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232 7 QUANTUM THEORY
P7C.4 An operator Ω̂ is hermitian if ∫ ψ∗i Ω̂ψ j dτ = [∫ ψ∗j Ω̂ψ i dτ]
∗
, [7C.7–253]. Pro-
ceed by integrating by parts (�e chemist’s toolkit 15 in Topic 7C on page 254)
to give
∫
2π
0
ψ∗i (
ħ
i
d
dϕ
)ψ j dϕ =
ħ
i
⎛
⎝
ψ∗i ψ j ∣
2π
0
´¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¶
A
−∫
2π
0
ψ j
dψ∗i
dϕ
dϕ
⎞
⎠
�e term A is zero because the wavefunction must be single valued, requiring
ψ i(ϕ) = ψ i(ϕ + 2π), and so ψ i(0) = ψ i(2π). It follows that
∫
2π
0
ψ∗i (
ħ
i
d
dϕ
)ψ j dϕ = −
ħ
i ∫
2π
0
ψ j
dψ∗i
dϕ
dϕ
�e term of the right is written as a complex conjugate to give
∫
2π
0
ψ∗i (
ħ
i
d
dϕ
)ψ j dϕ = [ħ
i ∫
2π
0
ψ∗j
dψ i
dϕ
dϕ]
∗
= [∫
2π
0
ψ∗j (
ħ
i
d
dϕ
)ψ i dϕ]
∗
Note that because the complex conjugate of the whole term is taken, to com-
pensate for this the complex conjugate of the terms inside the bracket need to
be taken too ψ = [ψ∗]∗; i∗ = −i is also used. �is �nal equation is consistent
with [7C.7–253] and so demonstrates that the angular momentum operator is
hermitian.
P7C.6 �e expectation value is given by [7C.11–256], ⟨Ω⟩ = ∫ ψ∗Ω̂ψ dτ, where ψ is
normalized. However, if ψ is an eigenfunction of Ω̂ each measurement gives
the corresponding eigenvalue, and this is therefore also the expectation value.
(a) �e function N exp(ikx) is an eigenfunction of the linear momentum
operator p̂x = (ħ/i)(d/dx)
ħ
i
d
dx
Neikx = ħ
i
× N ikeikx = ħk × Neikx
and the eigenvalue is ħk . Hence, the expectation value is equal to this.
(b) �e wavefunction N cos kx is not an eigenfunction of the linear momen-
tum operator, so the expectation value has to be computed by evaluating
the integral. First consider the e�ect of applying this operator to thewave-
function
p̂xψ = (ħ/i)(d/dx)N cos(kx) = −(ħk/i)N sin kx
It follows that the expectation value is given by
⟨px⟩ = −(ħk/i)N2 ∫
∞
−∞
cos(kx) sin(kx)dx = 0
�e integrand is an odd function, meaning that its value at −x is the
negative of that at x, whichmeans that its integral over a symmetric range
is zero.