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In quantum mechanics, the Byers–Yang theorem states that all physical properties of a doubly connected system (an annulus) enclosing a magnetic flux $\Phi$ through the opening are periodic in the flux with period $\Phi _{0}=hc/e$ (the magnetic flux quantum). The theorem was first stated and proven by Nina Byers and Chen-Ning Yang (1961),^[1] and further developed by Felix Bloch (1970).^[2]

Proof

An enclosed flux $\Phi$ corresponds to a vector potential $A(r)$ inside the annulus with a line integral ${\textstyle \oint _{C}A\cdot dl=\Phi }$ along any path $C$ that circulates around once. One can try to eliminate this vector potential by the gauge transformation

\psi '(\{r_{n}\})=\exp \left({\frac {ie}{\hbar }}\sum _{j}\chi (r_{j})\right)\psi (\{r_{n}\})

of the wave function $\psi (\{r_{n}\})$ of electrons at positions $r_{1},r_{2},\ldots$ . The gauge-transformed wave function satisfies the same Schrödinger equation as the original wave function, but with a different magnetic vector potential $A'(r)=A(r)+\nabla \chi (r)$ . It is assumed that the electrons experience zero magnetic field $B(r)=\nabla \times A(r)=0$ at all points $r$ inside the annulus, the field being nonzero only within the opening (where there are no electrons). It is then always possible to find a function $\chi (r)$ such that $A'(r)=0$ inside the annulus, so one would conclude that the system with enclosed flux $\Phi$ is equivalent to a system with zero enclosed flux.

However, for any arbitrary $\Phi$ the gauge transformed wave function is no longer single-valued: The phase of $\psi '$ changes by

\delta \phi =(e/\hbar )\oint _{C}\nabla \chi (r)\cdot dl=-(e/\hbar )\oint _{C}A(r)\cdot dl=-2\pi \Phi /\Phi _{0}

whenever one of the coordinates $r_{n}$ is moved along the ring to its starting point. The requirement of a single-valued wave function therefore restricts the gauge transformation to fluxes $\Phi$ that are an integer multiple of $\Phi _{0}$ . Systems that enclose a flux differing by a multiple of $h/e$ are equivalent.

Applications

An overview of physical effects governed by the Byers–Yang theorem is given by Yoseph Imry.^[3] These include the Aharonov–Bohm effect, persistent current in normal metals, and flux quantization in superconductors.

References

^ Byers, N.; Yang, C. N. (1961). "Theoretical Considerations Concerning Quantized Magnetic Flux in Superconducting Cylinders". Physical Review Letters. 7 (2): 46–49. Bibcode:1961PhRvL...7...46B. doi:10.1103/PhysRevLett.7.46.
^ Bloch, F. (1970). "Josephson Effect in a Superconducting Ring". Physical Review B. 2 (1): 109–121. Bibcode:1970PhRvB...2..109B. doi:10.1103/PhysRevB.2.109.
^ Imry, Y. (1997). Introduction to Mesoscopic Physics. Oxford University Press. ISBN 0-19-510167-7.

[1] Byers, N.; Yang, C. N. (1961). "Theoretical Considerations Concerning Quantized Magnetic Flux in Superconducting Cylinders". Physical Review Letters. 7 (2): 46–49. Bibcode:1961PhRvL...7...46B. doi:10.1103/PhysRevLett.7.46.

[2] Bloch, F. (1970). "Josephson Effect in a Superconducting Ring". Physical Review B. 2 (1): 109–121. Bibcode:1970PhRvB...2..109B. doi:10.1103/PhysRevB.2.109.

[3] Imry, Y. (1997). Introduction to Mesoscopic Physics. Oxford University Press. ISBN 0-19-510167-7.

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+{{Short description|Theorem in quantum mechanics}}
-In [[quantum mechanics]], the '''Byers–Yang theorem''' states that all physical properties of a doubly connected system (an annulus) enclosing a magnetic flux <math>\Phi</math> through the opening are periodic in the flux with period <math>\Phi_0=hc/e</math> (the [[magnetic flux quantum]]). The theorem was first stated and proven by [[Nina Byers]] and [[Chen-Ning Yang]] (1961),<ref>{{cite journal |last1 = Byers |first1 = N. |author1-link = Nina Byers |last2 = Yang |first2 = C. N. |author2-link = Yang Chen-Ning |title=Theoretical Considerations Concerning Quantized Magnetic Flux in Superconducting Cylinders |journal=[[Physical Review Letters]] |year=1961|volume=7|issue=2|pages=46–49 |doi = 10.1103/PhysRevLett.7.46 |bibcode = 1961PhRvL...7...46B }}</ref> and further developed by [[Felix Bloch]] (1970).<ref>{{cite journal |last=Bloch|first=F.|year=1970|title=Josephson Effect in a Superconducting Ring |journal=[[Physical Review B]]|volume=2|pages=109–121|doi=10.1103/PhysRevB.2.109 |bibcode = 1970PhRvB...2..109B }}</ref>
+In [[quantum mechanics]], the '''Byers–Yang theorem''' states that all physical properties of a doubly connected system (an annulus) enclosing a magnetic flux <math>\Phi</math> through the opening are periodic in the flux with period <math>\Phi_0=hc/e</math> (the [[magnetic flux quantum]]). The theorem was first stated and proven by [[Nina Byers]] and [[Chen-Ning Yang]] (1961),<ref>{{cite journal |last1 = Byers |first1 = N. |author1-link = Nina Byers |last2 = Yang |first2 = C. N. |author2-link = Yang Chen-Ning |title=Theoretical Considerations Concerning Quantized Magnetic Flux in Superconducting Cylinders |journal=[[Physical Review Letters]] |year=1961|volume=7|issue=2|pages=46–49 |doi = 10.1103/PhysRevLett.7.46 |bibcode = 1961PhRvL...7...46B }}</ref> and further developed by [[Felix Bloch]] (1970).<ref>{{cite journal |last=Bloch|first=F.|year=1970|title=Josephson Effect in a Superconducting Ring |journal=[[Physical Review B]]|volume=2|issue=1 |pages=109–121|doi=10.1103/PhysRevB.2.109 |bibcode = 1970PhRvB...2..109B }}</ref>
 ==Proof==
-An enclosed flux <math>\Phi</math> corresponds to a vector potential <math>A(r)</math> inside the annulus with a line integral <math>\oint_C A\cdot dl=\Phi</math> along any path <math>C</math> that circulates around once. One can try to eliminate this vector potential by the [[gauge transformation]]
+An enclosed flux <math>\Phi</math> corresponds to a vector potential <math>A(r)</math> inside the annulus with a line integral <math display="inline">\oint_C A\cdot dl=\Phi</math> along any path <math>C</math> that circulates around once. One can try to eliminate this vector potential by the [[gauge transformation]]
 : <math>\psi'(\{r_n\})=\exp\left(\frac{ie}{\hbar}\sum_j\chi(r_j)\right)\psi(\{r_n\})</math>
 of the [[wave function]] <math>\psi(\{r_n\})</math> of electrons at positions <math>r_1,r_2,\ldots</math>. The gauge-transformed wave function satisfies the same [[Schrödinger equation]] as the original wave function, but with a different [[magnetic vector potential]] <math>A'(r)=A(r)+\nabla\chi(r)</math>. It is assumed that the electrons experience zero magnetic field <math>B(r)=\nabla\times A(r)=0</math> at all points <math>r</math> inside the annulus, the field being nonzero only within the opening (where there are no electrons). It is then always possible to find a function <math>\chi(r)</math> such that <math>A'(r)=0</math> inside the annulus, so one would conclude that the system with enclosed flux <math>\Phi</math> is equivalent to a system with zero enclosed flux.
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 {{DEFAULTSORT:Byers-Yang theorem}}
-[[Category:Quantum mechanics]]
 [[Category:Theorems in quantum mechanics]]