18
1. The Particles and Forces of the Standard Model
FIGURE 1.1
Yukawa’s single-U exchange mechanism for the n–p interaction. (a) U
− exchange. (b) U
+ exchange.
The relation (1.23) may be interpreted as follows (we abridge the careful
discussion in section 44 of Landau and Lifshitz (1977)). Imagine an ‘energymeasuring device’ set up to measure the energy of a quantum system. To do
this, the device must interact with the quantum system for a certain length of
time Δt. If the energy of a sequence of identically prepared quantum systems
is measured, only in the limit Δt → ∞ will the same energy be obtained
each time. For finite Δt, the measured energies will necessarily fluctuate by
an amount ΔE as given by (1.23); in particular, the shorter the time over
which the energy measurement takes place, the larger the fluctuations in the
measured energy.
Wick (1938) applied (1.23) to Yukawa’s theory, and thereby shed new light
on the relation (1.20). Suppose a device is set up capable of checking to see
whether energy is, in fact, conserved while the U
± crosses over in figure 1.1.
The crossing time t must be at least r/c, where r is the distance apart of the
nucleons. However, the device must be capable of operating on a time scale
smaller than t (otherwise it will not be in a position to detect the U
± ), but
it need not be very much less than this. Thus the energy uncertainty in the
reading by the device will be
3
ħc
ΔE ∼ .
(1.24)
r
As r decreases, the uncertainty ΔE in the measured energy increases. If we
3 In this kind of argument, the ‘∼’ sign should be understood as meaning that numerical
factors of order 1 (such as 2 or π) are not important. The coincidence between (1.25) and
(1.20) should not be taken too literally. Nevertheless, the physics of (1.25) is qualitatively
correct.
1. The Particles and Forces of the Standard Model
FIGURE 1.1
Yukawa’s single-U exchange mechanism for the n–p interaction. (a) U
− exchange. (b) U
+ exchange.
The relation (1.23) may be interpreted as follows (we abridge the careful
discussion in section 44 of Landau and Lifshitz (1977)). Imagine an ‘energymeasuring device’ set up to measure the energy of a quantum system. To do
this, the device must interact with the quantum system for a certain length of
time Δt. If the energy of a sequence of identically prepared quantum systems
is measured, only in the limit Δt → ∞ will the same energy be obtained
each time. For finite Δt, the measured energies will necessarily fluctuate by
an amount ΔE as given by (1.23); in particular, the shorter the time over
which the energy measurement takes place, the larger the fluctuations in the
measured energy.
Wick (1938) applied (1.23) to Yukawa’s theory, and thereby shed new light
on the relation (1.20). Suppose a device is set up capable of checking to see
whether energy is, in fact, conserved while the U
± crosses over in figure 1.1.
The crossing time t must be at least r/c, where r is the distance apart of the
nucleons. However, the device must be capable of operating on a time scale
smaller than t (otherwise it will not be in a position to detect the U
± ), but
it need not be very much less than this. Thus the energy uncertainty in the
reading by the device will be
3
ħc
ΔE ∼ .
(1.24)
r
As r decreases, the uncertainty ΔE in the measured energy increases. If we
3 In this kind of argument, the ‘∼’ sign should be understood as meaning that numerical
factors of order 1 (such as 2 or π) are not important. The coincidence between (1.25) and
(1.20) should not be taken too literally. Nevertheless, the physics of (1.25) is qualitatively
correct.