Yukava potensiali

---

Zarrachalar, atom va kondensatsiyalangan muhitlar fizikasida Yukava potensiali (shuningdek, ekranlangan Kulon potensiali deb ham ataladi) yapon fizigi Xideki Yukava nomi bilan atalgan potensialdir. Potentsial quyidagi shaklga ega:





V

Yukawa


(
r
)
=
−

g

2





e

−
α
m
r


r


,


{\displaystyle V_{\text{Yukawa}}(r)=-g^{2}{\frac {e^{-\alpha mr}}{r}},}


Bu yerda g - masshtab shkalasi konstantasi, ya'ni potentsial amplitudasi, m - zarrachaning massasi, r - zarrachagacha bo'lgan radial masofa va α - boshqa masshtab konstantasi, shuning uchun



r
≈



1

α
m






{\displaystyle r\approx {\tfrac {1}{\alpha m}}}

taxminiy diapazondir. Potensial r -da monoton ravishda ortib bormoqda va u salbiy, bu kuchning tortishish kuchiligini bildiradi. XBS tizimida Yukava potentsialining birligi (1/metr).

Elektromagnetizmning Kulon potentsiali Yukava potentsialiga misol bo'la oladi, bunda




e

−
α
m
r




{\displaystyle e^{-\alpha mr}}

faktor 1 ga teng. Buni foton massasi m 0 ga teng deb talqin qilish mumkin. Foton o'zaro ta'sir qiluvchi, zaryadlangan zarralar orasidagi ta'sir tashuvchisidir.

Mezon maydoni va fermion maydoni o'rtasidagi o'zaro ta'sirlarda g doimiysi ushbu maydonlar orasidagi o'lchov juftligi konstantasiga teng. Yadro kuchiga kelsak, fermionlar proton bilan boshqa proton yoki neytron bo'lishi mumkin.

Tarixi

---

Xideki Yukavaning 1935-yildagi maqolasigacha fiziklar Jeyms Chadvikning kichik yadro ichida joylashgan, radiusi 10 metr boʻlgan musbat zaryadlangan proton va neytronlardan iborat atom modeli natijalarini tushuntirish uchun kurashdilar. Fiziklar bu masshtabdagi elektromagnit kuchlar ta'siri protonlarning bir-birini itarishi va natijada yadro parchalanishiga olib kelishini bilishgan. Shunday qilib, elementar zarralar orasidagi o'zaro ta'sirlarni yanada tushuntirish uchun motivatsiya paydo bo'ldi. 1932-yilda Verner Heisenberg yadro ichidagi neytronlar va protonlar o'rtasidagi "Platzwechsel" (migratsiya) o'zaro ta'sirini taklif qildi, unga ko'ra neytronlar proton va elektronlardan iborat kompozit zarralar edi. Ushbu kompozit neytronlar elektronlarni chiqaradi, protonlar bilan tortishish kuchi hosil qiladi va keyin o'zlari protonlarga aylanadi. 1933-yilda Solvay konferentsiyasida Geyzenberg o'zining o'zaro ta'sirini taklif qilganda,kuch qisqa masofaliligi tufayli, fiziklar bu ikki shaklda bo'lishi mumkin deb taxmin qilishdi:




J
(
r
)
=
a

e

−
b
r





or



J
(
r
)
=
a

e

−
b

r

2






{\displaystyle J(r)=ae^{-br}\quad {\textrm {or}}\quad J(r)=ae^{-br^{2}}}


Biroq, uning nazariyasida ko'p muammolar bor edi. Ya'ni, spini 1/2bo'lgan elektron va protonning spini 1/2 bo'lgan neytron hosil qilishi mumkin emasdi. Geyzenberg bu masalaga munosabati izospin g'oyalarini shakllantirishga davom etadi.

Kulon potentsialiga bog'liqligi

---

Agar zarrachaning massasi bo'lmasa (ya'ni, m = 0), u holda Yukava potensiali Kulon potensialiga kamayadi va diapazon cheksiz deyiladi. Aslida, bizda quyidagi chiqadi:




m
=
0
⇒

e

−
α
m
r


=

e

0


=
1.


{\displaystyle m=0\Rightarrow e^{-\alpha mr}=e^{0}=1.}


Natijada, tenglama ko'rinishi





V

Yukawa


(
r
)
=
−

g

2






e

−
α
m
r


r




{\displaystyle V_{\text{Yukawa}}(r)=-g^{2}\;{\frac {e^{-\alpha mr}}{r}}}


Kulon potentsiali shakliga soddalashadi





V

Coulomb


(
r
)
=
−

g

2





1
r


.


{\displaystyle V_{\text{Coulomb}}(r)=-g^{2}\;{\frac {1}{r}}.}


bu erda biz masshtab konstantasini o'rnatamiz:





g

2


=




q

1



q

2




4
π

ε

0







{\displaystyle g^{2}={\frac {q_{1}q_{2}}{4\pi \varepsilon _{0}}}}


Yukava va Kulon uchun uzoq masofali potentsial kuchning taqqoslanishi 2-rasmda ko'rsatilgan. Ko'rinib turibdiki, Kulon potentsiali uzoqroq masofaga ta'sir qiladi, Yukava potentsiali esa nolga tez yaqinlashadi. Biroq, har qanday Yukava potentsiali yoki Kulon potensiali har qanday katta r uchun nolga teng emas.

Furye konvertatsiyasi

---

Yukava potentsialining massiv maydon bilan bog'liqligini tushunishning eng oson yo'li uning Furye konvertatsiyasini o'rganishdir.




V
(

r

)
=



−

g

2




(
2
π

)

3





∫

e

i

k
⋅
r






4
π



k

2


+
(
α
m

)

2








d


3


k


{\displaystyle V(\mathbf {r} )={\frac {-g^{2}}{(2\pi )^{3}}}\int e^{i\mathbf {k\cdot r} }{\frac {4\pi }{k^{2}+(\alpha m)^{2}}}\,\mathrm {d} ^{3}k}


bu yerda integral k 3-vektor momentining barcha mumkin bo'lgan qiymatlari bo'yicha bajariladi. Ushbu shaklda va masshtablash faktorini birga teng deb belgilash,



α
=
1


{\displaystyle \alpha =1}

, kasr






4
π



k

2


+

m

2







{\textstyle {\frac {4\pi }{k^{2}+m^{2}}}}

Klein-Gordon tenglamasining targ'ibotchisi yoki Grin funktsiyasi sifatida ko'riladi.

Feynman amplitudasi

---

Yukava potentsialini juft fermionning o'zaro ta'sirining eng past tartibli amplitudasi sifatida olish mumkin. Yukava o'zaro ta'siri fermion maydonini birlashtiradi



ψ
(
x
)


{\displaystyle \psi (x)}

mezon maydoniga



ϕ
(
x
)


{\displaystyle \phi (x)}









L




i
n
t



(
x
)
=
g
 


ψ
¯


(
x
)
 
ϕ
(
x
)
 
ψ
(
x
)
 
.


{\displaystyle {\mathcal {L}}_{\mathrm {int} }(x)=g~{\overline {\psi }}(x)~\phi (x)~\psi (x)~.}


ulanish termini bilan birlashtiradi. Ikki fermion uchun tarqalish amplitudasi, biri




p

1




{\displaystyle p_{1}}

ikkinchisi esa




p

2




{\displaystyle p_{2}}

boshlang'ich impulsga ega, moment k bilan mezonni almashishi, o'ngdagi Feynman diagrammasi bilan berilgan.

Har bir cho'qqi uchun Feynman qoidalari g faktorini amplituda bilan bog'laydi; bu diagrammaning ikkita cho'qqisi borligi sababli, umumiy amplituda faktorga ega bo'ladi




g

2




{\displaystyle g^{2}}

. Ikki fermion chizig'ini bog'laydigan o'rtadagi chiziq mezon almashinuvini ifodalaydi. Zarrachalar almashinuvi uchun Feynman qoidasi - tarqatuvchidan foydalanish; massiv mezonning targ'ibotchisi hisoblanadi






−
4
π


 

k

2


+

m

2


 





{\textstyle {\frac {-4\pi }{~k^{2}+m^{2}~}}}

. Shunday qilib, biz ushbu grafik uchun Feynman amplitudasi boshqa narsa emasligini ko'ramiz

Shredinger tenglamasining xususiy qiymatlari

---

Yukava potentsialiga ega bo'lgan radial Shredinger tenglamasini perturbativ tarzda echish mumkin..  Radial Shredinger tenglamasidan quyidagi shaklda foydalanamiz





[





d


2




d


r

2





+

k

2


−



ℓ
(
ℓ
+
1
)


r

2




−
V
(
r
)

]

Ψ

(

ℓ
,
k
;

r

)

=
0
,


{\displaystyle \left[{\frac {\mathrm {d} ^{2}}{\mathrm {d} r^{2}}}+k^{2}-{\frac {\ell (\ell +1)}{r^{2}}}-V(r)\right]\Psi \left(\ell ,k;\,r\right)=0,}


va kuch bilan kengaytirilgan shaklda Yukawa potensiali:




V
(
r
)
=

∑

j
=
−
1


∞



M

j
+
1



(
−
r

)

j


,


{\displaystyle V(r)=\sum _{j=-1}^{\infty }M_{j+1}\,(-r)^{j},}


va



K
=
j
k


{\displaystyle K=jk}

sozlash kiritamiz, burchak momenti uchun



ℓ


{\displaystyle \ell }

ifoda olinadi:




ℓ
+
n
+
1
=
−





Δ

n


(
K
)



2
K





{\displaystyle \ell +n+1=-{\frac {\,\Delta _{n}(K)\,}{2K}}}






|

K

|

→
∞


{\displaystyle |K|\to \infty }

uchun,










Δ

n


(
K
)
=

M

0


−


1


2

K

2








[



n
(
n
+
1
)


M

2


+

M

0




M

1





]


−




2
n
+
1



4

K

3







M

0




M

2


 
+









+


1


8

K

4









[



3
(
n
−
1
)
n
(
n
+
1
)
(
n
+
2
)


M

4


+
2
(
3

n

2


+
3
n
−
1
)


M

3




M

0


 
+










 
+
 
6
n
(
n
+
1
)


M

2




M

1


+
2


M

2




M

0


2


+
3

M

1


2




M

0





]


 
+









+




2
n
+
1




8

K

5









[



3
(

n

2


+
n
−
1
)


M

4




M

0


+
3


M

3




M

0


2


+
n
(
n
+
1
)


M

2


2


+
4


M

2




M

1




M

0





]


 
+









+
 


O


⁡


(





1



K

7









)


 
.






{\displaystyle {\begin{aligned}&\Delta _{n}(K)=M_{0}-{\frac {1}{\,2K^{2}\,}}{\Bigl [}\,n(n+1)\,M_{2}+M_{0}\,M_{1}\,{\Bigr ]}-{\frac {\,2n+1\,}{4K^{3}}}\,M_{0}\,M_{2}~+\\&\qquad \qquad \quad +{\frac {1}{\,8K^{4}\,}}\,{\Bigl [}\,3(n-1)n(n+1)(n+2)\,M_{4}+2(3n^{2}+3n-1)\,M_{3}\,M_{0}~+\\&\qquad \qquad \qquad \qquad \qquad ~+~6n(n+1)\,M_{2}\,M_{1}+2\,M_{2}\,M_{0}^{2}+3M_{1}^{2}\,M_{0}\,{\Bigr ]}~+\\&\qquad \qquad \quad +{\frac {\,2n+1\,}{\,8K^{5}\,}}\,{\Bigl [}\,3(n^{2}+n-1)\,M_{4}\,M_{0}+3\,M_{3}\,M_{0}^{2}+n(n+1)\,M_{2}^{2}+4\,M_{2}\,M_{1}\,M_{0}\,{\Bigr ]}~+\\&\qquad \qquad \quad +~\operatorname {\mathcal {O}} {\Bigl (}\,{\frac {1}{\,K^{7}\,}}\,{\Bigr )}~.\end{aligned}}}


Orbital burchak momenti yoki Regge traektoriyasi uchun yuqoridagi kengayish



ℓ
(
K
)


{\displaystyle \ell (K)}

energiyaning xos qiymatlarini yoki





|


K



|



2




{\displaystyle {\bigl |}K{\bigr |}^{2}}

ekvivalentini olish uchun qaytarilishi mumkin.












|


K



|



2


 
=
 
−

M

1


 
+
 


1


4
(
ℓ
+
n
+
1

)

2









{




M

0


2


−
4
n
(
n
+
1
)
(
ℓ
+
n
+
1

)

2




M

2




M

0


+
4
(
2
n
+
1
)
(
ℓ
+
n
+
1

)

2





M

2





M

0






 
+







+
 
4




(
ℓ
+
n
+
1

)

4





M

0


3







[



3
(
n
−
1
)
n
(
n
+
1
)
(
n
+
2
+
3
)


M

4




M

0


 
+












−
 
3

n

2


(
n
+
1

)

2




M

2


2


+
2
(
3

n

2


+
3
n
−
1
)


M

3




M

0


2


+
2


M

2




M

0


3





]


 
+







−
 
24




(
2
n
+
1
)
(
ℓ
+
n
+
1

)

5





M

0


4







[



(

n

2


+
n
−
1
)


M

0




M

4


+

M

0


3




M

3


−
n
(
n
+
1
)


M

2


2





]


 
+







−
 
4





(
ℓ
+
n
+
1

)

6





M

0


7







[


 
10
(
n
−
2
)
(
n
−
1
)
n
(
n
+
1
)
(
n
+
2
)
(
n
+
3
)


M

6




M

0


2


 
+










+
 
4


M

3




M

0


5


+
2


(



5
n
(
n
+
1
)
(
3

n

2


+
3
n
−
10
)
+
12



)




M

5




M

0


3


 
+










+
 
2
(
6

n

2


+
6
n
−
11
)


M

4




M

0


4


+
2
(
9

n

2


+
9
n
−
1
)


M

2


2




M

0


3


 
+










−
 
10
n
(
n
+
1
)
(
3

n

2


+
3
n
+
2
)


M

3




M

2




M

0


2


+
20

n

3


(
n
+
1

)

3




M

2


3


 
+










−
 
30
(
n
−
1
)

n

2


(
n
+
1

)

2


(
n
+
2
)


M

4




M

2




M

0





]



+

⋯


}



.






{\displaystyle {\begin{aligned}&{\bigl |}K{\bigr |}^{2}~=~-M_{1}~+~{\frac {1}{\,4(\ell +n+1)^{2}\,}}\,{\biggl \{}\;M_{0}^{2}-4n(n+1)(\ell +n+1)^{2}\,M_{2}\,M_{0}+4(2n+1)(\ell +n+1)^{2}{\frac {M_{2}}{\;M_{0}\,}}~+\\&\quad +~4{\frac {\;(\ell +n+1)^{4}\,}{M_{0}^{3}}}\,{\Bigl [}\,3(n-1)n(n+1)(n+2+3)\,M_{4}\,M_{0}~+\\&\qquad \qquad \qquad \qquad \qquad \qquad -~3n^{2}(n+1)^{2}\,M_{2}^{2}+2(3n^{2}+3n-1)\,M_{3}\,M_{0}^{2}+2\,M_{2}\,M_{0}^{3}\,{\Bigr ]}~+\\&\quad -~24{\frac {\,(2n+1)(\ell +n+1)^{5}\,}{M_{0}^{4}}}\,{\Bigl [}\,(n^{2}+n-1)\,M_{0}\,M_{4}+M_{0}^{3}\,M_{3}-n(n+1)\,M_{2}^{2}\,{\Bigr ]}~+\\&\quad -~4\,{\frac {\,(\ell +n+1)^{6}\,}{M_{0}^{7}}}\,{\Bigl [}~10(n-2)(n-1)n(n+1)(n+2)(n+3)\,M_{6}\,M_{0}^{2}~+\\&\qquad \qquad \qquad \qquad +~4\,M_{3}\,M_{0}^{5}+2{\Bigl (}\,5n(n+1)(3n^{2}+3n-10)+12\,{\Bigr )}\,M_{5}\,M_{0}^{3}~+\\&\qquad \qquad \qquad \qquad +~2(6n^{2}+6n-11)\,M_{4}\,M_{0}^{4}+2(9n^{2}+9n-1)\,M_{2}^{2}\,M_{0}^{3}~+\\&\qquad \qquad \qquad \qquad -~10n(n+1)(3n^{2}+3n+2)\,M_{3}\,M_{2}\,M_{0}^{2}+20n^{3}(n+1)^{3}\,M_{2}^{3}~+\\&\qquad \qquad \qquad \qquad -~30(n-1)n^{2}(n+1)^{2}(n+2)\,M_{4}\,M_{2}\,M_{0}\,{\Bigr ]}\quad +\quad \cdots {\biggr \}}\quad .\end{aligned}}}


Yuqoridagi burchak momentining asimptotik kengayishi



ℓ
(
K
)


{\displaystyle \ell (K)}

ning kamayib borayotgan kuchlarida



K


{\displaystyle K}

WKB metodi bilan ham olinishi mumkin. Biroq, u holda, Kulon potentsialidagi kabi



ℓ
(
ℓ
+
1
)


{\displaystyle \ell (\ell +1)}

ifoda Shredinger tenglamasining markazdan qochma hadi





(

ℓ
+



1
2




)


2




{\displaystyle \left(\ell +{\tfrac {1}{2}}\right)^{2}}

bilan almashtirilishi kerak.

Ko'ndalang kesim

---

Yukava potentsialidan foydalanib, proton va neytron yoki pion o'rtasidagi differentsial kesimni hisoblashimiz mumkin. Biz Born yaqinlashuvidan foydalanamiz, bu bizga sferik simmetrik potentsialda biz chiquvchi tarqoq to'lqin funksiyasini kiruvchi tekislik to'lqin funktsiyasi va kichik chetlashishning yig'indisi sifatida taxmin qilishimiz mumkinligini aytadi:




ψ
(



r
→



)
≈
A

[

(

e

i
p
r


)
+



e

i
p
r


r


f
(
θ
)

]



{\displaystyle \psi ({\vec {r}})\approx A\left[(e^{ipr})+{\frac {e^{ipr}}{r}}f(\theta )\right]}


bu yerda






p
→



=
p



z
^





{\displaystyle {\vec {p}}=p{\hat {z}}}

zarrachaning kirishdagi impulsi. Funktsiya



f
(
θ
)


{\displaystyle f(\theta )}

quyidagicha berilgan:




f
(
θ
)
=



−
2
μ



ℏ

2



|




p
→



−




p
→



′


|






∫

0


∞


r

V
(
r
)

sin
⁡

(


|




p
→



−




p
→



′


|

r

)

 

d

r


{\displaystyle f(\theta )={\frac {-2\mu }{\hbar ^{2}\left|{\vec {p}}-{\vec {p}}'\right|}}\,\int _{0}^{\infty }r\,V(r)\,\sin \left(\left|{\vec {p}}-{\vec {p}}'\right|r\right)~\mathrm {d} r}


bu yerda







p
→



′

=
p



r
^





{\displaystyle {\vec {p}}'=p{\hat {r}}}

zarrachaning chiqishdagi tarqoq impulsi va



μ


{\displaystyle \mu }

kiruvchi zarralarning massasi (pion massasi



m
,


{\displaystyle m,}

bilan adashtirmaslik kerak). Biz




V

Yukawa




{\displaystyle V_{\text{Yukawa}}}

ga



f
(
θ
)


{\displaystyle f(\theta )}

ni kiritish orqali hisoblaymiz :




f
(
θ
)
=



2
μ



ℏ

2



|




p
→



−




p
→



′


|






g

2



∫

0


∞



e

−
α
m
r



sin
⁡

(


|




p
→



−




p
→



′


|


r

)



d

r


{\displaystyle f(\theta )={\frac {2\mu }{\hbar ^{2}\left|{\vec {p}}-{\vec {p}}'\right|}}\,g^{2}\int _{0}^{\infty }e^{-\alpha mr}\,\sin \left(\left|{\vec {p}}-{\vec {p}}'\right|\,r\right)\,\mathrm {d} r}


Integralni baholash quyidagini beradi




f
(
θ
)
=



2
μ

g

2





ℏ

2




[

(
α
m

)

2


+


|




p
→



−




p
→



′


|


2



]






{\displaystyle f(\theta )={\frac {2\mu g^{2}}{\hbar ^{2}\,\left[(\alpha m)^{2}+\left|{\vec {p}}-{\vec {p}}'\right|^{2}\right]}}}


Energiyani tejash nazarda tutadiki:






|





p
→





|


=


|






p
→



′



|


=
p
 


{\displaystyle {\bigl |}{\vec {p}}{\bigr |}={\bigl |}{\vec {p}}'{\bigr |}=p~}


Shuning uchun:





|




p
→



−




p
→



′


|

=
2

p

sin
⁡

(




1
2



θ

)

 


{\displaystyle \left|{\vec {p}}-{\vec {p}}'\right|=2\,p\,\sin \left({\tfrac {1}{2}}\theta \right)~}


o'rniga qo'yish orqali quyidagini olamiz:




f
(
θ
)
=



2
μ

g

2





ℏ

2



[

(
α
m

)

2


+
4


p

2




sin

2


⁡

(



1
2


θ

)


]






{\displaystyle f(\theta )={\frac {2\mu g^{2}}{\hbar ^{2}\left[(\alpha m)^{2}+4\,p^{2}\,\sin ^{2}\left({{\frac {1}{2}}\theta }\right)\right]}}}


Shunday qilib, biz differensial kesimni olamiz:








d

σ



d

Ω



=


|

f
(
θ
)

|


2


=



4

μ

2



g

4





ℏ

4


 


[

(
α
m

)

2


+
4

p

2



sin

2


⁡

(



1
2