Physical Foundations of Cosmology

4.5 Instantons, sphalerons and the early universe 181

a(E) b(E)

qm

E

(^00)
q
instanton
q
sphaleron
V −V
Fig. 4.12.
First, we neglect the thermal fluctuations and assume that the particle, withfixed
energyE<V(qm), is initially localized in the potential well. The only way to
escape from this well is via subbarrier tunneling. If the tunneling probability is
small, the energyEis anapproximateeigenvalue of the Hamiltonian, and we can
use the stationary Schr ̈odinger equation
(
−

1

2 M

∂^2

∂q^2

+V(q)

)

E (4.143)

toestimatethe tunneling amplitude. The approximate semiclassical solution of this

equation is

∝exp

(

i

∫ √

2 M(E−V)dq

)

. (4.144)

In the classically allowed regions, whereE>V,the wave function simply os-

cillates, while under the barrier it decays exponentially. Hence, in the region

q>b(E),the wave function is suppressed by the factor

exp

⎛

⎝−

∫b(E)

a(E)

√

2 M(V−E)dq

⎞

⎠ (4.145)

compared to its value inside the potential well. Expression (4.145) accounts for the

dominant exponential contribution to the tunneling amplitude.