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240 7 QUANTUM THEORY
�e probability density is zero when the wavefunction is zero, which is when
the argument of the sine function is a multiple of π
5πx/L = 0, π, 2π, ... hence x = 0, L/5, 2L/5, 3L/5, 4L/5, L
E7D.10(b) �e energy levels of a particle in a cubical box with side L are given by [7D.13b–
267], En = h2n2/8mL2, with n2 = n2x + n2y + n2z . When the side is decreased to
0.9 × L the energies become E′n = h2n2/8m(0.9 × L)2 = h2n2/8m(0.81 × L2)
giving a fractional change of
E′n − En
En
= [h2n2/8m(0.81 × L2)] − [h2n2/8mL2]
h2n2/8mL2
= 1
0.81
− 1 = 0.235
E7D.11(b) �e energy levels of a particle in a box of length L are given by [7D.6–263],
En = h2n2/8mL2. �is energy is equal to the average thermal energy when
kT/2 = h2n2/8mL2, leading to n = (2L/h)(mkT)1/2 .
For an argon atom, mass 39.95 mu in a box for length 0.1 cm this evaluates as
n = 2(0.1 × 10
−2 m)
6.6261 × 10−34 J s
× [(39.95 × 1.6605 × 10−27 kg) × (1.3806 × 10−23 JK−1) × (298 K)]1/2
= 5.0 × 107
E7D.12(b) �e wavefunction of a particle in a square box of side length L with quantum
numbers n1 = 1, n2 = 3 is ψ1,3(x , y) = (2/L) sin (πx/L) sin (3πy/L).�e cor-
responding probability density is P1,3(x , y) = (2/L)2 sin2 (πx/L) sin2 (3πy/L).
�e probability density is maximized when sin2 (πx/L) × sin2 (3πy/L) = 1
which occurs only when each sin term is equal to ±1.�e term in x is equal to
1 when πx/L = π/2 and hence x = L/2. For the term in y
sin (3πy/L) = ±1
3πy/L = π/2, 3π/2, 5π/2 hence y = L/6, L/2, 5L/6.
Hence the maxima occur at (x , y) = (L/2, L/6), (L/2, L/2), (L/2, 5L/6)
Nodes occur when the wavefunction passes through zero, which is when either
of the sin terms are zero, excluding the boundaries at x = 0, L y = 0, L because at
these points the wavefunction does not pass through zero.�ere are therefore
no nodes associated with the function sin (πx/L). In the y-direction there are
nodes when
sin (3πy/L) = 0
3πy/L = π, 2π hence y = L/3, 2L/3
�ere is thus a node when y = L/3 and for any value of x, that is a nodal line at
y = L/3 and parallel to the x-axis . Likewise the node at y = 2L/3 corresponds
to a nodal line at y = 2L/3 and parallel to the x-axis .