高等量子力学 (3).pdf

文档介绍：第五章
1.(1)
Z
~ ~ ~
ψˆδ(r) = hr|e−iδ·pˆ/~|ψi = dphr|e−iδ·pˆ/~|pihp|ψi
Z Z
~ 1 ~
= dphr|pie−iδ·p/~hp|ψi = dp ei(r−δ)·p/~hp|ψi
(2π~)3/2
Z
= dphr −~δ|pihp|ψi = hr −~δ|ψi = ψ(r −~δ) (1)
(2)
~ ~ ~ ~
ψˆδ(p) = hp|e−iδ·pˆ/~|ψi = e−iδ·pˆ/~hp|ψi = e−iδ·pˆ/~ψ(p) (2)
2.(1)
ˆ
ψˆg(r) = hr|e−iθ0ln |ψi (3)
2
其中ln是l沿转轴n的分量,θ0为转角。选取l 及ln 对角化的基矢|ηlmi,m是ln的量子数,
设η为分立值,则:
X X
ψˆg(r) = hr|e−iθ0ln |ηlmihηlm|ψi = e−imθ0 hr|ηlmihηlm|ψi (4)
ηlm ηlm
imϕ−1
相对于n轴把r表示为(r, θ, ϕ)时,有hr|ηlmi = e uηlm(r, θ),g r应当表示为(r, θ, ϕ−θ0).故
−imθ0 im(ϕ−θ0) −1
e hr|ηlmi = e uηlm(r, θ) = hg r|ηlmi (5)
X
ψˆg(r) = hg−1r|ηlmihηlm|ψi = hg−1r|ψi = ψ(g−1r) (6)
ηlm
(2)
Z Z
ψˆg(p) = hp|ψˆgi = drhp|rihr|ψˆgi = drhp|rihg−1r|ψi
Z Z
= drhp|grihr|ψi = drhg−1p|rihr|ψi
= hg−1p|ψi = ψ(g−1p) (7)
1
3.
ψˆ(r) = hr|Πψi = h−r|ψi = ψ(−r) (8)
ψˆ(p) = hp|Πψi = h−p|ψi = ψ(−p) (9)
4.
Z
ˆ 0 0 0
ψ(r) = hr|T0ψi = hr|T0{ dr |r ihr |ψi}
Z Z
= hr| dr0|r0ihr0|ψi∗= dr0δ(r − r0)hr0|ψi∗
= hr|ψi∗= ψ∗(r) (10)
Z
ˆ∗
ψ(p) = hp|T0ψi = hp|{ dr|riψ(r)}
Z Z
= drhp|riψ∗(r) = { drh−p|riψ(r)}∗= ψ∗(−p) (11)
2
或利用T0 = 1也可证明。
5.
T J± = T (Jx ± iJy) = (−Jx ± iJy)T (12)
−iΠJy −iΠJy
e T J± = e (−Jx ± iJy)T
−iΠJy iΠJy −iΠJy
= (−e Jxe ± iJy)e T
−iΠJy
= (Jx ± iJy)e T (13)
6.
动量算符在时间反演下变号,它的一次幂必须与自旋算符ˆs相乘才能构成时间反演不变的
厄米算符。电子的自旋算符只能以一次幂出现(不存在高次幂的独立形式),它在空间反射下
不变,在转动下是