Find a few of the Bohr energies for hydrogen by “wagging the dog” (Problem 2.55), starting with Equation 4.53—or, better yet, Equation 4.56; in fact, why not use Equation 4.68 to set ρ0 = 2n , and tweak n? We know that the correct solutions occur when n is a positive integer, so you might start with

Question

Find a few of the Bohr energies for hydrogen by “wagging the dog” (Problem 2.55), starting with Equation 4.53—or, better yet, Equation 4.56; in fact, why not use Equation 4.68 to set $ρ_0 = 2n$ , and tweak n? We know that the correct solutions occur when n is a positive integer, so you might start with n = 0.9 , 1.9 , 2.9 , etc. and increase it in small increments—the tail should wag when you pass 1, 2, 3, …. Find the lowest three ns, to four significant digits, first for $\ell = 0$ , and then for $\ell = 1$ and $\ell = 2$ Warning: Mathematica doesn’t like to divide by zero, so you might change ρ to $(\rho+0.000001)$ in the denominator. Note: $u(0)=0 \text { in all cases, but } u^{\prime}(0)=0 \text { only for } \ell \geq 1$ (Equation 4.59). So for $\ell = 0$ you can use $u(0)=0, u^{\prime}(0)=1 . \text { For } \ell>0$ you might be tempted to use $u(0)=0 \text { and } u^{\prime}(0)=0$ , but Mathematica is lazy, and will go for the trivial solution $u(\rho) \equiv 0$ ; better, therefore, to use (say) $u(1)=1 \text { and } u^{\prime}(0)=0$ .

$-\frac{\hbar^{2}}{2 m_{e}} \frac{d^{2} u}{d r^{2}}+\left[-\frac{e^{2}}{4 \pi \epsilon_{0}} \frac{1}{r}+\frac{\hbar^{2}}{2 m_{e}} \frac{\ell(\ell+1)}{r^{2}}\right] u=E u$ (4.53).

$\frac{d^{2} u}{d \rho^{2}}=\left[1-\frac{\rho_{0}}{\rho}+\frac{\ell(\ell+1)}{\rho^{2}}\right] u$ (4.56).

$\rho_{0}=2 n$ (4.68).

$u(\rho) \sim C \rho^{\ell+1}$ (4.59).

[latex] -\frac{\hbar^{2}}{2 m_{e}} \frac{d^{2} u}{d r^{2}}+\left[-\frac{e^{2}}{4 \pi \epsilon_{0}} \frac{1}{r}+\frac{\hbar^{2}}{2 m_{e}} \frac{\ell(\ell+1)}{r^{2}} ight] u=E u [/latex] (4.53).

[latex] \frac{d^{2} u}{d ho^{2}}=\left[1-\frac{ ho_{0}}{ ho}+\frac{\ell(\ell+1)}{ ho^{2}} ight] u [/latex] (4.56).

[latex] ho_{0}=2 n [/latex] (4.68).

[latex] u( ho) \sim C ho^{\ell+1} [/latex] (4.59).

Accepted Answer

[latex] \underline{\ell=0} [/latex].

[latex] ext { Plot [Evaluate }\left[\mathrm { u } [ \mathrm { x } ] / . ext { NDSolve } \left[\left\{ u ^{\prime \prime}[ x ]-(1-(2 \star 0.9999 ight. ight. ight. [/latex].

[latex] ) / (x + 0.000001)) u[x] ⩵ 0, u[0] ⩵ 0, u'[0] ⩵ 1\}, u[x], \{x, 10^(-8), 10\} [/latex],

[latex] MaxSteps → 10 000]], \{x, 0.01, 10\}, PlotRange → \{-0.02, 0.5\}][/latex].

[latex] ext { Plot [Evaluate }\left[\mathrm { u } [ \mathrm { x } ] / . ext { NDSolve } \left[\left\{ u ^{\prime \prime}[ x ]-(1-(2 \star 1.0001 ight. ight. ight. [/latex].

[latex] ) / (x + 0.000001)) u[x] ⩵ 0, u[0] ⩵ 0, u'[0] ⩵ 1\}, u[x], \{x, 10^(-8), 10\} [/latex],

[latex] MaxSteps → 10 000]], \{x, 0.01, 10\}, PlotRange → \{-0.1, 0.5\}][/latex].

So the lowest n is between 0.9999 and 1.0001.

[latex] ext { Plot [Evaluate }\left[\mathrm { u } [ \mathrm { x } ] / . ext { NDSolve } \left[\left\{ u ^{\prime \prime}[ x ]-(1-(2 \star 2.0001 ight. ight. ight. [/latex].

[latex] ) / (x + 0.000001)) u[x] ⩵ 0, u[0] ⩵ 0, u'[0] ⩵ 1\}, u[x], \{x, 10^(-8), 15\} [/latex],

[latex] MaxSteps → 10 000]], \{x, 0.01, 10\}, PlotRange → \{-0.4, 0.2\}][/latex].

[latex] ext { Plot [Evaluate }\left[\mathrm { u } [ \mathrm { x } ] / . ext { NDSolve } \left[\left\{ u ^{\prime \prime}[ x ]-(1-(2 \star 1.9999 ight. ight. ight. [/latex].

[latex] ) / (x + 0.000001)) u[x] ⩵ 0, u[0] ⩵ 0, u'[0] ⩵ 1\}, u[x], \{x, 10^(-8), 15\} [/latex],

[latex] MaxSteps → 10 000]], \{x, 0.01, 15\}, PlotRange → \{-0.4, 0.2\}][/latex].

So the next n is between 1.9999 and 2.0001.

[latex] ext { Plot [Evaluate }\left[\mathrm { u } [ \mathrm { x } ] / . ext { NDSolve } \left[\left\{ u ^{\prime \prime}[ x ]-(1-(2 \star 1.9999 ight. ight. ight. [/latex].

[latex] ) / (x + 0.000001)) u[x] ⩵ 0, u[0] ⩵ 0, u'[0] ⩵ 1\}, u[x], \{x, 10^(-8), 15\} [/latex],

[latex] MaxSteps → 10 000]], \{x, 0.01, 15\}, PlotRange → \{-0.4, 0.2\}][/latex].

[latex] ext { Plot [Evaluate }\left[\mathrm { u } [ \mathrm { x } ] / . ext { NDSolve } \left[\left\{ u ^{\prime \prime}[ x ]-(1-(2 \star 2.9999 ight. ight. ight. [/latex].

[latex] ) / (x + 0.000001)) u[x] ⩵ 0, u[0] ⩵ 0, u'[0] ⩵ 1\}, u[x], \{x, 10^(-8), 17\} [/latex],

[latex] MaxSteps → 10 000]], \{x, 0.01, 17\}, PlotRange → \{-0.2, 0.3\}][/latex].

[latex] ext { Plot [Evaluate }\left[\mathrm { u } [ \mathrm { x } ] / . ext { NDSolve } \left[\left\{ u ^{\prime \prime}[ x ]-(1-(2 \star 3.00001 ight. ight. ight. [/latex].

[latex] ) / (x + 0.000001)) u[x] ⩵ 0, u[0] ⩵ 0, u'[0] ⩵ 1\}, u[x], \{x, 10^(-8), 20\} [/latex],

[latex] MaxSteps → 10 000]], \{x, 0.01, 20\}, PlotRange → \{-0.2, 0.3\}][/latex].

So the third n is between 2.9999 and 3.00001.

[latex] \underline{\ell=1} [/latex].

This time there is no solution for n = 1; the lowest state (no nodes) is around n = 2:

[latex] Plot[Evaluate[u[x] /. NDSolve [\{u''[x] - (1 - ((2 * 1.999)/(x + 0.000001)) + [/latex].

[latex] (2 /(x^2 + 0.000000001))) u[x] ⩵ 0, u[1] ⩵ 1, u'[0] ⩵ 0\}, [/latex].

[latex] u[x], \{x, 10^(-8), 10\}, MaxSteps → 10 000]], \{x, 0.1, 10\}] [/latex].

[latex] Plot[Evaluate[u[x] /. NDSolve [\{u''[x] - (1 - ((2 * 2.0001)/(x + 0.000001)) + [/latex].

[latex] (2/(x^2 + 0.000000001))) u[x] ⩵ 0, u[1] ⩵ 1, u'[0] ⩵ 0\}, [/latex].

[latex] u[x], \{x, 10^(-8), 15\}, MaxSteps → 10 000]], \{x, 0.1, 15\}] [/latex].

So the lowest n is between 1.999 and 2.0001.

[latex] Plot[Evaluate[u[x] /. NDSolve [\{u''[x] - (1 - ((2 * 2.9999)/(x + 0.000001)) + [/latex].

[latex] (2 /(x^2 + 0.000000001))) u[x] ⩵ 0, u[1] ⩵ 1, u'[0] ⩵ 0\}, [/latex].

[latex] u[x], \{x, 10^(-8), 18\}, MaxSteps → 10 000]], \{x, 0.1, 18\}] [/latex].

[latex] Plot[Evaluate[u[x] /. NDSolve [\{u''[x] - (1 - ((2 * 3.0001)/(x + 0.000001)) + [/latex].

[latex] (2 /(x^2 + 0.000000001))) u[x] ⩵ 0, u[1] ⩵ 1, u'[0] ⩵ 0\}, [/latex].

[latex] u[x], \{x, 10^(-8), 20\}, MaxSteps → 10 000]], \{x, 0.1, 20\}] [/latex].

So the next n is between 2.9999 and 3.0001.

[latex] Plot[Evaluate[u[x] /. NDSolve [\{u''[x] - (1 - ((2 * 3.9999)/(x + 0.000001)) + [/latex]

[latex] (2 /(x^2 + 0.000000001))) u[x] ⩵ 0, u[1] ⩵ 1, u'[0] ⩵ 0\}, [/latex].

[latex] u[x], \{x, 10^(-8), 20\}, MaxSteps → 10 000]], \{x, 0.1, 20\}] [/latex].

[latex] Plot[Evaluate[u[x] /. NDSolve [\{u''[x] - (1 - ((2 * 4.0001)/(x + 0.000001)) + [/latex]

[latex] (2 /(x^2 + 0.000000001))) u[x] ⩵ 0, u[1] ⩵ 1, u'[0] ⩵ 0\}, [/latex].

[latex] u[x], \{x, 10^(-8), 20\}, MaxSteps → 10 000]], \{x, 0.1, 20\}] [/latex].

So the third n is between 3.9999 and 4.0001.

[latex] \underline{\ell=2} [/latex].

This time there is no solution for n = 1 or n = 2; the lowest state (no nodes) is around n = 3:

[latex] Plot[Evaluate[u[x] /. NDSolve [\{u''[x] - (1 - ((2 * 2.999)/(x + 0.000001)) + [/latex].