Particle in a Box: Probability Density and Positional Variance

​
quantum number n
1
The square of the absolute value of the Schrödinger wavefunction equals the probability density of finding a particle at any point in a box. It is relatively straightforward to show that for all values of
n
, the average or expectation value of the position of the particle will be at the exact midpoint of the box (
a/2
, where
a
is the length of the box)[1]. The variance, however, increases with increasing quantum number, to a limiting value of
2
a
/12
. This Demonstration plots the probability density (in blue) and a normal distribution (in yellow) with mean
a
2
and variance
2
a
12
1-
6
2
n
2
π
to represent the change in the variance with increasing
n
.

Details

The variance is
2
σ
=
2
x
-
2
〈x〉
, where the expectation values

2
x

and
〈x〉
are obtained from
a
∫
0
2
x
p(x)dx
and
a
∫
0
xp(x)dx
, respectively, where
a
is the length of the box. The (normalized) probability function for the one-dimensional particle in a box is given by[1]:
p(x)=
2
a
2
sin
nπx
a
.

References

[1] D. A. McQuarrie and J. D. Simon, Physical Chemistry: A Molecular Approach, Sausalito, CA: University Science Books, 1997.

External Links

Particles in 1D and 3D Boxes
Time-Dependent Superposition of Particle-in-a-Box Eigenstates
Two Electrons in a Box: Energies
Wave Packets for Particle in a Box
Time-Dependent Superposition of 2D Particle-in-a-Box Eigenstates

Permanent Citation

Daniel Barr
​
​"Particle in a Box: Probability Density and Positional Variance"​
​http://demonstrations.wolfram.com/ParticleInABoxProbabilityDensityAndPositionalVariance/​
​Wolfram Demonstrations Project​
​Published: September 23, 2020