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P. Xiang and Y. A. Wang
following form:
̃
𝜌(𝐫) = N⟨ ̃
Ψ| ̃
Ψ⟩ N−1
= (1 − 𝛽
2
)N⟨Ψ 0 |Ψ 0 ⟩ N−1 + 𝜆
2 N⟨Ψ t |Ψ t ⟩ N−1 + 2𝜆
√
1 − 𝛽 2 NRe
( ⟨Ψ 0 |Ψ t ⟩ N−1
)
= (1 − 𝛽
2
)𝜌 0 (𝐫) + 𝜆
2
𝜌 t (𝐫) + 2𝜆
√
1 − 𝛽 2 NRe
( ⟨Ψ 0 |Ψ t ⟩ N−1
) .
(116)
At any point, the trial density is identical to the path density to ensure that the density
variation is actually along the path we designed:
̃
𝜌(𝐫) = 𝜌 p (𝐫) → 𝜌 0 (𝐫) .
(117)
Therefore, we have
⟨̃ 𝜌(𝐫)⟩ =
⟨
𝜌 p (𝐫)
⟩ .
(118)
Substituting Eqs. (114) and (116) into Eq. (118) and simplifying the result, one
derives
𝛽
2
= 𝜆
2
+ 2𝜆c 0
√
1 − 𝛽 2 .
(119)
At one specific point on the variational path, the value of 𝛽 is fixed, we can solve 𝜆
in terms of 𝛽 based on Eq. (119):
𝜆 = −c 0
√
1 − 𝛽 2 ±
√
c
2
0
(1 − 𝛽 2 ) + 𝛽 2 .
(120)
Near the end of the variational path, when 𝛽 → 0 and c 0 ≠ 0,
𝜆 → −c 0
√
1 − 𝛽 2 ±
[
c 0
√
1 − 𝛽 2 +
1
2c 0
𝛽
2
+ ⋯
]
.
(121)
Again (see Appendix 3), the positive sign is chosen in Eq. (121), and we have
𝜆 →
1
2c 0
𝛽
2
+ ⋯ , as 𝛽 → 0 .
(122)
Immediately, we can conclude that towards the end of variational path, 𝜆 is of the
same magnitude of 𝛽 2 ∕c 0 . In other words, 𝜆 also approaches zero at nearly the same
rate as 𝛽 2 ∕c 0 approaches zero.
Because of Eqs. (114), (116), and (117), we obtain
𝜆
2
𝜌 t + 2N𝜆
√
1 − 𝛽 2 Re
(⟨ Ψ 0 |
| Ψ t
⟩
N−1
) = 𝛽
2
𝜌  .
(123)