Editing Bound state (section)

==Properties==
{{See also|Spectrum (physical sciences) #Continuous versus discrete spectra}}
As finitely normalizable states must lie within the [[Spectrum_(functional_analysis)#Classification_of_points_in_the_spectrum|pure point part]] of the spectrum, bound states must lie within the pure point part. However, as [[John von Neumann|Neumann]] and [[Wigner]] pointed out, it is possible for the energy of a bound state to be located in the continuous part of the spectrum. This phenomenon is referred to as [[bound state in the continuum]].<ref>{{cite journal | last1=Stillinger | first1=Frank H. | last2=Herrick | first2=David R. | title=Bound states in the continuum | journal=Physical Review A | publisher=American Physical Society (APS) | volume=11 | issue=2 | year=1975| issn=0556-2791 | doi=10.1103/physreva.11.446 | pages=446–454}}</ref><ref>{{cite journal | last1=Hsu | first1=Chia Wei | last2=Zhen | first2=Bo | last3=Stone | first3=A. Douglas | last4=Joannopoulos | first4=John D. | last5=Soljačić | first5=Marin | title=Bound states in the continuum | journal=Nature Reviews Materials | publisher=Springer Science and Business Media LLC | volume=1 | issue=9 | date=2016 | url=https://dspace.mit.edu/handle/1721.1/108400 | issn=2058-8437 | doi=10.1038/natrevmats.2016.48| hdl=1721.1/108400 | hdl-access=free }}</ref>

===Position-bound states===
Consider the one-particle Schrödinger equation. If a state has energy <math display="inline"> E < \max{\left(\lim_{x\to\infty}{V(x)}, \lim_{x\to-\infty}{V(x)}\right)}</math>, then the wavefunction {{mvar|ψ}} satisfies, for some <math>X > 0</math>

:<math>\frac{\psi^{\prime\prime}}{\psi}=\frac{2m}{\hbar^2}(V(x)-E) > 0\text{ for }x > X</math>

so that {{mvar|ψ}} is exponentially suppressed at large {{mvar|x}}. This behaviour is well-studied for smoothly varying potentials in the [[WKB approximation]] for wavefunction, where an oscillatory behaviour is observed if the right hand side of the equation is negative and growing/decaying behaviour if it is positive.<ref>{{Cite book |last=Hall |first=Brian C. |title=Quantum theory for mathematicians |date=2013 |publisher=Springer |isbn=978-1-4614-7115-8 |series=Graduate texts in mathematics |location=New York Heidelberg$fDordrecht London |page=316-320}}</ref> Hence, negative energy-states are bound if <math>V(x)</math> vanishes at infinity. 

=== Non-degeneracy in one-dimensional bound states ===
One-dimensional bound states can be shown to be non-degenerate in energy for well-behaved wavefunctions that decay to zero at infinities. This need not hold true for wavefunctions in higher dimensions. Due to the property of non-degenerate states, one-dimensional bound states can always be expressed as real wavefunctions.

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!Proof
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Consider two energy eigenstates states <math display="inline"> \Psi_1</math> and <math display="inline"> \Psi_2</math> with same energy eigenvalue.

Then since, the Schrodinger equation, which is expressed as:<math display="block">E  = - \frac 1 {\Psi_i(x,t)} \frac{\hbar^2}{2m}\frac{\partial^2\Psi_i(x,t) }{\partial x^2} + V(x,t) </math>is satisfied for i = 1 and 2, subtracting the two equations gives:<math display="block">\frac 1 {\Psi_1(x,t)} \frac{\partial^2\Psi_1(x,t) }{\partial x^2} - \frac 1 {\Psi_2(x,t)} \frac{\partial^2\Psi_2(x,t) }{\partial x^2} = 0     </math>which can be rearranged to give the condition:<math display="block">
\frac{\partial }{\partial x} \left(\frac{\partial \Psi_1}{\partial x}\Psi_2\right)-\frac{\partial }{\partial x} \left(\frac{\partial \Psi_2}{\partial x}\Psi_1\right)=0  </math>Since <math display="inline">
\frac{\partial \Psi_1}{\partial x}(x)\Psi_2(x)- 
\frac{\partial \Psi_2}{\partial x}(x)\Psi_1(x)= C  </math>, taking limit of x going to infinity on both sides, the wavefunctions vanish and gives <math display="inline">
C = 0  </math>.


Solving for <math display="inline">
\frac{\partial \Psi_1}{\partial x}(x)\Psi_2(x) =
\frac{\partial \Psi_2}{\partial x}(x)\Psi_1(x) </math>, we get: <math display="inline">
\Psi_1(x) = k \Psi_2(x)       </math> which proves that the energy eigenfunction of a 1D bound state is unique.



Furthermore it can be shown that these wavefunctions can always be represented by a completely real wavefunction. Define real functions <math display="inline">
\rho_1(x) </math> and <math display="inline">
\rho_2(x)   </math> such that <math display="inline">
\Psi(x) = \rho_1(x) + i \rho_2(x) </math>. Then, from Schrodinger's equation:<math display="block">\Psi'' = - \frac{2m(E-V(x))}{\hbar^2}\Psi </math> we get that, since the terms in the equation are all real values:<math display="block">\rho_i'' = - \frac{2m(E-V(x))}{\hbar^2}\rho_i   </math>applies for i = 1 and 2. Thus every 1D bound state can be represented by completely real eigenfunctions. Note that real function representation of wavefunctions from this proof applies for all non-degenerate states in general. 
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=== Node theorem ===
Node theorem states that <math>n\text{th}</math> bound wavefunction ordered according to increasing energy has exactly <math>n-1</math> nodes, i.e., points <math>x=a</math> where <math>\psi(a)=0 \neq \psi'(a)</math>. Due to the form of Schrödinger's time independent equations, it is not possible for a physical wavefunction to have <math>\psi(a) = 0 = \psi'(a)</math> since it corresponds to <math>\psi(x)=0</math> solution.<ref>{{Cite book |last=Berezin |first=F. A. |url=http://archive.org/details/schrodingerequat0000bere |title=The Schrödinger equation |publisher=Dordrecht ; Boston : Kluwer Academic Publishers |year=1991 |isbn=978-0-7923-1218-5 |pages=64–66}}</ref>