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

ψ(x) = Ae^−x(1−e^−x)

i) The normalization condition

1 = $\int^∞_0∣ψ(x)∣$²dx

1 = A² $∫ _ 0^ ∞ ​ e$^-2x(1 - e^-x)²dx = A² / 12

A = 2$\sqrt{3}$

(ii) The expectation value of x

⟨x⟩ = 12$\int_0^∞$​xe^−2x(1−e^−x)²dx = 13 / 12 ​= 1.08$​$