# Ground state: Difference between revisions

(simplify; explanation is not necessary if logarithm isn't mentioned: unique state + absolute zero -> zero entropy; no need to dig deeper here) |
en>Double sharp m (→Examples: wow! apparently I've suddenly become active instead of passing by and scramming! even if I'm just adding hyphens) |
||

(2 intermediate revisions by 2 users not shown) | |||

Line 1: | Line 1: | ||

{{For|the ''Angel'' episode|Ground State (Angel)}} | {{For|the ''Angel'' episode|Ground State (Angel)}} | ||

[[Image: | [[Image:Energy levels.svg|thumb|right|[[Energy level]]s for an [[electron]] in an [[atom]]: '''ground state''' and [[excited state]]s. After absorbing [[energy]], an electron may [[Atomic electron transition|jump]] from the ground state to a higher energy excited state.]] | ||

The '''ground state''' of a [[quantum mechanics|quantum mechanical]] system is its lowest-[[energy]] [[stationary state|state]]; the energy of the ground state is known as the [[zero-point energy]] of the system. An [[excited state]] is any state with energy greater than the ground state. The ground state of a [[quantum field theory]] is usually called the [[vacuum state]] or the [[vacuum#The quantum-mechanical vacuum|vacuum]]. | The '''ground state''' of a [[quantum mechanics|quantum mechanical]] system is its lowest-[[energy]] [[stationary state|state]]; the energy of the ground state is known as the [[zero-point energy]] of the system. An [[excited state]] is any state with energy greater than the ground state. The ground state of a [[quantum field theory]] is usually called the [[vacuum state]] or the [[vacuum#The quantum-mechanical vacuum|vacuum]]. | ||

If more than one ground state exists, they are said to be [[degenerate energy level|degenerate]]. Many systems have degenerate ground states. | If more than one ground state exists, they are said to be [[degenerate energy level|degenerate]]. Many systems have degenerate ground states. Degeneracy occurs whenever there exists a [[unitary operator]] which acts non-trivially on a ground state and [[commutator|commutes]] with the [[Hamiltonian (quantum mechanics)|Hamiltonian]] of the system. | ||

According to the [[third law of thermodynamics]], a system at [[absolute zero]] [[temperature]] exists in its ground state; thus, its [[entropy]] is determined by the degeneracy of the ground state. Many systems, such as a perfect [[crystal lattice]], have a unique ground state and therefore have zero entropy at absolute zero. It is also possible for the highest excited state to have [[absolute zero]] [[temperature]] for systems that exhibit [[negative temperature]]. | According to the [[third law of thermodynamics]], a system at [[absolute zero]] [[temperature]] exists in its ground state; thus, its [[entropy]] is determined by the degeneracy of the ground state. Many systems, such as a perfect [[crystal lattice]], have a unique ground state and therefore have zero entropy at absolute zero. It is also possible for the highest excited state to have [[absolute zero]] [[temperature]] for systems that exhibit [[negative temperature]]. | ||

{{clear}} | {{clear}} | ||

==1D ground state has no nodes== | |||

In 1D the ground state of the [[Schrödinger equation]] has no nodes. This can be proved considering an average energy in the state with a node at <math>x=0</math>, i.e. <math>\psi(0)=0</math>. Consider the average energy in this state | |||

<math> | |||

\left\langle\psi| H|\psi\right\rangle=\int dx\; \left(-\frac{\hbar^2}{2m}\psi^* \frac{d^2\psi}{dx^2}+V(x)|\psi(x)|^2\right) | |||

</math> | |||

where <math>V(x)</math> is the potential. Now consider a small interval around <math>x=0</math>, i.e. <math>x\in[-\epsilon,\epsilon]</math>. Take a new wavefunction <math>\psi'(x)</math> to be defined as <math>\psi'(x)=\psi(x),x<-\epsilon</math> and <math>\psi'(x)=-\psi(x),x>\epsilon</math> and constant for <math>x\in[-\epsilon,\epsilon]</math>. If epsilon is small enough then this is always possible to do so that <math>\psi'(x)</math> is continuous. So assuming <math>\psi(x)\approx-cx</math> around <math>x=0</math>, we can write the new function as | |||

<math> | |||

\psi'(x)=N\left\{\begin{array}{ll} | |||

|\psi(x)| & |x|>\epsilon\\ | |||

c\epsilon & |x|\le\epsilon | |||

\end{array}\right. | |||

</math> | |||

where <math>N=\frac{1}{\sqrt{1+|c|^2\epsilon^3/3}}</math> is the norm. Note that the kinetic energy density <math>|d\psi'/dx|^2<|d\psi/dx|^2</math> everywhere because of the normalization. Now consider the potential energy. For definiteness let us choose <math>V(x)\ge 0</math>. Then it is clear that outside the interval <math>x\in[-\epsilon,\epsilon]</math> the potential energy density is smaller for the <math>\psi'</math> because <math>|\psi'|<|\psi|</math> there. On the other hand in the interval <math>x\in[-\epsilon,\epsilon]</math> we have | |||

<math> | |||

{V^\epsilon_{avg}}'=\int_{-\epsilon}^\epsilon dx\; V(x)|\psi'|^2=\frac{\epsilon^3|c|^2}{1+|c|^2\epsilon^3/3}\int_{-\epsilon}^\epsilon V(x)\approx \frac{2\epsilon^4|c|^2}{3}V(0)+\dots\;. | |||

</math> | |||

which is correct to this order of <math>\epsilon</math> and <math>\dots</math> indicate higher order corrections. On the other hand the potential energy in the <math>\psi</math> state is | |||

<math> | |||

V^\epsilon_{avg}=\int_{-\epsilon}^\epsilon dx\; V(x)|\psi|^2=\int_{-\epsilon}^\epsilon dx\; |c|^2|x|^2V(x)\approx\frac{2\epsilon^4 |c|^2}{3}V(0)+\dots\;. | |||

</math> | |||

which is the same as that of the <math>\psi'</math> state to the order shown. | |||

Therefore the potential energy unchanged to leading order in <math>\epsilon</math> by deforming the state with a node <math>\psi</math> into a state without a node <math>\psi'</math>. We can do this by removing all nodes thereby reducing the energy, which implies that the ground state energy must not have a node. This completes the proof. | |||

==Examples== | ==Examples== | ||

[[File:particle in a box wavefunctions 2.svg|thumb|right|upright|Initial wave functions for the first four states of a one-dimensional particle in a box]] | [[File:particle in a box wavefunctions 2.svg|thumb|right|upright|Initial wave functions for the first four states of a one-dimensional particle in a box]] | ||

*The [[wave function]] of the ground state of a [[particle in a box|particle in a one-dimensional well]] is a half-period [[sine wave]] which goes to zero at the two edges of the well. The energy of the particle is given by <math>\frac{h^2 n^2}{8 m L^2}</math>, where ''h'' is the [[Planck constant]], ''m'' is the mass of the particle, ''n'' is the energy state (''n'' = 1 corresponds to the ground-state energy), and ''L'' is the width of the well. | *The [[wave function]] of the ground state of a [[particle in a box|particle in a one-dimensional well]] is a half-period [[sine wave]] which goes to zero at the two edges of the well. The energy of the particle is given by <math>\frac{h^2 n^2}{8 m L^2}</math>, where ''h'' is the [[Planck constant]], ''m'' is the mass of the particle, ''n'' is the energy state (''n'' = 1 corresponds to the ground-state energy), and ''L'' is the width of the well. | ||

*The wave function of the ground state of a hydrogen atom is a spherically-symmetric distribution centred on the [[atomic nucleus|nucleus]], which is largest at the center and reduces [[exponential distribution|exponentially]] at larger distances. The [[electron]] is most likely to be found at a distance from the nucleus equal to the [[Bohr radius]]. This function is known as the 1s [[atomic orbital]]. For hydrogen (H), an electron in the ground state has energy {{val|-13.6|u=eV}}, relative to | *The wave function of the ground state of a hydrogen atom is a spherically-symmetric distribution centred on the [[atomic nucleus|nucleus]], which is largest at the center and reduces [[exponential distribution|exponentially]] at larger distances. The [[electron]] is most likely to be found at a distance from the nucleus equal to the [[Bohr radius]]. This function is known as the 1s [[atomic orbital]]. For hydrogen (H), an electron in the ground state has energy {{val|-13.6|u=eV}}, relative to the [[Ionization energy|ionization threshold]]. In other words, 13.6 eV is the energy input required for the electron to no longer be [[Bound state|bound]] to the atom. | ||

== | *The exact definition of one [[second]] of [[time]] since 1997 has been the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two [[hyperfine]] levels of the ground state of the [[caesium]]-133 atom at rest at a temperature of 0 K.<ref> | ||

* {{cite book|title=The Feynman Lectures on Physics|volume=3| first1=Richard |last1=Feynman|author1-link=Richard Feynman|first2=Robert |last2=Leighton|first3=Matthew|last3= Sands|year=1965|chapter=see section 2-5 for energy levels, 19 for the hydrogen atom}} | {{cite web | ||

|title=Unit of time (second) | |||

|work=SI Brochure | |||

|url=http://www.bipm.org/en/si/si_brochure/chapter2/2-1/second.html | |||

|publisher=[[BIPM]] | |||

|accessdate=2013-12-22}} | |||

</ref> | |||

==Notes== | |||

{{Reflist}} | |||

==Bibliography== | |||

* {{cite book|title=The Feynman Lectures on Physics|volume=3| first1=Richard |last1=Feynman|author1-link=Richard Feynman|first2=Robert |last2=Leighton|first3=Matthew|last3= Sands|year=1965|chapter=see section 2-5 for energy levels, 19 for the hydrogen atom|url=http://www.feynmanlectures.caltech.edu/III_toc.html}} | |||

<!-- Covered in any QM textbook - from the Feynman lectures, see section 2-5 for energy levels, 19 for the hydrogen atom --> | <!-- Covered in any QM textbook - from the Feynman lectures, see section 2-5 for energy levels, 19 for the hydrogen atom --> | ||

[[Category:Quantum mechanics]] | [[Category:Quantum mechanics]] | ||

## Latest revision as of 15:03, 27 December 2014

{{#invoke:Hatnote|hatnote}}

The **ground state** of a quantum mechanical system is its lowest-energy state; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state. The ground state of a quantum field theory is usually called the vacuum state or the vacuum.

If more than one ground state exists, they are said to be degenerate. Many systems have degenerate ground states. Degeneracy occurs whenever there exists a unitary operator which acts non-trivially on a ground state and commutes with the Hamiltonian of the system.

According to the third law of thermodynamics, a system at absolute zero temperature exists in its ground state; thus, its entropy is determined by the degeneracy of the ground state. Many systems, such as a perfect crystal lattice, have a unique ground state and therefore have zero entropy at absolute zero. It is also possible for the highest excited state to have absolute zero temperature for systems that exhibit negative temperature.

## 1D ground state has no nodes

In 1D the ground state of the Schrödinger equation has no nodes. This can be proved considering an average energy in the state with a node at , i.e. . Consider the average energy in this state

where is the potential. Now consider a small interval around , i.e. . Take a new wavefunction to be defined as and and constant for . If epsilon is small enough then this is always possible to do so that is continuous. So assuming around , we can write the new function as

where is the norm. Note that the kinetic energy density everywhere because of the normalization. Now consider the potential energy. For definiteness let us choose . Then it is clear that outside the interval the potential energy density is smaller for the because there. On the other hand in the interval we have

which is correct to this order of and indicate higher order corrections. On the other hand the potential energy in the state is

which is the same as that of the state to the order shown. Therefore the potential energy unchanged to leading order in by deforming the state with a node into a state without a node . We can do this by removing all nodes thereby reducing the energy, which implies that the ground state energy must not have a node. This completes the proof.

## Examples

- The wave function of the ground state of a particle in a one-dimensional well is a half-period sine wave which goes to zero at the two edges of the well. The energy of the particle is given by , where
*h*is the Planck constant,*m*is the mass of the particle,*n*is the energy state (*n*= 1 corresponds to the ground-state energy), and*L*is the width of the well. - The wave function of the ground state of a hydrogen atom is a spherically-symmetric distribution centred on the nucleus, which is largest at the center and reduces exponentially at larger distances. The electron is most likely to be found at a distance from the nucleus equal to the Bohr radius. This function is known as the 1s atomic orbital. For hydrogen (H), an electron in the ground state has energy Template:Val, relative to the ionization threshold. In other words, 13.6 eV is the energy input required for the electron to no longer be bound to the atom.

- The exact definition of one second of time since 1997 has been the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 atom at rest at a temperature of 0 K.
^{[1]}

## Notes

## Bibliography

- {{#invoke:citation/CS1|citation

|CitationClass=book }}