Introduction to quantum field theory
Mathematical Prerequisites for the radial solution to H atom
The radial behaviour of the eigenstates of H-like (i.e mono-electron) atoms are represented in terms of associated Laguerre polynomials, [math]\displaystyle{ L_{p}^{k}(x) }[/math],
[math]\displaystyle{ R^{\ell}_{n}(x) = x^{\ell} e^{-\frac{1}{2}x} L^{2\ell+1}_{n+\ell}(x)\,, }[/math]
where, [math]\displaystyle{ a = \hbar^{2}/me^2, }[/math] and
[math]\displaystyle{ L_{r}^{s}(x) = \frac{d^{s}}{dx^{s}} \left[ e^{x} \frac{d^{r}}{dx^{r}} \left( e^{-x}x^r \right) \right]. }[/math]
The necessary properties of these functions are summarised in any standard text on differential equations, but also, e.g. in Appendix B, Section I.2 in Messiah's book on Quantum Mechanics.
Grading assignment
To be submitted before the commencement of the discussion session involving Exercise AB3
Using the wave function for the electron in the Hydrogen atom in any general state [math]\displaystyle{ \mid n\ell m \rangle }[/math], i.e.
[math]\displaystyle{ \Psi_{n\ell m}(r,\theta,\varphi) = - \left[ \frac{4(n - \ell -1)!}{(na)^3 n [(n+\ell)!]^3} \right]^{1/2} F_{n\ell}\left(\frac{2r}{na}\right) Y_{\ell}^{m}(\theta, \varphi)\,, }[/math]
where,
[math]\displaystyle{ F_{n\ell}(x) = x^{\ell} L_{n+\ell}^{2\ell+1}(x)e^{-x/2}\,\\ }[/math]
evaluate:
- [math]\displaystyle{ \langle r \rangle }[/math],
- [math]\displaystyle{ \left\langle r^2 \right\rangle }[/math], and
- [math]\displaystyle{ \Delta r / \langle r \rangle }[/math].
What is the value of [math]\displaystyle{ \langle r \rangle }[/math] and [math]\displaystyle{ \Delta r / \langle r \rangle }[/math] for large values of [math]\displaystyle{ n }[/math]? Compare these results to the Bohr radius for the same [math]\displaystyle{ n }[/math].
Hint
You should be able to derive the more general recursion relation:
[math]\displaystyle{
\frac{p+1}{n^2}\left\langle r^{p+1} \right\rangle - (2p+1)a \left\langle r^p \right\rangle + \frac{p}{4}\left[ (2\ell + 1)^{2}- p^{2} \right]a^2\left\langle r^{p-1} \right\rangle = 0\,,\\
}[/math]
for
[math]\displaystyle{ p \geqslant -2\ell - 1. }[/math]
Exercise AB3
17:17, 31 October 2017 (UTC)
- For the ground state of the Hydrogen atom (assuming no spin), compute the probability that the electron is farther away from the proton than its Bohr radius. What is the probability that it is inside the nucleus, assuming a nuclear radius of about 0.5 fermi?
- Explicitly compute [math]\displaystyle{ \left\langle \tfrac{1}{r} \right\rangle }[/math] for an electron in a general [math]\displaystyle{ \Psi_{n\ell m} }[/math] state of the Hydrogen atom (assuming no spin).
- Making use of the fact that the expectation value of the operator [math]\displaystyle{ \hat{\mathbf{x}}\cdot\hat{\mathbf{p}} }[/math] is conserved for any eigenstate of the Hamiltonian [math]\displaystyle{ \hat{H} = \hat{p}^{2}/2m + V(\mathbf{x}) }[/math], prove the virial theorem [math]\displaystyle{ 2\langle \hat{T} \rangle = \langle x \cdot \nabla V \rangle }[/math], where [math]\displaystyle{ T = \mathbf{p}^2/2m }[/math]. Use this result and the result from problem 2 above, to compute [math]\displaystyle{ \left\langle p^2 \right\rangle }[/math] for an electron in a general [math]\displaystyle{ \Psi_{n\ell m} }[/math] state of the Hydrogen atom (assuming no spin).
Hint for problem 3
First convince yourself that for a potential [math]\displaystyle{ V(r) \propto r^{n} }[/math], this implies [math]\displaystyle{ 2\langle T \rangle = n\left\langle V(r) \right\rangle\,. }[/math]
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