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{{Author|Harold Foppele}}
{{Physics}}
{{Author|Harold Foppele}}
{{Physics}}
{{learning project}}
←[[Quantum]]
[[File:Quantum course overview.svg|650px]]

= Overview: Quantum Mechanisms of Cohesion =
== Classical vs Quantum Explanation for Matter Stability ==
[[File:Artistic impression of an atom 9.png|thumb|Artistic impression of an atom 9]]
Opposite charges attract, but that explanation cannot account for the stability of atoms. <br>If electrons were point particles, they would collapse into the nucleus.

The stability of matter is quantum mechanical.<br>[[W:Electron|Electrons]] are described by [[W:Wave function|wavefunctions]]. <br>These waves:

* Spread through space
* Interfere with one another
* Obey symmetry principles such as the [[W:Pauli exclusion principle|Pauli exclusion principle]]

Waves do not hold matter together by acting like glue. Instead, cohesion arises from the overlap, interference, and symmetry constraints of quantum wavefunctions, together with the balance between kinetic energy and Coulomb forces.

In this course we examine how:

* The Schrödinger equation already encodes stability
* Wavefunction overlap produces chemical bonds
* Fermi statistics prevents collapse
* Collective coherence gives rise to metals and superconductors
* Decoherence in open systems can modify binding

Matter holds together because particles behave as quantum waves.

== Classical vs Quantum Picture. ==

* Cohesion in matter exists because particles have wave character, and '''wavefunctions overlap, interfere, and obey symmetry constraints.'''

Let’s build this step by step.

==1️⃣ Starting Point: The Schrödinger Equation==
[[File:Schrodinger equation.png|thumb|Schrodinger equation time dependant]]
The '''Schrödinger equation''' a [[W:partial differential equation|partial differential equation]] that overlooks the [[W:wave function|wave function]] of non-relativistic quantum-mechanical systems.<ref>{{cite book |last=Griffiths| first=David J.|title=Introduction to Quantum Mechanics (2nd ed.)|title-link=Introduction to Quantum Mechanics (book)|publisher=Prentice Hall| year=2004|isbn=978-0-13-111892-8|location=|pages=|author-link=W:David J. Griffiths}}</ref>{{rp|1–2}} Its discovery was significant in the development of [[W:quantum mechanics|quantum mechanics]]. It is named after [[W:Erwin Schrödinger|Erwin Schrödinger]],  who postulated the equation in 1925 and published it in 1926. It forms the basis for the work that leaded to his [[W:Nobel Prize in Physics|Nobel Prize in Physics]] in 1933.<ref>{{cite news|title=Physicist Erwin Schrödinger's Google doodle marks quantum mechanics work|url=https://www.theguardian.com/technology/2013/aug/12/erwin-schrodinger-google-doodle|access-date=25 August 2013|newspaper=[[W:The Guardian|The Guardian]]|date=13 August 2013}}</ref><ref name = sch>
{{cite journal
 | last         = Schrödinger | first = E.
 | title        = An Undulatory Theory of the Mechanics of Atoms and Molecules
 | url          = http://home.tiscali.nl/physis/HistoricPaper/Schroedinger/Schroedinger1926c.pdf
 | archive-url  = https://web.archive.org/web/20081217040121/http://home.tiscali.nl/physis/HistoricPaper/Schroedinger/Schroedinger1926c.pdf
 | archive-date = 17 December 2008
 | journal      = [[W:Physical Review|Physical Review]]
 | volume       = 28
 | issue        = 6
 | pages        = 1049–70
 | year         = 1926
 | doi          = 10.1103/PhysRev.28.1049
 |bibcode       = 1926PhRv...28.1049S
}}</ref><ref>[https://onlinelibrary.wiley.com/doi/10.1002/andp.19263840404      E Schrödinger,  Quantisierung als Eigenwertproblem, "Erste Mitteilung", 'Ann Phys' ''79'' (1926) 361]</ref>

For an electron in a potential <math>V(\mathbf r)</math>:

<math>
i\hbar \frac{\partial \psi(\mathbf r,t)}{\partial t}
=
\left(
-\frac{\hbar^2}{2m}\nabla^2 + V(\mathbf r)
\right)\psi(\mathbf r,t)
</math>

The first term,

<math>
-\frac{\hbar^2}{2m}\nabla^2 \psi
</math>

represents the kinetic energy operator and is purely wave-like (it depends on the spatial curvature of the wavefunction).

The second term,

<math>
V(\mathbf r)\psi
</math>

represents the interaction energy (e.g., the Coulomb potential).

Cohesion emerges from the interplay between:

* Wave-like delocalization (kinetic term <math>-\frac{\hbar^2}{2m}\nabla^2</math>)
* Coulomb attraction between charges encoded in <math>V(\mathbf r)</math>
[[File:Constructive interference of overlapping wavefunctions leading to enhanced probability density between nuclei and quantum cohesion.png|599px|center|Constructive interference of overlapping wavefunctions leading to enhanced probability density between nuclei and quantum cohesion]]

== 2️⃣ Wave Nature of Particles and Quantum Stability ==
[[File:Propagation of a de broglie wave.svg|290px|right|thumb|Propagation of '''de Broglie waves''' in one dimension – real part of the [[W:complex number|complex]] amplitude is blue, imaginary part is green. The probability (shown as the color [[W:opacity (optics)|opacity]]) of finding the particle at a given point {{math|''x''}} is spread out like a waveform; there is no definite position of the particle. As the amplitude increases above zero the [[W:slope|slope]] decreases, so the amplitude diminishes again, and vice versa. The result is an alternating amplitude: a wave. Top: [[W:plane wave|plane wave]]. Bottom: [[W:wave packet|wave packet]].]]
'''Matter waves,''' a central part of the theory of [[W:quantum mechanics|quantum mechanics]], being half of [[W:wave–particle duality|wave–particle duality]]. All scales where measurements have been practical, [[W:matter|matter]] exhibits [[W:wave|wave]]-like behavior. A beam of electrons can be [[W:diffraction|diffracted]] like a beam of light or a water wave.

Matter behaves like a wave was proposed by French physicist [[W:Louis de Broglie|Louis de Broglie]] in 1924, and matter waves are also known as '''de Broglie waves'''.

If electrons were classical point particles, they would collapse into the nucleus and atoms would not be stable.

To find the [[wavelength]] equivalent to a moving body, de Broglie set the [[W:Energy–momentum relation#Connection to E = mc2|total energy]] from [[special relativity]] for that body equal to {{math | ''h&nu;''}}:<math display="block">E = \frac{mc^2}{\sqrt{1-\frac{v^2}{c^2}}} = h\nu</math>

(Modern physics no longer uses this form of the total energy; the [[W:energy–momentum relation:energy–momentum relation]] has proven more useful.) De Broglie identified the velocity of the particle, <math>v</math>, with the wave [[W:group velocity|group velocity]] in free space:
<math display="block"> v_\text{g} \equiv \frac{\partial \omega}{\partial k} = \frac{d\nu}{d(1/\lambda)} </math>

(The modern definition of group velocity uses angular frequency {{mvar|ω}} and wave number {{mvar|k}}). By applying the differentials to the energy equation and identifying the [[W:Momentum#Relativistic|relativistic momentum]]:
<math display="block"> p = \frac{mv}{\sqrt{1-\frac{v^2}{c^2}}} </math>

then integrating, de Broglie arrived at his formula for the relationship between the [[W:wavelength|wavelength]], {{mvar|λ}}, associated with an electron and the modulus of its [[W:momentum|momentum]], {{math|''p''}}, through the [[W:Planck constant|Planck constant]], {{math|''h''}}:<ref>{{cite book |title=Introducing Quantum Theory |author1=McEvoy, J. P. |author2=Zarate, Oscar  |publisher=Totem Books |year=2004 |isbn=978-1-84046-577-8 |pages=110–114}}</ref><math display="block"> \lambda = \frac{h}{p}.</math>
The quantum kinetic energy associated with spatial confinement scales as

<math> E_{\text{kin}} \sim \frac{\hbar^2}{2m L^2} </math>

where <math>L</math> is the characteristic localization length.

As <math>L</math> becomes smaller (stronger localization), the kinetic energy increases rapidly. This follows from the Laplacian term in the Schrödinger equation and is closely related to the uncertainty principle,

<math> \Delta x \, \Delta p \ge \frac{\hbar}{2} </math>

Thus, wave behavior prevents atomic collapse.

== 3️⃣ Covalent Bonding and Wavefunction Overlap ==
[[File:H2 molecular orbitals.svg|thumb|H2 molecular orbitals]]
[[File:Covalent bond hydrogen.svg|thumb|250px|A covalent bond forming H<sub>2</sub> (right) where two [[hydrogen atom]]s share the two electrons]]
'''Covalent bonding''' a [[W:chemical bond|chemical bond]] that involves [[W:electrons|electrons]] to form [[W:electron pair|electron pair]]s between [[W:atom|atom]]s. These electron pairs are known as '''shared pairs''' or '''bonding pairs'''. Stable balance of attracting and repulsive forces between atoms, when they share electrons, is known as covalent bonding.<ref>{{cite book |last1=Whitten |first1=Kenneth W. |last2=Gailey |first2=Kenneth D. |last3=Davis |first3=Raymond E. |title=General Chemistry |date=1992 |publisher=Saunders College Publishing |isbn=0-03-072373-6 |page=264 |edition=4th |chapter=7-3 Formation of covalent bonds}}</ref> For [[W:molecule|molecule]]s, to share electrons allows each atom to attain the equivalent of a full valence shell, responding to an electronic configuration. In organic chemistry, covalent bonding is more common than [[W:ionic bonding|ionic bonding]].

Consider two hydrogen atoms. Their atomic orbitals can combine into symmetric and antisymmetric superpositions:

Bonding combination:

<math> \psi_{+} = \frac{1}{\sqrt{2}}\left(\psi_A + \psi_B\right) </math>

Antibonding combination:

<math> \psi_{-} = \frac{1}{\sqrt{2}}\left(\psi_A - \psi_B\right) </math>

The bonding state <math>\psi_{+}</math> has increased probability density between the nuclei:

<math> |\psi_{+}|^2 > |\psi_A|^2 + |\psi_B|^2 </math>

in the internuclear region. 

This lowers the total energy through: 
* Electron delocalization 
* Reduced Coulomb repulsion 
* Lower effective kinetic energy due to spreading

Bonding is therefore an energetic consequence of constructive wave interference.

== 4️⃣ Fermi Statistics and Pauli Effects ==
[[File:Fermi sphere and Pauli pressure schematic.svg|thumb|Fermi sphere in momentum space illustrating Pauli exclusion. States are filled up to the Fermi momentum <math>p_F</math>; increasing density enlarges <math>p_F</math>, raising kinetic energy and generating Fermi pressure.]]
A degenerate Fermi gas represents a state of matter where particles (fermions) are packed at high density and low temperatures, causing quantum effects. The pressure of a degenerate Fermi gas, known as degeneracy pressure, differs between high-density (high-pressure) and low-density (low-pressure) regimes, because the pressure depends on density (<math>n</math>) rather than temperature (<math>T</math>)<br>Electrons are fermions, so the total wavefunction must be antisymmetric under exchange:

<math> \Psi(\mathbf r_1, \mathbf r_2) = - \Psi(\mathbf r_2, \mathbf r_1) </math>

This implies the Pauli exclusion principle:

<math> n_i \le 1 </math>

for each single-particle quantum state <math>i</math>.

As electrons fill higher momentum states, the Fermi momentum <math>p_F</math> increases, giving rise to Fermi pressure. The associated energy density scales as

<math> E \sim \frac{p_F^2}{2m} </math>

This quantum pressure stabilizes matter and prevents collapse.

== 5️⃣ Exchange Interaction and Collective Order ==
[[File:Exchange interaction spin alignment schematic.svg|thumb|Exchange interaction: antisymmetric <math>\Psi(\mathbf r_1,\mathbf r_2)</math> couples spin configuration to spatial overlap, changing Coulomb energy and allowing ferro- or antiferromagnetic ordering.]]
Antisymmetry, the total energy includes an exchange term. In simplified form, the exchange contribution between two states can be written as

<math> E_{\text{ex}} \sim \int \int \psi_i^*(\mathbf r_1)\psi_j^*(\mathbf r_2) \frac{e^2}{|\mathbf r_1 - \mathbf r_2|} \psi_i(\mathbf r_2)\psi_j(\mathbf r_1) \, d\mathbf r_1 \, d\mathbf r_2 </math>

This term has no classical analogue and arises purely from wavefunction symmetry.

It explains phenomena such as:
* Ferromagnetism
* Antiferromagnetism
* Additional molecular stabilization

==6️⃣ Metallic Bonding: Bloch Waves==
[[File:Bloch theorem metallic bonding band diagram.svg|thumb|Bloch wave and band formation in a periodic lattice. States of the form <math>\psi_{nk}(\mathbf r)=e^{i\mathbf k\cdot\mathbf r}u_{nk}(\mathbf r)</math> are delocalized across the crystal and give rise to metallic energy bands.]]
'''Metallic bonding''' is a [[W:chemical bond|chemical bond]]ing that arises from the electrostatic attractive force between [[W:conduction electrons|conduction electrons]] (in the form of an electron cloud of [[W:delocalized electron|delocalized electron]]s) and positively charged [[W:metal|metal]] [[W:ion|ion]]s. It is the sharing of ''free'' electrons among a [[W:crystal structure|crystal structure]] of positively charged ions ([[W:cation|cation]]s). Metallic bonding accounts for [[W:physical property|physical properties]] of metals, such as [[W:Strength of materials|strength]], [[W:ductility|ductility]], [[W:thermal conductivity|thermal]] and [[W:electrical resistivity and conductivity]], [[W:Opacity (optics)|opacity]], and [[W:lustre (mineralogy)|lustre]].<ref>[http://www.chemguide.co.uk/atoms/bonding/metallic.html Metallic bonding]. chemguide.co.uk</ref><ref>[http://www.chemguide.co.uk/atoms/structures/metals.html Metal structures]. chemguide.co.uk</ref><ref>[http://hyperphysics.phy-astr.gsu.edu/hbase/chemical/bond.html Chemical Bonds]. chemguide.co.uk</ref><ref>[https://web.archive.org/web/19991018204506/http://www.physics.ohio-state.edu/%7Eaubrecht/physics133.html "Physics 133 Lecture Notes" Spring, 2004. Marion Campus]. physics.ohio-state.edu</ref>

In a periodic lattice potential <math>V(\mathbf r)</math>, electron eigenstates take the Bloch form:

<math> \psi_{n\mathbf k}(\mathbf r) = e^{i \mathbf k \cdot \mathbf r} u_{n\mathbf k}(\mathbf r) </math>

where

<math> u_{n\mathbf k}(\mathbf r + \mathbf R) = u_{n\mathbf k}(\mathbf r) </math>

for lattice vectors <math>\mathbf R</math>.

These delocalized wave functions extend across the crystal and lower total energy collectively, producing metallic cohesion.

== 7️⃣ Superconductivity as a Cohesive Quantum Phenomenon ==
[[File:Superconductivity Cooper pairing and phase coherence schematic.svg|thumb|Electron–phonon interaction induces an effective attraction, forming Cooper pairs. A macroscopic coherent state emerges with the order parameter <math>\Delta \sim \langle c_{-k\downarrow} c_{k\uparrow} \rangle</math>.]]
'''Superconductivity,''' physical properties observed in '''superconductors''': materials where [[W:Electrical resistance and conductance|electrical resistance]] vanishes and [[W:Magnetic field|magnetic fields]] are expelled from the material. Unlike a standard metallic [[W:Electrical conductor|conductor]], whose resistance decreases gradually as its temperature is lowered, even down to near [[W:absolute zero|absolute zero]], a superconductor characteristics are [[W:Phase transition|critical temperature]] below which the resistance drops abruptly to zero.<ref name="Combescot2">{{cite book |last1=Combescot |first1=Roland |url=https://books.google.com/books?id=lRhdEAAAQBAJ&pg=PA1 |title=Superconductivity |date=2022 |publisher=Cambridge University Press |isbn=978-1-108-42841-5 |pages=1–2 |doi= |id=}}</ref><ref name="Fossheim2">{{cite book |last1=Fossheim |first1=Kristian |url=https://books.google.com/books?id=Ep1MLS9YQX8C&pg=PA13 |title=Superconductivity: Physics and Applications |last2=Sudboe |first2=Asle |date=2005 |publisher=John Wiley and Sons |isbn=978-0-470-02643-4 |pages=7 |doi= |id=}}</ref> An [[W:electric current|electric current]] through a loop of [[W:superconducting wire|superconducting wire]] can persist indefinitely with no power source.<ref>{{cite journal |author=Bardeen |first1=John |url=https://books.google.com/books?id=_QKPGDG-cuAC&q=%22persist+indefinitely&pg=PA76 |bibcode-access=free |doi-access=free |s2cid-access=free |title=Theory of Superconductivity |last2=Cooper |first2=Leon |last3=Schrieffer |first3=J. R. |date=December 1, 1957 |journal=Physical Review |volume=108 |page=1175 |bibcode=1957PhRv..108.1175B |doi=10.1103/physrev.108.1175 |access-date=June 6, 2014 |issue=5 |s2cid=73661301}} Reprinted in {{cite book | editor-first=Nikolaĭ Nikolaevich | editor-last=Bogoliubov | date=1963 | url=https://books.google.com/books?id=_QKPGDG-cuAC&pg=PA73 | title=The Theory of Superconductivity | volume=4 | publisher=Gordon and Breach | oclc=537010 | page=73}}</ref><ref name="Daintith2">{{cite book |author=Daintith |first=John |url=https://books.google.com/books?id=VdEVdJo3CDgC&pg=PA238 |title=The Facts on File Dictionary of Physics |date=2009 |publisher=Infobase Publishing |isbn=978-1-4381-0949-7 |edition=4th |page=238 |language=en}}</ref><ref name="Gallop2">{{cite book |author=Gallop |first=John C. |url=https://books.google.com/books?id=ad8_JsfCdKQC |title=SQUIDS, the Josephson Effects and Superconducting Electronics |date=1990 |publisher=[[W:CRC Press|CRC Press]] |isbn=978-0-7503-0051-3 |pages=1, 20 |language=en}}</ref><ref name="Durrant2">{{cite book |last1=Durrant |first1=Alan |url=https://books.google.com/books?id=F0JmHRkJHiUC&q=%22persist+indefinitely&pg=PA103 |title=Quantum Physics of Matter |date=2000 |publisher=CRC Press |isbn=978-0-7503-0721-5 |pages=102–103}}</ref>



Through electron-phonon interaction, an effective attractive interaction can arise:

<math> H_{\text{int}} = \sum_{k,k'} V_{kk'} c_{k\uparrow}^\dagger c_{-k\downarrow}^\dagger c_{-k'\downarrow} c_{k'\uparrow} </math>

This leads to Cooper pairing and a macroscopic coherent state with order parameter

<math> \Delta \sim \langle c_{-k\downarrow} c_{k\uparrow} \rangle </math>

Cohesion here emerges from collective quantum phase coherence.

== 8️⃣ Stability of Matter: Mathematical Perspective ==
[[File:Theory of chemical reaction (Static theory of atomic model).jpg|thumb|Atom model. The structure of the atom. Static theory. №1 Series 12 photos.]]
'''Stability of matter''' refers to the ability of a large number of [[W:charged particle|charged particle]]s, such as electrons and [[W:proton|proton]]s, to create macroscopic objects without collapsing or blowing apart due to [[W:Electromagnetism|electromagnetic]] interactions. [[W:Classical physics|Classical physics]] predicts that such systems should be inherently unstable due to attractive and repulsive [[W:electrostatic forces|electrostatic forces]] between charges, and thus the stability of matter was a theoretical problem that required a [[W:Quantum mechanics|quantum mechanical]] explanation. 

The first solution to this problem was provided by [[W:Freeman Dyson|Freeman Dyson]] and Andrew Lenard in Freeman Dyson 1967–1968,<ref name="DysLen1">{{cite journal |last1=Dyson |first1=Freeman J. |last2=Lenard |first2=A. |title=Stability of Matter. I |journal=Journal of Mathematical Physics |date=March 1967 |volume=8 |issue=3 |pages=423–434 |doi=10.1063/1.1705209|bibcode=1967JMP.....8..423D }}</ref><ref name="DysLen2">{{cite journal |last1=Lenard |first1=A. |last2=Dyson |first2=Freeman J. |title=Stability of Matter. II |journal=Journal of Mathematical Physics |date=May 1968 |volume=9 |issue=5 |pages=698–711 |doi=10.1063/1.1664631|bibcode=1968JMP.....9..698L |doi-access=free }}</ref> but a shorter and more conceptual proof was found later by [[W:Elliott Lieb|Elliott Lieb]] in 1975 using the [[W:Lieb–Thirring inequality|Lieb–Thirring inequality]].<ref name="LT">{{cite journal |last1=Lieb |first1=Elliott H. |last2=Thirring |first2=Walter E. |title=Bound for the Kinetic Energy of Fermions Which Proves the Stability of Matter |journal=[[W:Physical Review Letters|Physical Review Letters]] |date=15 September 1975 |volume=35 |issue=11 |pages=687–689 |doi=10.1103/PhysRevLett.35.687|bibcode=1975PhRvL..35..687L }}</ref> The stability of matter is partly due to the [[W:uncertainty principle|uncertainty principle]] and the [[W:Pauli exclusion principle|Pauli exclusion principle]].<ref>{{Cite book |last=Marder |first=Michael P. |url=https://books.google.com/books?id=ijloadAt4BQC&q=marder+condensed+matter |title=Condensed Matter Physics |date=2010-11-17 |publisher=John Wiley & Sons |isbn=978-0-470-94994-8 |language=en}}</ref>

The stability of matter results from the balance:

<math> E_{\text{total}} = T_{\text{quantum}} + V_{\text{Coulomb}} </math>

The kinetic term originates from wave curvature and the Coulomb term from electromagnetic interaction.

Without the quantum kinetic term, the energy would decrease without bound and matter would collapse.

== 9️⃣ Decoherence Effects on Binding in Open Quantum Systems. ==
[[File:Open quantum systems coherence and cohesion schematic.svg|thumb|Open quantum systems: Redfield/Lindblad decoherence suppresses off-diagonal terms of <math>\rho</math>, reducing delocalization and potentially changing effective binding; cohesion is sensitive to coherence.]]
In [[physics]], an '''open quantum system''' is a [[Quantum mechanics|quantum mechanical]] system that interacts with an external [[W:quantum system|quantum system]],  "the ''environment"'' or a ''[[W:Quantum dissipation|"bath]]"''. In general, interactions significantly change the dynamics of the system, the information contained in the system is then lost to its environment. No quantum system is completely isolated from its surroundings,<ref>{{cite book |last1=Breuer |first1=H.-P. |title=The Theory of Open Quantum Systems |last2=Petruccione |first2=F. |publisher=Oxford University Press |year=2007 |isbn=978-0-19-921390-0 |page=vii |quote="Quantum mechanical systems must be regarded as open systems".}}</ref> it is important to develop a [[W:Quantum field theory|theoretical framework]] for treating these interactions to obtain an accurate understanding of quantum systems.

In open systems:

* Cohesion is related to coherence.
* Redfield/Lindblad dynamics describe loss of coherence.
* Decoherence can influence binding (e.g., exciton dynamics).

Under strong dissipation:

* Delocalization decreases
* Effective binding can change

So:

* Cohesion is sensitive to quantum coherence.

== 🔬 Summary: What Holds Matter Together? ==

Do waves in matter create cohesion?

Not as classical vibrations.

But at the quantum level:

* Yes — the wave character of electrons is essential for binding.
* Interference, antisymmetry, and delocalization make stable matter possible.
* Without quantum wave mechanics, matter would not exist.

==See also==
{{:Quantum/See also}}
==References==
{{reflist|3}}