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Transition Probability

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  • #1
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At page 190 of Bransden & Joachain (see the page from http://books.google.com/books?id=i5IPWXDQlcIC&printsec=frontcover&dq=B.+H.+Bransden,+Charles+Jean+Joachain&hl=da#PPA190,M1"), there are 2 expressions for the transition probability, (4.38) and it's absolute value squared in (4.39).
Is it just me or are the 2 term dimensionally different? Obviously everything from (4.38) is squared in (4.39) except the [itex] d\omega [/itex]. Hence they can't be the same dimensionally. How can this be right?
 
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  • #2
Cyosis
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I am not sure what the question is, but they are both dimensionless. Probability doesn't have a unit.
 
  • #3
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Cyosis: Yes they are supposed to be dimensionless, but I just looked at the difference between the 2 expressions.

Note that every term from (4.38) appears squared in (4.39) except for the [itex]d\omega [/itex] term which has the same power in both, hence (4.39) is short by a factor of 1/sec.
 
  • #4
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In any case (4.38) is not the probability, it's the amplitude.
 
  • #5
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But it's supposed to be dimensionless like the probability, hence must have the same dimensions.

EDIT:
Ok to make it more clear, then (4.38) has the form

[tex] c_b = \int_0^{\infty} f(\omega) \, d\omega [/tex]

while (4.39) has the form

[tex] |c_b|^2 = \int_0^{\infty} |f(\omega)|^2 \, d\omega [/tex]

which certainly can't be right. What's going on?
 
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  • #6
Redbelly98
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That's weird, it certainly does look dimensionally inconsistent, regardless of what cb represents.

I'm assuming ω means what it usually does and has units of s-1.
 
  • #7
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