Question

Normalize the equations

a) psi(x)=ae^(-kx) from zero to infinity

b) psi(x)=ae^(ikx) from 0 to L

Answer 1

Solution is in explanation

Step-by-step explanation:

part a)

For normalization we have

`\int_(0)^(\infty )f(x)dx=1\\\\\therefore \int_(0)^(\infty )ae^(-kx)dx=1\\\\\Rightarrow a\int_(0)^(\infty )e^(-kx)dx=1\\\\(a)/(-k)[(1)/(e^(kx))]_(0)^(\infty )=1\\\\(a)/(-k)[0-1]=1\\\\\therefore a=k`

Part b)

`\int_(0)^(L )f(x)dx=1\\\\\therefore Re(\int_(0)^(L )ae^(-ikx)dx)=1\\\\\Rightarrow Re(a\int_(0)^(L )e^(-ikx)dx)=1\\\\\therefore Re((a)/(-ik)[(1)/(e^(ikx))]_(0)^(L))=1\\\\\Rightarrow Re((a)/(-ik)(e^(-ikL)-1))=1\\\\(a)/(k)Re((1)/(-i)(cos(-kL)+isin(-kL)-1))=1`

`(a)/(k)Re((1)/(-i)(cos(-kL)+isin(-kL)-1))=1\\\\(a)/(k)Re(icos(-kL)+sin(kL)+(1)/(i))=1\\\\(a)/(k)sin(kL)=1\\\\a=(k)/(sin(kL))`