Question

Radius of hydrogen atom. Let $\Psi_{n,l,m}(r, \theta, \phi) = \Psi_{n,l}(r)Y_{l,m}(\theta, \phi)$ be the normalized eigen-\ functions of the Hamiltonian\ $\hat{H} = -\frac{1}{2m_e}\Delta - \frac{e^2}{r}$, as described in the lectures. (a) (This question is bonus material) Show the recursion relation $4(k+1)\langle r^k \rangle - 4(2k+1)a^2n^2\langle r^{k-1} \rangle + a^2n^2k[(2l+1)^2 - k^2]\langle r^{k-2} \rangle = 0$, (1) where $\langle ... \rangle$ means expectation value in $\Psi_{n,l,m}$, and where $a = 1/(m_ee^2)$ is the Bohr radius. (b) Using the recursion relation, show $\langle r \rangle = \frac{1}{2}[3n^2 - l(l+1)]a$ $\langle r^2 \rangle = \frac{1}{2}[5n^2 + 1 - 3l(l+1)]n^2a^2$, (2) Compare your results with the radii obtained via Bohr's original method ('old quantum\ mechanics'). (c) Show that the variance of r ('radius operator') in the states $\Psi_{n,l,m}$ is given by $\Delta r = \frac{a}{2}\sqrt{n^4 + 2n^2 - l^2(l+1)^2}$. (3)

          Radius of hydrogen atom. Let $\Psi_{n,l,m}(r, \theta, \phi) = \Psi_{n,l}(r)Y_{l,m}(\theta, \phi)$ be the normalized eigen-\ functions of the Hamiltonian\ 
$\hat{H} = -\frac{1}{2m_e}\Delta - \frac{e^2}{r}$,
as described in the lectures.
(a) (This question is bonus material) Show the recursion relation
$4(k+1)\langle r^k \rangle - 4(2k+1)a^2n^2\langle r^{k-1} \rangle + a^2n^2k[(2l+1)^2 - k^2]\langle r^{k-2} \rangle = 0$,
(1)
where $\langle ... \rangle$ means expectation value in $\Psi_{n,l,m}$, and where $a = 1/(m_ee^2)$ is the Bohr radius.
(b) Using the recursion relation, show
$\langle r \rangle = \frac{1}{2}[3n^2 - l(l+1)]a$  $\langle r^2 \rangle = \frac{1}{2}[5n^2 + 1 - 3l(l+1)]n^2a^2$,
(2)
Compare your results with the radii obtained via Bohr's original method ('old quantum\ mechanics').
(c) Show that the variance of r ('radius operator') in the states $\Psi_{n,l,m}$ is given by
$\Delta r = \frac{a}{2}\sqrt{n^4 + 2n^2 - l^2(l+1)^2}$.
(3)

Radius of hydrogen atom. Let Ψn,l,m(r, θ, ϕ) = Ψn,l(r)Yl,m(θ, ϕ) be the normalized eigen- functions of the Hamiltonian
Ĥ = -(1)/(2me)Δ - (e^2)/(r),
as described in the lectures.
(a) (This question is bonus material) Show the recursion relation
4(k+1)⟨ r^k ⟩ - 4(2k+1)a^2n^2⟨ r^k-1⟩ + a^2n^2k[(2l+1)^2 - k^2]⟨ r^k-2⟩ = 0,
(1)
where ⟨ ... ⟩ means expectation value in Ψn,l,m, and where a = 1/(mee^2) is the Bohr radius.
(b) Using the recursion relation, show
⟨ r ⟩ = (1)/(2)[3n^2 - l(l+1)]a ⟨ r^2 ⟩ = (1)/(2)[5n^2 + 1 - 3l(l+1)]n^2a^2,
(2)
Compare your results with the radii obtained via Bohr's original method ('old quantum mechanics').
(c) Show that the variance of r ('radius operator') in the states Ψn,l,m is given by
Δ r = (a)/(2)√(n^4 + 2n^2 - l^2(l+1)^2).
(3)

Added by Carlos M.

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