Chapter 7, Problem 7C.2P

(a)

Interpretation Introduction

Interpretation:

The identity $\sin A\sin B=\frac{1}{2}\left\{\cos \left(A-B\right)-\cos \left(A+B\right)\right\}$ has to be used to write the given integrand as a sum of two terms.

Concept introduction:

The integral of a wave function and the conjugate of another wavefunction over a restricted region represent the orthogonality of both the wave functions. If the integral is zero, the wavefunctions are orthogonal to each other. Trigonometric identities are used to simplify the integrals.

(b)

Interpretation Introduction

Interpretation:

Considering the case $n=2$, $m=1$, two separate sketch graphs of the two terms identifies in (a) have to be made in the range $x=0$ to $x=L$.

Concept introduction:

The cosine graph is a periodic graph.  It is an even function.  The area under the curve above the x axis is equal to the area under the curve below the x axis.

(c)

Interpretation Introduction

Interpretation:

The area enclosed between the curves and x axis is zero and hence the integral is zero has to be shown in both cases.

Concept introduction:

The area under the curve of a wavefunction is equal to the value of integral for that function.  The integral of a wave function and the conjugate of another wavefunction over a restricted region represent the orthogonality of both the wave functions. If the integral is zero the wavefunctions are orthogonal to each other.

(d)

Interpretation Introduction

Interpretation:

The argument in (c) has to be generalized for the case of arbitrary $n$ and $m$ $\left(n\ne m\right)$.

Concept introduction:

The integral of a wave function and the conjugate of another wavefunction over a restricted region represent the orthogonality of both the wave functions. If the integral is zero the wavefunctions are orthogonal to each other..