Phase gap

"Rank Osatsu" is transferred to this item. Please see "phase" (linguistics) about the phase difference in the linguistics.

While incident state and dispersion are in a state by dispersion, phase gap (rank Osatsu, phase shift, phase-shift) is Osatsu in a quantum-mechanical dispersion theory so as to occur.

Table of contents

The partial wave development that is in an incident state

When a partial wave presents incident plane wave, I can express it as follows (formula of Reilly).

${\displaystyle e^{i\mathbf {k} \cdot \mathbf {r} }=\sum _{l=0}^{\infty }(2l+1)i^{l}j(kr)P_{l}(\cos \theta)}$ 

This${\displaystyle r\}$ But, in the place very big,

${\displaystyle e^{i\mathbf {k} \cdot \mathbf {r} }\to \sum _{l=0}^{\infty }{\frac {(2l+1)}{2ik}}({\frac {e^{ikr}}{r}}-(-1)^{l}{\frac {e^{-ikr}}{r}})P_{l}(\cos \theta) \quad (r\to \infty) \quad \cdots (1)}$ 

Partial wave development of the dispersion state

I think the dispersion state to be incident plane wave and help laying upon of the dispersion spherical wave.

${\displaystyle \psi ^{+}(r,\theta) =e^{i\mathbf {k} \cdot \mathbf {r} }+f(\theta) {\frac {e^{ikr}}{r}}\quad \cdots (2)}$ 

In addition, it is ルジャンドル multinomial expression${\displaystyle P_{l}(\cos \theta) \}$ I make は 完全系.

${\displaystyle \int _{0}^{\pi }P_{l}(\cos \theta) P_{l'}(\cos \theta) \sin \theta d\theta ={\frac {2}{2l+1}}\delta _{l,l'}}$ 

Thus, the dispersion amplitude${\displaystyle P_{l}(\cos \theta) \}$ I can express it by の linear combination. The development coefficient${\displaystyle a_{l}\}$ とすると,

${\displaystyle f(\theta) =\sum _{l=0}^{\infty }{\frac {(2l+1)}{2ik}}a_{l}P_{l}(\cos \theta) \quad \cdots (3)}$ 

Thus, when I substitute (1), a (3) type for a (2) type,${\displaystyle r\}$ But, the dispersion in the very big place is in a state${\displaystyle \psi ^{+}(r,\theta) \}$ Oh,

${\displaystyle \psi ^{+}(r,\theta) \to \sum _{l=0}^{\infty }{\frac {(2l+1)}{2ik}}{\Big [}(1+a_{l}){\frac {e^{ikr}}{r}}-(-1)^{l}{\frac {e^{-ikr}}{r}}{\Big ]}P_{l}(\cos {\theta })\quad (r\to \infty) \quad \cdots (4)}$ 

In the first item in this parenthesis, the second item expresses an inner direction spherical wave by an outward arrow spherical wave each.

Preservation, phase gap of the probability

By preservation of the probability, the absolute value of the amplitude of the direction spherical wave in an outward arrow spherical wave must equal. In other words,

${\displaystyle \left|1+a_{l}\right|=1}$ 

The gap that is phase here${\displaystyle \delta _{l}\}$ (actual number) I define it to be less than を.

${\displaystyle 1+a_{l}=e^{i2\delta _{l}}}$ 

(1)An expression, a (4) type${\displaystyle a_{l}\}$ Then there is not it; この${\displaystyle \delta _{l}\}$ When I use を and rewrite it,

${\displaystyle e^{i\mathbf {k} \cdot \mathbf {r} }\to \sum _{l=0}^{\infty }{\frac {(2l+1)}{kr}}i^{l}\sin(kr-{\frac {l\pi }{2}}+\delta _{l})P_{l}(\cos {\theta })\quad (r\to \infty) \quad \cdots (1)'}$ 

${\displaystyle \psi ^{+}(r,\theta) \to \sum _{l=0}^{\infty }{\frac {(2l+1)}{kr}}e^{i\delta _{l}}i^{l}\sin(kr-{\frac {l\pi }{2}}+\delta _{l})P_{l}(\cos {\theta })\quad (r\to \infty) \quad \cdots (4)'}$ 

Thus, than a state incident as for the dispersion state phase${\displaystyle \delta _{l}\}$ Hold it and slip off.

References

• Shigenobu Sunakawa "quantum theory Iwanami Shoten of the dispersion", 1977. ISBN 4000212133。

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This article is taken from the Japanese Wikipedia Phase gap

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