Prob(x_{0} < x < x_{0} + dx) = |Ψ(x_{0})|^{2}dx

\begin{aligned}Prob(r_{0} < r < r_{0} + dr) & = |Ψ(r_{0})|^{2}dV \\ & = |Ψ(r_{0})|^{2}r^{2}dr \end{aligned}

Prob(R_{1} < r < R_{2}) = ∫_{R_{1}}^{R_{2}}|Ψ(r)|^{2}r^{2}dr

E = −\frac{h̄^{2}}{2ma^{2}} = −\frac{m_{e}^{2}e^{4}}{32π^{2}ϵ_{0}^{2}h̄^{2}}

Ψ(r) = Qe^{−r/a}

Prob (r_{1} < r < r_{2}) = ∫_{r_{1}}^{r_{2}} Ψ(r)^{2} dr = ∫_{r_{1}}^{r_{2}} Q^{2}e^{−2r/a} dr

dV = dx

dV = dx dy

dV = dr rdθ

dV = dx dy dz

\begin{aligned}dV & = (dr) (rdθ) (r sin θ dϕ) \\ & = r^{2}dr sin θdθ dϕ \end{aligned}

Prob(r, θ, ϕ) = ∫ (Ψ(r, θ, ϕ))^{2} dV
= ∫ (Ψ(r, θ, ϕ))^{2} r^{2}dr sin θdθ dϕ

Prob(r, θ, ϕ) = ∫ (R(r)Θ(θ)Φ(ϕ))^{2} r^{2}dr sin θdθ dϕ

= ∫ (R^{2}(r)r^{2}dr) (Θ^{2}(θ) sin θdθ) (Φ^{2}(ϕ)dϕ)

= ∫ R^{2}(r)r^{2}dr ∫ Θ^{2}(θ) sin θdθ ∫ Φ^{2}(ϕ)dϕ

∫ 1 sin θdθ

∫ 1 dϕ
R(r) = Qe^{−r/a}

Prob(r_{1} < r < r_{2}) = 4π∫_{r_{1}}^{r_{2}} R^{2}(r)r^{2}dr

Prob(r_{1} < r < r_{2}) = 4π (Q^{2} \frac{a^{3}}{8} ) ∫_{2r_{1}/a}^{2r_{2}/a} u^{2}e^{−u}du

Prob(r_{1} < r < r_{2}) = 4π (Q^{2} \frac{a^{3}}{8} ) [−e^{−u}(u^{2} + 2u + 2)]_{2r_{1}/a}^{2r_{2}/a} < r > = \begin{matrix}∫ & Ψ^{2} r dV \\ ∫ & Ψ^{2} dV \end{matrix}

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