# Particle in a box ground states

The ground state wave function for a particle in a box confined to a 1-D box of length L is Ψ=(2/L)^(1/2) sin (pi x/ L)

Suppose that the box is 10.0 nm long. Calculate the probability that the particle is

a) between x=9.90 nm and x= 10.0 nm

b) in the right half of the box

https://brainmass.com/physics/wavefunction/particle-box-ground-states-302022

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We start with the simple trigonometric identity:

(1.1)

Therefore:

(1.2)

The probability to find the particle between and is given by:

(1.3)

For the ground state:

(1.4)

Therefore:

(1.5)

Simple substitution:

(1.6)

And we get normalization of the box width (we set L=1) and the interval is now measured in the fraction of the box's size

(1.7)

For we obtain the probability:

For we obtain the probability:

As expected, due to the symmetry of the ground state the particle has 50% to be in the right 50% of the box.

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