Physics:Quantum Partition function - Revision history

WikiHarold: Restore missing Quantum reference definitions

2026-05-24T00:31:23Z

Restore missing Quantum reference definitions

← Older revision		Revision as of 00:31, 24 May 2026
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	</math>		</math>

	Here <math>k_B</math> is Boltzmann’s constant, <math>T</math> is the absolute temperature, and the trace is taken over the system’s Hilbert space.<ref name="Pathria"/><ref name="Reif">{{cite book		Here <math>k_B</math> is Boltzmann’s constant, <math>T</math> is the absolute temperature, and the trace is taken over the system’s Hilbert space.<ref name="Pathria">{{cite book
			\|last1=Pathria
			\|first1=R. K.
			\|last2=Beale
			\|first2=Paul D.
			\|title=Statistical Mechanics
			\|edition=3
			\|publisher=Elsevier
			\|year=2011
			\|isbn=9780123821881
			}}</ref><ref name="Reif">{{cite book
	\|last=Reif		\|last=Reif
	\|first=Frederick		\|first=Frederick
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	</math>		</math>

	This shows that each energy level contributes with a Boltzmann weight determined by its energy.<ref name="Tolman"/><ref name="Pathria"/>		This shows that each energy level contributes with a Boltzmann weight determined by its energy.<ref name="Tolman">{{cite book
			\|last=Tolman
			\|first=Richard C.
			\|title=The Principles of Statistical Mechanics
			\|publisher=Dover Publications
			\|year=1979
			\|isbn=9780486638966
			}}</ref><ref name="Pathria"/>

	==Physical meaning==		==Physical meaning==
	The partition function summarizes the thermal accessibility of all possible quantum states of the system.<ref name="LandauLifshitz"/> Low-energy states contribute most strongly at low temperature, while many higher-energy states become thermally populated as the temperature increases.<ref name="Reif"/>		The partition function summarizes the thermal accessibility of all possible quantum states of the system.<ref name="LandauLifshitz">{{cite book
			\|last1=Landau
			\|first1=L. D.
			\|last2=Lifshitz
			\|first2=E. M.
			\|title=Statistical Physics, Part 1
			\|edition=3
			\|publisher=Butterworth-Heinemann
			\|year=1980
			\|isbn=9780750633727
			}}</ref> Low-energy states contribute most strongly at low temperature, while many higher-energy states become thermally populated as the temperature increases.<ref name="Reif"/>

	Once <math>Z</math> is known, the equilibrium probability of finding the system in eigenstate <math>\|n\rangle</math> is		Once <math>Z</math> is known, the equilibrium probability of finding the system in eigenstate <math>\|n\rangle</math> is
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	{{#invoke:PhysicsQC\|tocHeadingAndList\|Physics:Quantum basics/See also}}		{{#invoke:PhysicsQC\|tocHeadingAndList\|Physics:Quantum basics/See also}}

	==References==		= References =
	{{reflist\|3}}		{{reflist\|3}}
	{{Author\|Harold Foppele}}		{{Author\|Harold Foppele}}

Maintenance script: Normalize quantum page header order

2026-05-22T11:33:45Z

Normalize quantum page header order

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	{{Quantum book backlink\|Statistical mechanics and kinetic theory}}		{{Quantum book backlink\|Statistical mechanics and kinetic theory}}
	{{Quantum article nav\|previous=Physics:Quantum Statistical mechanics\|previous label=Statistical mechanics\|next=Physics:Quantum Distribution functions\|next label=Distribution functions}}		{{Quantum article nav\|previous=Physics:Quantum Statistical mechanics\|previous label=Statistical mechanics\|next=Physics:Quantum Distribution functions\|next label=Distribution functions}}

	<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

2026-05-22T11:12:03Z

Clean Book label remnants and backlink spacing

← Older revision		Revision as of 11:12, 22 May 2026
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	{{Short description\|Quantum Collection topic on Quantum Partition function}}		{{Short description\|Quantum Collection topic on Quantum Partition function}}

	{{Quantum book backlink\|Statistical mechanics and kinetic theory}}		{{Quantum book backlink\|Statistical mechanics and kinetic theory}}
	{{Quantum article nav\|previous=Physics:Quantum Statistical mechanics\|previous label=Statistical mechanics\|next=Physics:Quantum Distribution functions\|next label=Distribution functions}}		{{Quantum article nav\|previous=Physics:Quantum Statistical mechanics\|previous label=Statistical mechanics\|next=Physics:Quantum Distribution functions\|next label=Distribution functions}}

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

2026-05-22T11:04:16Z

Remove hidden BOM characters and direct Book label after Short description

← Older revision		Revision as of 11:04, 22 May 2026
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	{{Short description\|Quantum Collection topic on Quantum Partition function}}~~Book I~~		{{Short description\|Quantum Collection topic on Quantum Partition function}}

	{{Quantum book backlink\|Statistical mechanics and kinetic theory}}		{{Quantum book backlink\|Statistical mechanics and kinetic theory}}

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

2026-05-22T10:49:17Z

Clean Short description prefix and add Book I label

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	~~~~{{Short description\|Quantum Collection topic on Quantum Partition function}}		{{Short description\|Quantum Collection topic on Quantum Partition function}}Book I

	{{Quantum book backlink\|Statistical mechanics and kinetic theory}}		{{Quantum book backlink\|Statistical mechanics and kinetic theory}}

Maintenance script: Apply Quantum previous-next navigation

2026-05-20T12:26:29Z

Apply Quantum previous-next navigation

← Older revision		Revision as of 12:26, 20 May 2026
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	{{Short description\|Quantum Collection topic on Quantum Partition function}}		{{Short description\|Quantum Collection topic on Quantum Partition function}}

	{{Quantum book backlink\|Statistical mechanics and kinetic theory}}		{{Quantum book backlink\|Statistical mechanics and kinetic theory}}
			{{Quantum article nav\|previous=Physics:Quantum Statistical mechanics\|previous label=Statistical mechanics\|next=Physics:Quantum Distribution functions\|next label=Distribution functions}}

	<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

2026-05-20T09:05:19Z

Improve Book I intro quality

← Older revision		Revision as of 09:05, 20 May 2026
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	{{Short description\|Quantum Collection topic on Quantum Partition function}}		{{Short description\|Quantum Collection topic on Quantum Partition function}}

	{{Quantum book backlink\|Statistical mechanics and kinetic theory}}		{{Quantum book backlink\|Statistical mechanics and kinetic theory}}
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	<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;">
	'''Partition function''' ~~is a Book I topic in the Quantum Collection. Quantum~~ partition function is the central quantity of quantum statistical mechanics, encoding how the energy eigenstates of a quantum system are thermally populated at temperature T. It provides the bridge between microscopic quantum states and macroscopic thermodynamic quantities such as internal energy, entropy, Helmholtz free energy, and heat capacity. Quantum partition function is the central quantity of quantum statistical mechanics, encoding how the energy eigenstates of a quantum system are thermally populated at temperature T.~~{{cite book }} It provides the bridge between microscopic quantum states and macroscopic thermodynamic quantities such as internal energy, entropy, Helmholtz free energy, and heat capacity.{{cite book~~ For a quantum system in thermal equilibrium with Hamiltonian H, the canonical partition function is defined by Here k_B is Boltzmann’s constant, T is the absolute temperature, and the trace is taken over the system’s Hilbert space.{{cite book If the Hamiltonian has discrete energy eigenstates \|n\rangle with eigenvalues E_n, then the trace becomes This shows that each energy level contributes with a Boltzmann weight determined by its energy.		'''Partition function''' quantum partition function is the central quantity of quantum statistical mechanics, encoding how the energy eigenstates of a quantum system are thermally populated at temperature T. It provides the bridge between microscopic quantum states and macroscopic thermodynamic quantities such as internal energy, entropy, Helmholtz free energy, and heat capacity. Quantum partition function is the central quantity of quantum statistical mechanics, encoding how the energy eigenstates of a quantum system are thermally populated at temperature T. For a quantum system in thermal equilibrium with Hamiltonian H, the canonical partition function is defined by This shows that each energy level contributes with a Boltzmann weight determined by its energy.
	</div>		</div>

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

2026-05-20T08:17:00Z

Clean Book I red links, intro, and image slots

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 {{Short description|Quantum Collection topic on Quantum Partition function}}
 {{Quantum book backlink|Statistical mechanics and kinetic theory}}
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 <div style="flex:1; line-height:1.45; color:#006b45; column-count:2; column-gap:32px; column-rule:1px solid #b8d8c8;">
-'''Quantum partition function''' is the central quantity of [[Physics:Quantum Statistical mechanics|quantum statistical mechanics]], encoding how the energy eigenstates of a quantum system are thermally populated at temperature <math>T</math>.<ref name="Pathria">{{cite book
+'''Partition function''' is a Book I topic in the Quantum Collection. Quantum partition function is the central quantity of quantum statistical mechanics, encoding how the energy eigenstates of a quantum system are thermally populated at temperature T. It provides the bridge between microscopic quantum states and macroscopic thermodynamic quantities such as internal energy, entropy, Helmholtz free energy, and heat capacity. Quantum partition function is the central quantity of quantum statistical mechanics, encoding how the energy eigenstates of a quantum system are thermally populated at temperature T.{{cite book }} It provides the bridge between microscopic quantum states and macroscopic thermodynamic quantities such as internal energy, entropy, Helmholtz free energy, and heat capacity.{{cite book For a quantum system in thermal equilibrium with Hamiltonian H, the canonical partition function is defined by Here k_B is Boltzmann’s constant, T is the absolute temperature, and the trace is taken over the system’s Hilbert space.{{cite book If the Hamiltonian has discrete energy eigenstates |n\rangle with eigenvalues E_n, then the trace becomes This shows that each energy level contributes with a Boltzmann weight determined by its energy.
 </div>

Maintenance script: Normalize Book I Quantum page structure

2026-05-19T22:50:42Z

Normalize Book I Quantum page structure

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	[[Category:Statistical mechanics]]		[[Category:Statistical mechanics]]

	{{Sourceattribution\|Quantum ~~partition~~ function\|1}}		{{Sourceattribution\|Physics:Quantum Partition function\|1}}

Harold: Arrange page top as TOC lead image columns

2026-05-17T14:02:26Z

Arrange page top as TOC lead image columns

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 {{Quantum book backlink|Statistical mechanics and kinetic theory}}
 '''Quantum partition function''' is the central quantity of [[Physics:Quantum Statistical mechanics|quantum statistical mechanics]], encoding how the energy eigenstates of a quantum system are thermally populated at temperature <math>T</math>.<ref name="Pathria">{{cite book
   |last1=Pathria
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   |isbn=9780750633727
 }}</ref>
-[[File:Partition_function_visualization.jpg|thumb|400px|Partition function as a weighted sum over all quantum states, where each state's contribution depends exponentially on its energy.]]
+</div>
 ==Definition==
 For a quantum system in thermal equilibrium with Hamiltonian <math>H</math>, the canonical partition function is defined by