Keuzecollege Hoge EnergieFysica Katholieke Universiteit ... - EHEF
Keuzecollege Hoge EnergieFysica Katholieke Universiteit ... - EHEF
Keuzecollege Hoge EnergieFysica Katholieke Universiteit ... - EHEF
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2.5 Laat zien dat inderdaad alle factoren V in formule (2.41)<br />
⎛ ⎞<br />
1<br />
( dσ ----------------------------------------------------- M 2 ( 2π) 4 δ⎜<br />
) tegen elkaar<br />
4 ( p A<br />
⋅ p B<br />
) 2 2 2 ⎜<br />
p i<br />
– m AmB ∑<br />
– p A<br />
– p ⎟<br />
d 3 p i<br />
=<br />
B⎟∏<br />
----------------------<br />
⎝<br />
⎠ ( 2π) 3 2E i<br />
i = 1<br />
wegvallen. Hint: het kwadraat van de deltafunctie in<br />
kan worden geschreven als een deltafunctie<br />
maal een volume V . Laat dit ook zien.<br />
2<br />
T<br />
W fi<br />
fi<br />
= ---------- =<br />
TV<br />
N<br />
N<br />
Vd 3 p i<br />
= --------------------------------------------------- ; faseruimte<br />
;<br />
4 ( p A<br />
⋅ p B<br />
) 2 2 2<br />
– m AmB ∏----------------------<br />
( 2π) 3 V N d 3 p i<br />
= =<br />
2E i<br />
∏----------------------<br />
( 2π) 3 2E i<br />
N<br />
T fi<br />
2<br />
⎛<br />
⎞ ⎛ ⎞<br />
– i( 2π) 4 δ⎜<br />
∑<br />
p p i –<br />
A<br />
– p ⎟<br />
⎜<br />
B<br />
N<br />
⎟ A<br />
N ⎜<br />
B<br />
N ⎟<br />
∏ i<br />
M<br />
⎜ ⎟<br />
⎝<br />
⎠ ⎝ ⎠<br />
i = 1<br />
i = 1<br />
----------------------------------------------------------------------------------------------------------------<br />
TV<br />
⎛ ⎛<br />
⎞⎞ 2<br />
⎜( 2π) 4 δ⎜<br />
∑<br />
p p i –<br />
A<br />
–<br />
⎜<br />
p ⎟⎟<br />
1<br />
B<br />
--<br />
⎜<br />
⎟⎟<br />
⎝ ⎛ V⎠<br />
⎞ 2 ⎛ 1 ⎞ 2<br />
⎜ ------- ⎟<br />
∏<br />
M 2<br />
⎜<br />
⎝ ⎝<br />
⎠⎠<br />
⎝<br />
V⎟<br />
⎠<br />
i = 1<br />
i = 1<br />
= ----------------------------------------------------------------------------------------------------------------------<br />
TV<br />
N<br />
⎛ ⎛<br />
⎞⎞ TV⎜( 2π) 4 δ⎜<br />
⎜∑ p p i<br />
–<br />
A<br />
– p ⎟⎟<br />
⎜<br />
B⎟⎟<br />
⎛ 1<br />
⎝V -- ⎠<br />
⎞ 2 --<br />
1<br />
⎝ ⎛ V⎠<br />
⎞ N M 2<br />
⎝ ⎝<br />
⎠⎠<br />
i = 1<br />
= -------------------------------------------------------------------------------------------------------------- =<br />
TV<br />
1<br />
--------<br />
flux<br />
dσ<br />
V 2<br />
W fi<br />
faseruimte<br />
= -------------------------------- =<br />
flux<br />
N<br />
N<br />
N<br />
i = 1<br />
N<br />
N<br />
i = 1<br />
2<br />
N<br />
⎛<br />
⎞<br />
( 2π) 4 δ⎜<br />
∑<br />
p p i<br />
–<br />
A<br />
– p ⎟<br />
B<br />
M 2<br />
⎜<br />
⎟<br />
⎝<br />
⎠<br />
i = 1<br />
-----------------------------------------------------------------------<br />
V N + 2<br />
N<br />
i = 1<br />
⎛<br />
⎞<br />
( 2π) 4 δ⎜<br />
∑<br />
p p i –<br />
A<br />
– p ⎟<br />
⎜<br />
B⎟<br />
M 2<br />
⎝<br />
⎠<br />
i = 1<br />
V 2<br />
-----------------------------------------------------------------------<br />
V N + 2 ---------------------------------------------------V N<br />
4 ( p A<br />
⋅ p B<br />
) 2 2 2<br />
– m AmB N<br />
∏<br />
i = 1<br />
d 3 p i<br />
----------------------<br />
( 2π) 3 2E i<br />
=<br />
N<br />
⎛ ⎛<br />
⎞<br />
⎜( 2π) 4 δ⎜<br />
⎜∑ p p i – A – p ⎟<br />
⎜<br />
B⎟<br />
⎝ ⎝<br />
⎠<br />
i = 1<br />
M 2<br />
⎞<br />
⎟ 1<br />
---------------------------------------------------<br />
⎟<br />
⎠ 4 ( p A ⋅ p B ) 2 2 2<br />
– m AmB N<br />
∏<br />
i = 1<br />
d 3 p i<br />
----------------------<br />
( 2π) 3 2E i<br />
dσ e 4 ⎛<br />
2.6 Leid formule (2.48) (-------<br />
------------- 3 + 2m2 ⁄ p 2 + cos 2 θ⎞ 2 α 2 ⎛<br />
) af.<br />
dΩ 16π 2 ⎜------------------------------------------------<br />
s⎝<br />
sin 2 ⎟ ----- 3 + 2m2 ⁄ p 2 + cos 2 θ⎞ 2<br />
= = ⎜------------------------------------------------<br />
θ ⎠ s ⎝ sin 2 ⎟<br />
θ ⎠<br />
⎛<br />
⎞<br />
1<br />
dσ = ----------------------------------------------------- M 2 ( 2π) 4 δ⎜<br />
p i<br />
4 ( p A ⋅ p B ) 2 2 2<br />
– m AmB ∑<br />
– p A –<br />
⎜<br />
p ⎟<br />
B⎟<br />
⎝<br />
⎠<br />
2<br />
i = 1<br />
2<br />
∏<br />
i = 1<br />
d 3 p<br />
---------------------- i<br />
( 2π) 3 2E i<br />
144 Collegedictaat <strong>Hoge</strong> Energiefysica
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