Physics:Quantum state - Revision history

WikiHarold: Remove imported red links from Quantum page

2026-05-23T23:48:27Z

Remove imported red links from Quantum page

← Older revision		Revision as of 23:48, 23 May 2026
Line 23:		Line 23:

	== From the states of classical mechanics ==		== From the states of classical mechanics ==
	~~{{Quantum mechanics\|cTopic=Fundamental concepts}}~~
	As a tool for physics, quantum states grew out of states in classical mechanics. A classical dynamical state consists of a set of dynamical variables with well-defined real values at each instant of time.<ref name=messiah />{{rp\|3}} For example, the state of a cannon ball would consist of its position and velocity. The state values evolve under equations of motion and thus remain strictly determined. If we know the position of a cannon and the exit velocity of its projectiles, then we can use equations containing the force of gravity to predict the trajectory of a cannon ball precisely.		As a tool for physics, quantum states grew out of states in classical mechanics. A classical dynamical state consists of a set of dynamical variables with well-defined real values at each instant of time.<ref name=messiah />{{rp\|3}} For example, the state of a cannon ball would consist of its position and velocity. The state values evolve under equations of motion and thus remain strictly determined. If we know the position of a cannon and the exit velocity of its projectiles, then we can use equations containing the force of gravity to predict the trajectory of a cannon ball precisely.

Line 124:		Line 123:

	=== Spin ===		=== Spin ===
	The angular momentum has the same dimension (M·L{{sup\|2}}·T{{sup\|−1}}) as the Planck constant and, at quantum scale, behaves as a ''discrete'' degree of freedom of a quantum system. Most particles possess a kind of intrinsic angular momentum that does not appear at all in classical mechanics and arises from Dirac's relativistic generalization of the theory. Mathematically it is described with spinors. In non-relativistic quantum mechanics the group representations of the Lie group SU(2) are used to describe this additional freedom. For a given particle, the choice of representation (and hence the range of possible values of the spin observable) is specified by a non-negative number {{math\|''S''}} that, in units of the reduced Planck constant {{math\|''ħ''}}, is either an integer (0, 1, 2, ...) or a half-integer (1/2, 3/2, 5/2, ...). For a massive particle with spin {{math\|''S''}}, its spin quantum number {{math\|''m''}} always assumes one of the {{math\|2''S'' + 1}} possible values in the set		The angular momentum has the same dimension (M·L<sup>2</sup>·T<sup>−1</sup>) as the Planck constant and, at quantum scale, behaves as a ''discrete'' degree of freedom of a quantum system. Most particles possess a kind of intrinsic angular momentum that does not appear at all in classical mechanics and arises from Dirac's relativistic generalization of the theory. Mathematically it is described with spinors. In non-relativistic quantum mechanics the group representations of the Lie group SU(2) are used to describe this additional freedom. For a given particle, the choice of representation (and hence the range of possible values of the spin observable) is specified by a non-negative number {{math\|''S''}} that, in units of the reduced Planck constant {{math\|''ħ''}}, is either an integer (0, 1, 2, ...) or a half-integer (1/2, 3/2, 5/2, ...). For a massive particle with spin {{math\|''S''}}, its spin quantum number {{math\|''m''}} always assumes one of the {{math\|2''S'' + 1}} possible values in the set
	<math display="block">\{ -S, -S+1, \ldots, S-1, S \}</math>		<math display="block">\{ -S, -S+1, \ldots, S-1, S \}</math>

Line 167:		Line 166:

	=== Superposition of pure states ===		=== Superposition of pure states ===
	~~{{Main\|Physics:Quantum superposition}}~~
	As mentioned above, quantum states may be superposed. If <math>\|\alpha\rangle</math> and <math>\|\beta\rangle</math> are two kets corresponding to quantum states, the ket		As mentioned above, quantum states may be superposed. If <math>\|\alpha\rangle</math> and <math>\|\beta\rangle</math> are two kets corresponding to quantum states, the ket
	<math display="block">c_\alpha\|\alpha\rang + c_\beta\|\beta\rang</math>		<math display="block">c_\alpha\|\alpha\rang + c_\beta\|\beta\rang</math>
Line 230:		Line 228:
	* {{cite book \|title=Geometry of Quantum States \|title-link=Geometry of Quantum States \|publisher=Cambridge University Press \|location=Cambridge \|year=2006\|author1 = Bengtsson I\|author2 = Życzkowski K\|}}, [https://chaos.if.uj.edu.pl/~karol/geometry.htm second, revised edition (2017)]		* {{cite book \|title=Geometry of Quantum States \|title-link=Geometry of Quantum States \|publisher=Cambridge University Press \|location=Cambridge \|year=2006\|author1 = Bengtsson I\|author2 = Życzkowski K\|}}, [https://chaos.if.uj.edu.pl/~karol/geometry.htm second, revised edition (2017)]
	{{DEFAULTSORT:Quantum State}}		{{DEFAULTSORT:Quantum State}}
	~~[[Category:Quantum states\| ]]~~

	{{Author\|Harold Foppele}}		{{Author\|Harold Foppele}}

	{{Sourceattribution\|Physics:Quantum state\|1}}		{{Sourceattribution\|Physics:Quantum state\|1}}

WikiHarold: Clean Quantum page image and red links

2026-05-23T23:35:43Z

Clean Quantum page image and red links

← Older revision		Revision as of 23:35, 23 May 2026
Line 229:		Line 229:
	For a discussion of geometric aspects see:		For a discussion of geometric aspects see:
	* {{cite book \|title=Geometry of Quantum States \|title-link=Geometry of Quantum States \|publisher=Cambridge University Press \|location=Cambridge \|year=2006\|author1 = Bengtsson I\|author2 = Życzkowski K\|}}, [https://chaos.if.uj.edu.pl/~karol/geometry.htm second, revised edition (2017)]		* {{cite book \|title=Geometry of Quantum States \|title-link=Geometry of Quantum States \|publisher=Cambridge University Press \|location=Cambridge \|year=2006\|author1 = Bengtsson I\|author2 = Życzkowski K\|}}, [https://chaos.if.uj.edu.pl/~karol/geometry.htm second, revised edition (2017)]

	~~{{Quantum mechanics topics}}~~

	{{DEFAULTSORT:Quantum State}}		{{DEFAULTSORT:Quantum State}}
	[[Category:Quantum states\| ]]		[[Category:Quantum states\| ]]

WikiHarold: Use Quantum See also index module

2026-05-23T22:26:07Z

Use Quantum See also index module

@@ Line 208: / Line 208: @@
 == See also ==
-{{cols|colwidth=26em}}
+{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}
 == Notes  ==
 {{notelist}}
 == References ==

Maintenance script: Point hydrogen atom link to Quantum Collection page

2026-05-23T10:40:50Z

Point hydrogen atom link to Quantum Collection page

← Older revision		Revision as of 10:40, 23 May 2026
Line 61:		Line 61:
	[[File:Hydrogen Density Plots.png\|thumb\|450px\|Probability densities for the electron of a hydrogen atom in different quantum states.]]		[[File:Hydrogen Density Plots.png\|thumb\|450px\|Probability densities for the electron of a hydrogen atom in different quantum states.]]
	Numerical or analytic solutions in quantum mechanics can be expressed as '''pure states'''. These solution states, called eigenstates, are labeled with quantized values, typically quantum numbers.		Numerical or analytic solutions in quantum mechanics can be expressed as '''pure states'''. These solution states, called eigenstates, are labeled with quantized values, typically quantum numbers.
	For example, when dealing with the energy spectrum of the electron in a [[Physics:~~Hydrogen atom~~\|hydrogen atom]], the relevant pure states are identified by the principal quantum number {{math\|''n''}}, the angular momentum quantum number {{math\|''ℓ''}}, the magnetic quantum number {{math\|''m''}}, and the spin ''z''-component {{math\|''s''<sub>''z''</sub>}}. For another example, if the spin of an electron is measured in any direction, e.g. with a Stern–Gerlach experiment, there are two possible results: up or down. A pure state here is represented by a two-dimensional complex vector <math>(\alpha, \beta)</math>, with a length of one; that is, with		For example, when dealing with the energy spectrum of the electron in a [[Physics:Quantum atoms/hydrogen\|hydrogen atom]], the relevant pure states are identified by the principal quantum number {{math\|''n''}}, the angular momentum quantum number {{math\|''ℓ''}}, the magnetic quantum number {{math\|''m''}}, and the spin ''z''-component {{math\|''s''<sub>''z''</sub>}}. For another example, if the spin of an electron is measured in any direction, e.g. with a Stern–Gerlach experiment, there are two possible results: up or down. A pure state here is represented by a two-dimensional complex vector <math>(\alpha, \beta)</math>, with a length of one; that is, with
	<math display="block">\|\alpha\|^2 + \|\beta\|^2 = 1,</math>		<math display="block">\|\alpha\|^2 + \|\beta\|^2 = 1,</math>
	where <math>\|\alpha\|</math> and <math>\|\beta\|</math> are the absolute values of <math>\alpha</math> and <math>\beta</math>.		where <math>\|\alpha\|</math> and <math>\|\beta\|</math> are the absolute values of <math>\alpha</math> and <math>\beta</math>.

Maintenance script: Normalize quantum page header order

2026-05-22T11:31:04Z

Normalize quantum page header order

← Older revision		Revision as of 11:31, 22 May 2026
Line 2:		Line 2:
	{{Quantum book backlink\|Foundations}}		{{Quantum book backlink\|Foundations}}
	{{Quantum article nav\|previous=Physics:Quantum mechanics measurements\|previous label=Mechanics measurements\|next=Physics:Quantum system\|next label=System}}		{{Quantum article nav\|previous=Physics:Quantum mechanics measurements\|previous label=Mechanics measurements\|next=Physics:Quantum system\|next label=System}}


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

2026-05-22T11:01:34Z

Remove hidden BOM characters and direct Book label after Short description

← Older revision		Revision as of 11:01, 22 May 2026
Line 1:		Line 1:
	{{Short description\|Mathematical entity to describe the probability of each possible measurement on a system}}~~Book I~~		{{Short description\|Mathematical entity to describe the probability of each possible measurement on a system}}
	{{Quantum book backlink\|Foundations}}		{{Quantum book backlink\|Foundations}}
	{{Quantum article nav\|previous=Physics:Quantum mechanics measurements\|previous label=Mechanics measurements\|next=Physics:Quantum system\|next label=System}}		{{Quantum article nav\|previous=Physics:Quantum mechanics measurements\|previous label=Mechanics measurements\|next=Physics:Quantum system\|next label=System}}

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

2026-05-22T10:46:39Z

Clean Short description prefix and add Book I label

← Older revision		Revision as of 10:46, 22 May 2026
Line 1:		Line 1:
	~~~~{{Short description\|Mathematical entity to describe the probability of each possible measurement on a system}}		{{Short description\|Mathematical entity to describe the probability of each possible measurement on a system}}Book I
	{{Quantum book backlink\|Foundations}}		{{Quantum book backlink\|Foundations}}
	{{Quantum article nav\|previous=Physics:Quantum mechanics measurements\|previous label=Mechanics measurements\|next=Physics:Quantum system\|next label=System}}		{{Quantum article nav\|previous=Physics:Quantum mechanics measurements\|previous label=Mechanics measurements\|next=Physics:Quantum system\|next label=System}}

Maintenance script: Add Foundations previous-next navigation

2026-05-20T12:15:21Z

Add Foundations previous-next navigation

← Older revision		Revision as of 12:15, 20 May 2026
Line 1:		Line 1:
	~~~~{{Short description\|Mathematical entity to describe the probability of each possible measurement on a system}}		{{Short description\|Mathematical entity to describe the probability of each possible measurement on a system}}
	{{Quantum book backlink\|Foundations}}		{{Quantum book backlink\|Foundations}}
			{{Quantum article nav\|previous=Physics:Quantum mechanics measurements\|previous label=Mechanics measurements\|next=Physics:Quantum system\|next label=System}}

Maintenance script: Clean missing Book I links and template output

2026-05-20T08:46:34Z

Clean missing Book I links and template output

← Older revision		Revision as of 08:46, 20 May 2026
Line 1:		Line 1:
	{{Short description\|Mathematical entity to describe the probability of each possible measurement on a system}}		{{Short description\|Mathematical entity to describe the probability of each possible measurement on a system}}
	{{Quantum book backlink\|Foundations}}		{{Quantum book backlink\|Foundations}}

Line 36:		Line 36:
	== Measurements ==		== Measurements ==
	<!-- summary only -->		<!-- summary only -->
	~~{{main \| Physics~~:Measurement in quantum mechanics}}		''Related topic:'' Measurement in quantum mechanics

	Measurements, macroscopic operations on quantum states, filter the state.<ref name=messiah />{{rp\|196}} Whatever the input quantum state might be, repeated identical measurements give consistent values. For this reason, measurements 'prepare' quantum states for experiments, placing the system in a partially defined state. Subsequent measurements may either further prepare the system – these are compatible measurements – or it may alter the state, redefining it – these are called incompatible or complementary measurements. For example, we may measure the momentum of a state along the <math>x</math> axis any number of times and get the same result, but if we measure the position after once measuring the momentum, subsequent measurements of momentum are changed. The quantum state appears unavoidably altered by incompatible measurements. This is known as the uncertainty principle.		Measurements, macroscopic operations on quantum states, filter the state.<ref name=messiah />{{rp\|196}} Whatever the input quantum state might be, repeated identical measurements give consistent values. For this reason, measurements 'prepare' quantum states for experiments, placing the system in a partially defined state. Subsequent measurements may either further prepare the system – these are compatible measurements – or it may alter the state, redefining it – these are called incompatible or complementary measurements. For example, we may measure the momentum of a state along the <math>x</math> axis any number of times and get the same result, but if we measure the position after once measuring the momentum, subsequent measurements of momentum are changed. The quantum state appears unavoidably altered by incompatible measurements. This is known as the uncertainty principle.
Line 54:		Line 54:

	== Wave function representations ==		== Wave function representations ==
	~~{{main\|~~Wave function}}		''Related topic:'' Wave function
	Wave functions represent quantum states, particularly when they are functions of position or of momentum. Historically, definitions of quantum states used wavefunctions before the more formal methods were developed.<ref name="Whittaker">{{cite book \| last=Whittaker \| first=Sir Edmund \| title=A History of the Theories of Aether and Electricity \| publisher=Courier Dover Publications \| date=1989-01-01 \| isbn=0-486-26126-3 \| volume=2 \| page=87}}</ref>{{rp\|268}} The wave function is a complex-valued function of any complete set of commuting or compatible degrees of freedom. For example, one set could be the <math>x,y,z</math> spatial coordinates of an electron.		Wave functions represent quantum states, particularly when they are functions of position or of momentum. Historically, definitions of quantum states used wavefunctions before the more formal methods were developed.<ref name="Whittaker">{{cite book \| last=Whittaker \| first=Sir Edmund \| title=A History of the Theories of Aether and Electricity \| publisher=Courier Dover Publications \| date=1989-01-01 \| isbn=0-486-26126-3 \| volume=2 \| page=87}}</ref>{{rp\|268}} The wave function is a complex-valued function of any complete set of commuting or compatible degrees of freedom. For example, one set could be the <math>x,y,z</math> spatial coordinates of an electron.
	Preparing a system by measuring the complete set of compatible observables produces a '''pure quantum state'''. More common, incomplete preparation produces a '''mixed quantum state'''. Wave function solutions of Schrödinger's equations of motion for operators corresponding to measurements can readily be expressed as pure states; they must be combined with statistical weights matching experimental preparation to compute the expected probability distribution.<ref name=messiah />{{rp\|205}}		Preparing a system by measuring the complete set of compatible observables produces a '''pure quantum state'''. More common, incomplete preparation produces a '''mixed quantum state'''. Wave function solutions of Schrödinger's equations of motion for operators corresponding to measurements can readily be expressed as pure states; they must be combined with statistical weights matching experimental preparation to compute the expected probability distribution.<ref name=messiah />{{rp\|205}}
Line 117:		Line 117:
	== Formalism in quantum physics ==		== Formalism in quantum physics ==

	~~{{See also\|Physics~~:Mathematical formulation of quantum mechanics}}		''Related topic:'' Mathematical formulation of quantum mechanics

	=== Pure states as rays in a complex Hilbert space ===		=== Pure states as rays in a complex Hilbert space ===
Line 184:		Line 184:

	=== Mixed states ===		=== Mixed states ===
	~~{{Main\|~~Density matrix}}		''Related topic:'' Density matrix

	A ''pure quantum state'' is a state which can be described by a single ket vector, as described above. A ''mixed quantum state'' is a statistical ensemble of pure states (see ''[[Physics:Quantum statistical mechanics\|Quantum statistical mechanics]]'').<ref name="peres" />{{Rp\|page=73}}		A ''pure quantum state'' is a state which can be described by a single ket vector, as described above. A ''mixed quantum state'' is a statistical ensemble of pure states (see ''[[Physics:Quantum statistical mechanics\|Quantum statistical mechanics]]'').<ref name="peres" />{{Rp\|page=73}}

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

2026-05-20T08:33:50Z

Clean Book I red links, intro, and image slots

Show changes

← Older revision		Revision as of 10:40, 23 May 2026
Line 61:		Line 61:
	[[File:Hydrogen Density Plots.png\|thumb\|450px\|Probability densities for the electron of a hydrogen atom in different quantum states.]]		[[File:Hydrogen Density Plots.png\|thumb\|450px\|Probability densities for the electron of a hydrogen atom in different quantum states.]]
	Numerical or analytic solutions in quantum mechanics can be expressed as '''pure states'''. These solution states, called eigenstates, are labeled with quantized values, typically quantum numbers.		Numerical or analytic solutions in quantum mechanics can be expressed as '''pure states'''. These solution states, called eigenstates, are labeled with quantized values, typically quantum numbers.
	For example, when dealing with the energy spectrum of the electron in a [[Physics:~~Hydrogen atom~~\|hydrogen atom]], the relevant pure states are identified by the principal quantum number {{math\|''n''}}, the angular momentum quantum number {{math\|''ℓ''}}, the magnetic quantum number {{math\|''m''}}, and the spin ''z''-component {{math\|''s''<sub>''z''</sub>}}. For another example, if the spin of an electron is measured in any direction, e.g. with a Stern–Gerlach experiment, there are two possible results: up or down. A pure state here is represented by a two-dimensional complex vector <math>(\alpha, \beta)</math>, with a length of one; that is, with		For example, when dealing with the energy spectrum of the electron in a [[Physics:Quantum atoms/hydrogen\|hydrogen atom]], the relevant pure states are identified by the principal quantum number {{math\|''n''}}, the angular momentum quantum number {{math\|''ℓ''}}, the magnetic quantum number {{math\|''m''}}, and the spin ''z''-component {{math\|''s''<sub>''z''</sub>}}. For another example, if the spin of an electron is measured in any direction, e.g. with a Stern–Gerlach experiment, there are two possible results: up or down. A pure state here is represented by a two-dimensional complex vector <math>(\alpha, \beta)</math>, with a length of one; that is, with
	<math display="block">\|\alpha\|^2 + \|\beta\|^2 = 1,</math>		<math display="block">\|\alpha\|^2 + \|\beta\|^2 = 1,</math>
	where <math>\|\alpha\|</math> and <math>\|\beta\|</math> are the absolute values of <math>\alpha</math> and <math>\beta</math>.		where <math>\|\alpha\|</math> and <math>\|\beta\|</math> are the absolute values of <math>\alpha</math> and <math>\beta</math>.

← Older revision		Revision as of 11:01, 22 May 2026
Line 1:		Line 1:
	{{Short description\|Mathematical entity to describe the probability of each possible measurement on a system}}~~Book I~~		{{Short description\|Mathematical entity to describe the probability of each possible measurement on a system}}
	{{Quantum book backlink\|Foundations}}		{{Quantum book backlink\|Foundations}}
	{{Quantum article nav\|previous=Physics:Quantum mechanics measurements\|previous label=Mechanics measurements\|next=Physics:Quantum system\|next label=System}}		{{Quantum article nav\|previous=Physics:Quantum mechanics measurements\|previous label=Mechanics measurements\|next=Physics:Quantum system\|next label=System}}

← Older revision		Revision as of 10:46, 22 May 2026
Line 1:		Line 1:
	~~~~{{Short description\|Mathematical entity to describe the probability of each possible measurement on a system}}		{{Short description\|Mathematical entity to describe the probability of each possible measurement on a system}}Book I
	{{Quantum book backlink\|Foundations}}		{{Quantum book backlink\|Foundations}}
	{{Quantum article nav\|previous=Physics:Quantum mechanics measurements\|previous label=Mechanics measurements\|next=Physics:Quantum system\|next label=System}}		{{Quantum article nav\|previous=Physics:Quantum mechanics measurements\|previous label=Mechanics measurements\|next=Physics:Quantum system\|next label=System}}

← Older revision		Revision as of 12:15, 20 May 2026
Line 1:		Line 1:
	~~~~{{Short description\|Mathematical entity to describe the probability of each possible measurement on a system}}		{{Short description\|Mathematical entity to describe the probability of each possible measurement on a system}}
	{{Quantum book backlink\|Foundations}}		{{Quantum book backlink\|Foundations}}
			{{Quantum article nav\|previous=Physics:Quantum mechanics measurements\|previous label=Mechanics measurements\|next=Physics:Quantum system\|next label=System}}