Question

An electron has the following wave function:
¥ (x) = Ae-x (1-e -x)
Determine
i) the normalization constant A, and
ii) the expectation value of x

Answer 1

The wave function is

$$ψ(x)=Ae^{-x}(1-e^{ -x}).$$

Determine the normalization constant A:

$$1=\int^{+\infty}_{-\infty}\psi^*(x)\psi(x)dx,\\ \space\\ 1=\int^{+\infty}_{0}A^2e^{-2x}(1-e^{-x})2\space dx,\\ \space\\ 1=A^2\bigg(\frac{1}{12}\bigg),\\ A^2=12.\space\\$$

Determine the expectation value of x:

$$<\hat{A}>=\int^{+\infty}_{-\infty}\psi^*(x)\hat{A}\psi(x)dx,\\ \space\\ <x>=12\int^{+\infty}_{0}xe^{-2x}(1-e^{-x})2\space dx,\\ \space\\<x>=\frac{13}{12}.$$