Consider a particle moving in a one-dimensional box for which the walls at x = -L/2 and x = L/2. (a) Write the wave-functions and probability densities for the state n = 1 and excited states n = 2, and n = 3. (b) Sketch the wave-functions and probability densities.
Solution;
As the wave function is bounded by X = -L/2 and X = L/2 that's why it must follow the boundary conditions then
𝚿(X=-L/2) = 0 and 𝚿(X=L/2) = 0
And we know that the wave function for a particle in a box under boundary conditions is
𝚿(X) = √2/L Sin(nπx/L)
This wave function is zero when n is even (n=2,n=4,n=6...) But this is not zero when n is odd (n=1,n=3,n=5...)
But if we write wave function in term of Cosine then it is zero when n is odd
So when n is odd then wave function will be
𝚿(X) = √2/L Cos(nπx/L)
But when n is even then wave function will be
𝚿(X) = √2/L Sin(nπx/L)
Further detail is given in pictures
