Functions of the form cos(nπx/L), where n = 1, 3, 5 …, can be used to model the wavefunctions of particles confined to the region between x = −L/2 and x = +L/2. The integration is limited to the range −L/2 to +L/2 because the wavefunction is zero outside this range. Show that the wavefunctions are orthogonal for n = 1 and 3. You will find the necessary integral in the Resource section.

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Connection formulae for WKB approximation

In quantum mechanics, Schrödinger’s equation for certain systems with varying parameters cannot be solved directly. In such a case, an approximation method is used to obtain the approximate solution. WKB method is a special technique used to obtain the approximate solution for Schrödinger’s time-independent equation. In general, the WKB method is used to solve linear differential equations with varying coefficients. This method is called semiclassical approximation in quantum mechanics.