Question

The wavefunction of a particle in a box is ψ(x)=Cx(L−x)
(a) Determine the constant C

Answer 1

To find the constant C for the wave function of a particle in a box, one must use the normalization condition and integrate the square of the given wave function over the interval from 0 to L, set it equal to 1, and solve for C.

Step-by-step explanation:

The student has asked us to determine the constant C for the wave function ψ(x) = Cx(L-x) of a particle in a box by using the normalization condition. To find C, we must ensure that the integral of the square of the wave function over the region from 0 to L is equal to 1. This is the condition that the total probability of finding the particle within the box is certain (i.e., 100%).

The normalization condition is given by:

\(\int_{0}^{L} |ψ(x)|^2 dx = 1\)

We need to calculate the integral:

\(\int_{0}^{L} |Cx(L-x)|^2 dx\)

and solve for C which will yield:

C = √( ⁄ ( &int_0^L (x^2(L-x)^2 dx ) ) )