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A state decays at the Fermi-golden-rule rate
W
=
(
2
π
/
ℏ
)
∣
V
f
i
∣
2
ρ
W = (2\pi/\hbar)|V_{fi}|^2 \rho
W
=
(
2
π
/ℏ
)
∣
V
f
i
∣
2
ρ
with
ℏ
=
1
\hbar = 1
ℏ
=
1
,
V
f
i
=
0.05
V_{fi} = 0.05
V
f
i
=
0.05
,
ρ
=
80
\rho = 80
ρ
=
80
. After exactly one lifetime
τ
=
1
/
W
\tau = 1/W
τ
=
1/
W
, the survival probability is
P
=
e
−
W
τ
P = e^{-W\tau}
P
=
e
−
W
τ
. What is
P
P
P
?
0.5
e
−
2
≈
0.135
e^{-2} \approx 0.135
e
−
2
≈
0.135
1
−
1
/
e
≈
0.632
1 - 1/e \approx 0.632
1
−
1/
e
≈
0.632
(decay, not survival)
1
/
e
≈
0.368
1/e \approx 0.368
1/
e
≈
0.368
Check answer