19
1.3. Particle interactions in the Standard Model
k
k
′
g N
g N
FIGURE 1.2
Scattering by a static point-like U-source.
require ΔE = m U c
2 , then
ħ
r ∼
(1.25)
m U c
just as in (1.20). The ‘r’ in (1.25) is the extent of the separation allowed
between the n and the p, such that – in the time available – the U
± can
‘borrow’ the necessary energy to come into existence and cross from one to
the other. In this sense, r is the effective range of the associated force, as in
(1.20).
Despite the similarity to virtual intermediate states in ordinary quantum
mechanics, the Yukawa–Wick process is nevertheless truly revolutionary because it postulated an energy fluctuation ΔE great enough to create an as yet
unseen new particle, a new state of matter.
We proceed to explore further aspects of Yukawa’s force mechanism. The
reader should note that throughout the remainder of this book we shall generally (unless otherwise stated) use units such that ħ = c = 1: see Appendix B.
1.3.3 The one-quantum exchange amplitude
Consider a particle, carrying ‘strong charge’ g N , being scattered by an infinitely massive (static) point-like U-source also of ‘charge’ g N as pictured in
figure 1.2. From the previous section, we know that the potential energy in
the Schr¨ odinger equation for the scattered particle is precisely the U (r) from
(1.13). Treating this to its lowest order in U (r) (‘Born Approximation’ – see
appendix H), the scattering amplitude is proportional to the Fourier transform
of U (r):
∫
f (q) = e
iq·r U (r) d
3
r
(1.26)
where q is the momentum (or wavevector, since ħ = 1) transfer q = k − k
′ .
The transform is evaluated in appendix G equation (G.24), or in problem 1.1,
1.3. Particle interactions in the Standard Model
k
k
′
g N
g N
FIGURE 1.2
Scattering by a static point-like U-source.
require ΔE = m U c
2 , then
ħ
r ∼
(1.25)
m U c
just as in (1.20). The ‘r’ in (1.25) is the extent of the separation allowed
between the n and the p, such that – in the time available – the U
± can
‘borrow’ the necessary energy to come into existence and cross from one to
the other. In this sense, r is the effective range of the associated force, as in
(1.20).
Despite the similarity to virtual intermediate states in ordinary quantum
mechanics, the Yukawa–Wick process is nevertheless truly revolutionary because it postulated an energy fluctuation ΔE great enough to create an as yet
unseen new particle, a new state of matter.
We proceed to explore further aspects of Yukawa’s force mechanism. The
reader should note that throughout the remainder of this book we shall generally (unless otherwise stated) use units such that ħ = c = 1: see Appendix B.
1.3.3 The one-quantum exchange amplitude
Consider a particle, carrying ‘strong charge’ g N , being scattered by an infinitely massive (static) point-like U-source also of ‘charge’ g N as pictured in
figure 1.2. From the previous section, we know that the potential energy in
the Schr¨ odinger equation for the scattered particle is precisely the U (r) from
(1.13). Treating this to its lowest order in U (r) (‘Born Approximation’ – see
appendix H), the scattering amplitude is proportional to the Fourier transform
of U (r):
∫
f (q) = e
iq·r U (r) d
3
r
(1.26)
where q is the momentum (or wavevector, since ħ = 1) transfer q = k − k
′ .
The transform is evaluated in appendix G equation (G.24), or in problem 1.1,