Question

Consider a particle-in the box trial wavefunction φ(×)=x^α(L-x). Optimize the energy functional ε with respect to the adjustable parameter α.

Answer 1

Solution:

The solution of Schrodinger equation for this system is available in nearly any introductory textbook on quantum mechanics. The expressions for the wave-functions and corresponding energy levels are given here;

Wave-functions, where n=1, 2, 3, ......., and (2/L) 1/2 is normalization constant/.

Energy levels, where m is the mass of the particle.

To simplify our work, we will utilize atomic units where mass(m) is equal to 1 and Planck's constant (h) is equal to 2p. we will also define the length of the box (L) to equal a unit-less length of 1.

Then we will get

We can do this for any wave-function in same manner.