Subatomic Physics

262 P , C, CP, andT
What happens to a particle that is dropped into one potential well, say L, at
time t = 0? Equation (9.63) gives its state at t =0as
|ψ(0)〉 = |L〉 =
�
1
2 {|s〉 + |a〉}; (9.68)
the state does not have definite parity and is not an eigenstate of H. To investigate
the behavior of the particle at later times, we use the time-dependent Schrödinger
equation
and the expansion
i� d
dt |ψ(t)〉 =(H0 + Hint)|ψ(t)〉 (9.69)
|ψ(t)〉 = α(t)|L〉 + β(t)|R〉
|α(t)| 2 + |β(t)| 2 =1.
(9.70)
Inserting the expansion (9.70) into the Schrödinger equation (9.69) and multiplying
in turn from the left by 〈L| and 〈R|, yields a system of two coupled differential
equations for α(t) andβ(t):
i� ˙α(t) =(E0 + E ′ )α(t)+∆Eβ(t)
i� ˙ β(t) =∆Eα(t)+(E0 + E ′ )β(t).
(9.71)
The solution of these equations with the initial conditions α(0) = 1 and β(0) = 0
gives
�
−i(E0 + E
|ψ(t)〉 =exp
′ �� � � � � �
)t ∆Et
∆Et
cos |L〉−i sin |R〉 . (9.72)
�
�
�
The probability of finding the particle, dropped into well L at t =0,inwellR at a
time t is given by the absolute square of the expansion coefficient of |R〉, or
prob(R) =sin 2
� �
∆Et
. (9.73)
�
The particle hence oscillates between the two wells with a circular frequency 2ω,
where
ω = ∆E
�
1
= 〈L|Hint|R〉 . (9.74)
�