Question

An electron has the following wave function:

ψ(x)=A(e^-x)(1-e^-x)


Determine
i) the normalization constant A, and
ii) the expectation value of x .

Answer 1

The wave function of an electron

$$\psi(x)=Ae^{-x}(1-e^{-x}).$$

(i) The normalization condition

$$1=\int_{0}^{\infty}|\psi(x)|^2dx\\ 1=A^2\int_{0}^{\infty}e^{-2x}(1-e^{-x})^2dx=\frac{A^2}{12}\\ A=2\sqrt{3}.$$

(ii) The expectation value of $x$

$$\lang x\rang =12\int_{0}^{\infty}xe^{-2x}(1-e^{-x})^2dx=\frac{13}{12}=1.08$$