1 (235)

Prévia do material em texto

SOLUTIONSMANUAL TO ACCOMPANY ATKINS' PHYSICAL CHEMISTRY 227
P7B.8 �e probability of �nding the particle within the range x = a to x = b, and
y = c to y = d is
P(a → b, c → d) = ∫
b
a
∫
d
c
∣ψ(x , y)∣2 dxdy
= ∫
b
a
∫
d
c
(2/L)2 sin2(πx/L) sin2(πy/L)dxdy
�e integral separates into integrals over x and y
= ∫
b
a
(2/L) sin2(πx/L)dx × ∫
d
c
(2/L) sin2(πx/L)dx
Both integrals are evaluated using Integral T.2 to give the probability as
(b − a
L
− 1
2π
[sin(2πb
L
) − sin(2πa
L
)])(d − c
L
− 1
2π
[sin(2πd
L
) − sin(2πc
L
)])
Hence
(a) a = 0, b = L/2, c = 0, d = L/2; P = 1/4
(b) a = L/4, b = 3L/4, c = L/4, d = 3L/4; P = 1
4 (1 + 2/π)2 = 0.670
P7B.10 (a) Normalization requires �nding N such that N2 ∫
∞
−∞ exp(−x
2/a2)dx = 1.
Because the integrand is symmetric, the integral from −∞ to +∞ is twice
that from 0 to +∞. With this, the integral is evaluated using Integral G.1
2N2 ∫
∞
0
e−x
2/a2dx = 2N2 × 1
2 (πa2)1/2
Setting this equal to 1 gives N = (πa2)−1/4 . �e required probability is
given by
∫
a
−a
[(πa2)−1/4 exp(−x2/a2)]
2
dx = 2(πa2)−1/2 ∫
a
0
exp(−x2/a2)dx
= erf(1) = 0.843
�is integral has no analytical solution, but is easily evaluated usingmath-
ematical so�ware.
7C Operators and observables
Answers to discussion questions
D7C.2 In quantum mechanics an observable quantity (such as energy, position or
momentum) is represented by a particular operator Ω̂. If the wavefunction
is ψ the average value of the quantity represented by the operator Ω̂ is given by
⟨Ω⟩ = ∫ ψ∗Ω̂ψ dτ, called the expectation value. For the special case that ψ is
an eigenfunction of Ω̂, the expectation value is the eigenvalue corresponding
to this eigenfunction.