How Is Equation 15.5.7 Derived in Schaum's Outline of Quantum Mechanics?

[jhon]

in Schaum's Outline of Theory and Problems of Quantum Mechanics

how did they got eq 15.5.7

 

[jdwood983]

Did you try using the relation

$$\sin[x]=\frac{\exp[ix]-\exp[-ix]}{2i}$$

in Equation 15.5.6 and expanding to see what you get? It looks like you should get the result they obtain if you work at it using that relation.

 

[jhon]

the second eq II ?? it's equale zero why??

 

[jdwood983]

If you have expanded Equation 15.5.6 using the substitution given, try separating the resulting equation into the real and imaginary components. It looks to me like Equation I in 15.5.7 is the imaginary part and Equation II of 15.5.7 is the real part after substituting for $\sin[x]$.

 

[gabbagabbahey]

I don't think that the two parts of 15.5.7 are the real and imaginary parts of 15.5.6, but rather they result from the fact that $e^{ikr}$ and $e^{-ikr}$ are orthogonal functions of $r$, and hence the relation [tex]\alpha e^{ikr}+\beta e^{-ikr}=0[/itex] can only be true for all $r$ if $\alpha=\beta=0$.

 

[jhon]

jdwood983 and gabbagabbahey thanks for help