Question 1.5: Consider the wave function Ψ (x,t) = A e^-λ|x|e ^- iωt, wher...

(a)

1 = \int{\left|\Psi \right|^2 } dx = 2\left|A\right| ^2 \int_{0}^{\infty }{e^{-2\lambda x}}dx = 2\left|A\right| ^2 \biggl(\frac{e^{-2\lambda x}}{-2\lambda }\biggr) \mid ^\infty _0 = \frac{2\left|A\right| ^2}{λ} ; A = \sqrt{\lambda } .

(b)

\left\langle x\right\rangle = \int {\left|\Psi \right|^2 } dx  = \left|A\right| ^2 \int_{-\infty}^{\infty }{x e^{-2λ \left|\x \right|}}dx = 0 .    [Odd integrand.]

\left\langle x\right\rangle^2 = 2\left|A\right| ^2 \int_{0}^{\infty }{x^2 e^{-2λ x}}dx = 2λ \biggl[\frac{2}{(2λ)^3}\biggr]  = \frac{1}{2λ^2} .

(c)

σ^2 = \left\langle x^2\right\rangle – \left\langle x\right\rangle^2 = \frac{1}{2λ^2} ;    σ = \frac{1}{\sqrt{2} λ}.

\left|\Psi (\pm \sigma )\right| ^2 = \left|A\right|^2 e^{-2\lambda \sigma } = \lambda e^{-2\lambda /\sqrt{2}\lambda } = \lambda e^{-\sqrt{2} } = 0.2441 \lambda .

Probability outside:

2 \int_{σ}^{\infty }{\left|\Psi \right| ^2 }dx = 2 \left| A \right| ^2 \int_{σ}^{\infty }{e^{-2λx}}dx = 2 λ \Bigl(\frac{e^{-2\lambda x}}{-2\lambda }\Bigr)\mid ^\infty _\sigma = e^{-2\lambda \sigma } = e^{-\sqrt{2} } = 0.2431 .