Chapter 35
Problem 53
†
Solution
Find the quantum state that corresponds to a 0.303 probability of finding the particle in the left-hand
quarter of an infinite square well.
The normalized wave function for a particle in an infinite square well is
r
nπ 2
sin
x
ψ=
L
L
To find the probability integrate the square of the wave function over the interval of interest.
P (x) = ψ 2 dx
Z
L/4
P (lef t quarter) =
0
Z
r
nπ 2
sin
x
L
L
!2
dx
L/4
nπ 2
sin2
x dx
L
L
0
Z
2 L/4 1
2nπ
P (lef t quarter) =
x
dx
1 − cos
L 0
2
L
L/4
sin 2nπ
x
1
L
x−
P (lef t quarter) =
L
2nπ/L 0
1 L
L
2nπ L
L
2nπ
P (lef t quarter) =
−
sin
−0+
sin
0
L 4
2nπ
L 4
2nπ
L
1 L
L
2nπ
P (lef t quarter) =
−
sin
−0+0
L 4
2nπ
4
nπ 1
1
P (lef t quarter) = −
sin
4 2nπ
2
If n is a multiple of 2, the sine function will give a value of zero and the probability will be 1/4 or 25%
(as you would expect. If the value of n is odd, the following probabilities are calculated.
1
1
(1)π
n=1
P (lef t quarter) = −
sin
= 0.0908
(9.08%)
4 2(1)π
2
1
1
(3)π
n=3
P (lef t quarter) = −
sin
= 0.303
(30.3%)
4 2(3)π
2
1
1
(5)π
sin
= 0.218
(21.8%)
n=5
P (lef t quarter) = −
4 2(5)π
2
1
1
(7)π
sin
= 0.273
(27.3%)
n=7
P (lef t quarter) = −
4 2(7)π
2
P (lef t quarter) =
The answer is n = 3. The additional calculations were included so you could see that as nodd → ∞, the
probability → 0.25.
†
Problem from Essential University Physics, Wolfson