\|next label=Runge–Lenz vector}}

	<div style="display:flex; gap:24px; align-items:flex-start; max-width:1200px;">		<div style="display:flex; gap:24px; align-items:flex-start; max-width:1200px;">

Maintenance script: Improve Book I intro quality

2026-05-20T09:03:03Z

Improve Book I intro quality

← Older revision		Revision as of 09:03, 20 May 2026
Line 1:		Line 1:
	{{Short description\|Quantum-mechanical operator associated with rotational symmetry}}		{{Short description\|Quantum-mechanical operator associated with rotational symmetry}}

	{{Quantum book backlink\|Mathematical structure and systems}}		{{Quantum book backlink\|Mathematical structure and systems}}
Line 10:		Line 10:

	<div style="flex:1; line-height:1.45; color:#006b45; column-count:2; column-gap:32px; column-rule:1px solid #b8d8c8;">		<div style="flex:1; line-height:1.45; color:#006b45; column-count:2; column-gap:32px; column-rule:1px solid #b8d8c8;">
	~~In quantum mechanics, the~~ '''~~angular~~ momentum operator''' is the operator associated with rotational motion and rotational symmetry, and is the quantum analogue of angular momentum in classical mechanics. Classically, angular momentum is described by a vector <math>\mathbf{L} = (L_x, L_y, L_z)</math>, whose components can all be specified simultaneously. In quantum mechanics, these components become operators <math>\hat{L}_x, \hat{L}_y, \hat{L}_z</math>, representing measurements of angular momentum along each axis. However, unlike in classical physics, these operators do not commute, satisfying relations such as <math>[\hat{L}_x, \hat{L}_y] = i\hbar \hat{L}_z</math> (and cyclic permutations), which implies that the components <math>L_x, L_y, L_z</math> cannot be simultaneously known exactly. Instead, one can simultaneously determine the total angular momentum <math>\hat{L}^2 = \hat{L}_x^2 + \hat{L}_y^2 + \hat{L}_z^2</math> and a single component, usually <math>\hat{L}_z</math>. Quantum states are therefore labeled by the quantum numbers <math>\ell</math> (total angular momentum) and <math>m</math> (its projection along a chosen axis). Geometrically, this corresponds to a situation in which the magnitude of the angular momentum vector is well-defined while only one of its directional components is sharp, so the vector cannot be assigned a definite direction in three-dimensional space. Instead, it is often visualized as lying on a sphere of fixed radius (set by ~~<math>~~L^2~~</math>~~) with uncertainty in its orientation, forming a “cone” of possible directions. As an observable, angular momentum is represented by operators whose eigenstates correspond to states with definite angular momentum, and whose eigenvalues give the possible results of measurement; it plays a central role in atomic physics, molecular physics, spectroscopy, and quantum theory more generally.		'''Angular momentum operator''' in quantum mechanics, the angular momentum operator is the operator associated with rotational motion and rotational symmetry, and is the quantum analogue of angular momentum in classical mechanics. Geometrically, this corresponds to a situation in which the magnitude of the angular momentum vector is well-defined while only one of its directional components is sharp, so the vector cannot be assigned a definite direction in three-dimensional space. Instead, it is often visualized as lying on a sphere of fixed radius (set by L^2) with uncertainty in its orientation, forming a “cone” of possible directions. As an observable, angular momentum is represented by operators whose eigenstates correspond to states with definite angular momentum, and whose eigenvalues give the possible results of measurement; it plays a central role in atomic physics, molecular physics, spectroscopy, and quantum theory more generally. Angular momentum is one of the fundamental conserved quantities of motion, together with linear momentum and energy. Depending on context, the expression angular momentum operator may refer either to orbital angular momentum or to total angular momentum. For a closed system, the total angular momentum is conserved, in accordance with rotational symmetry and Noether's theorem. In quantum mechanics, angular momentum appears in three closely related forms: orbital angular momentum, spin angular momentum, and total angular momentum.

	Angular momentum is one of the fundamental conserved quantities of motion, together with linear momentum and energy.~~<ref name="Liboff">Introductory Quantum Mechanics, Richard L. Liboff, 2nd Edition, {{ISBN\|0-201-54715-5}}</ref> In quantum mechanics, several related angular momentum operators appear:~~

	* the '''orbital angular momentum''' operator, usually denoted <math>\mathbf{L}</math>,
	* the '''spin angular momentum''' operator, usually denoted <math>\mathbf{S}</math>,
	* the '''total angular momentum''' operator, usually denoted <math>\mathbf{J}</math>.

	~~These are related by~~
	~~<math display="block">\mathbf{J}=\mathbf{L}+\mathbf{S}.</math>~~

	Depending on context, the expression ''angular momentum operator'' may refer either to orbital angular momentum or to total angular momentum. For a closed system, the total angular momentum is conserved, in accordance with rotational symmetry and Noether's theorem.
	{{quantum mechanics}}
	</div>		</div>

Maintenance script: Clean missing Book I links and template output

2026-05-20T08:46:57Z

Clean missing Book I links and template output

Show changes

Maintenance script: Clean Book I red links, intro, and image slots

2026-05-20T08:12:38Z

Clean Book I red links, intro, and image slots

Show changes

Maintenance script: Normalize Book I Quantum page structure

2026-05-19T22:48:41Z

Normalize Book I Quantum page structure

← Older revision		Revision as of 22:48, 19 May 2026
Line 311:		Line 311:
	[[Category:Rotational symmetry]]		[[Category:Rotational symmetry]]

	{{Sourceattribution\|Quantum Angular momentum operator\|1}}		{{Sourceattribution\|Physics:Quantum Angular momentum operator\|1}}

← Older revision		Revision as of 11:09, 22 May 2026
Line 1:		Line 1:
	{{Short description\|Quantum-mechanical operator associated with rotational symmetry}}		{{Short description\|Quantum-mechanical operator associated with rotational symmetry}}

	{{Quantum book backlink\|Mathematical structure and systems}}		{{Quantum book backlink\|Mathematical structure and systems}}
	{{Quantum article nav\|previous=Physics:Quantum Noether theorem\|previous label=Noether theorem\|next=Physics:Quantum Runge–Lenz vector\|next label=Runge–Lenz vector}}		{{Quantum article nav\|previous=Physics:Quantum Noether theorem\|previous label=Noether theorem\|next=Physics:Quantum Runge–Lenz vector\|next label=Runge–Lenz vector}}