Scattering Lecture Note

# Scattering θm The deflection of the particle from Incident direction or change in phase is Called Scattering. Types of scattering → Ques:1 Page-100 Turning Pt at x = ± 10 V = finite V(x) = ax^6 V(x=10) = a*10^6 = E energy Quantization Cond'n 1/h ∫ √2m(E-ax^6) dx + π/4 + π/4 = (n+1)π, n=0,1,2,... \int_{-\infty}^{\infty} \sqrt{E-ax^6}dx = \left(\frac{n+\frac{1}{2}}{2}\right)\pi\hbar \Rightarrow \int_{-\infty}^{\infty} \sqrt{E}x^3\sqrt{1-\frac{ax^6}{E}}dx = \left(\frac{n+\frac{1}{2}}{2}\right)\pi\hbar Let \ \ \ 1-\frac{ax^6}{E} = y \Rightarrow -\frac{a}{E}6x^5dx = dy \Rightarrow dx = \frac{-E}{6a}x^{-5}dy $$ x = (\frac{E}{a}(1-y))^{\frac{1}{6}} let \ \ \ fm = \frac{-E}{6a}(\frac{E}{a})^{-5/6}(1-y)^{-5/6}dy \Rightarrow 2 \int_{0}^{1} \sqrt{E}E^{\frac{1}{2}}x^3y^{\frac{1}{2}}(1-y)^{-\frac{5}{6}}dy = \left(\frac{n+\frac{1}{2}}{2}\right)\pi\hbar \Rightarrow 2(\frac{1}{6}) \int_{0}^{1} E^{\frac{5}{6}}y^{\frac{1}{2}}(1-y)^{-\frac{5}{6}}dy = \left(\frac{n+\frac{1}{2}}{2}\right)\pi\hbar \Rightarrow (\frac{1}{3})E^{\frac{5}{6}} = \left(\frac{n+\frac{1}{2}}{2}\right)\pi\hbar for \ \ \ large \ \ n, n-n or \frac{ax^6}{E} = 1-y $$ 1-\frac{ax^6}{\epsilon} = Y\Rightarrow -(\frac{a}{\epsilon})6x^5dx = dy $$ \Rightarrow dn=\frac{-E}{6a}n^{-5}dy E 4/6-an En 3/2 (option-d) ques-5 (V(x)=a|x|,a>0) , aso page-100 [E_{n}(n=0,1,2,3\cdot\cdot\cdot)] 0 at \(x=\pm x_{0}\) \[E=v(n)=9x_{0}\] #Jam (E-a/ml)+m+ += (m+1), m=9h? -210 → (\alpha\int_{2mE}\int_{0}^{\eta r_{0}}\sqrt[1-\frac{\partial r}{E}}dn=(n+\frac{1}{2})\pi\hbar) [\alpha\int_{0}^{\chi_{0}}{\int2m(E-an)}=(n+\frac{1}{2})\pi\hbar,n=0,1,2...] \[1-\frac{ax}{E}=y\Rightarrow r_{i}=\frac{+E}{a}(i-y)\] \[8\times\sqrt{dmE}\int_{0}^{1}Y^{\frac{1}{2}}(\frac{E}{a_{1}})dy=(m+\frac{1}{2})\pi\hbar\] \[-d~dx=\frac{-E}{9}dy\] EY ⇒ 2xJJME = (m+)かち 3/2 b ont 3 => 4/3 * sqrt(2m/E) * E^(3/2) = (n + 1/2) * pi * h => E = (3/4 * pi * h * a / sqrt(2m)) * (n + 1/2)^(2/3), n = 0, 1, 2, 3... (option-b) Trick of H V(r) = a * r^m En = {[(m + 1/2)^(2m/(m+2))] / a^(m/(m+2))} * {[(m - 1/2)^(2m/(m+2))] / a^(m/(m+2))} * {[(m - 1)^(2m/(m+2))] / a^(m/(m+2))} * {[(m + 3/4)^(2m/(m+2))] / a^(m/(m+2))} @page-10 Same as above (@page-5) (option-b) @page-14 V(r) = lambda * r^4 En ~ (m + 1/2)^(2*r^4/(4+2)) => Integral from -pi* to +pi* of f(m)(E - lambda * r^4) dm = (m + 1/2) * pi * h^2 0 -> Vo E x1 x2 Classical for kidding region Transmission Coeff. T ~ exp [-2/h * integral from x1 to x2 of sqrt(2m(Vo - E)) dx] x1 -> to x2 => T ~ exp [-2/h * integral from 0 to x2 of sqrt(2m(Vo - E)) (m - m1)] 0 Yo L T ~ exp [-2kL] k = 1/h * sqrt(2m(Vo - E))