Phy5646/soham1

(Introduction to Qusntum Mechanics, Griffiths, 2e)Problem 7.14

If the photon has a nonzero mass Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (m_{\gamma} \neq 0)} , the Coulomb potential would be rep[laced by the Yukawa potential, $V(r)={\frac {e^{2}}{4\pi \epsilon _{0}}}{\frac {e^{-\mu r}}{r}}$ where $\mu =m_{\gamma }c\hbar$. With a trial wave function of your own devising, estimate the binding energy of a "hydrogen" atom with this potential. Assume $\mu a\ll 1$, and give your answer correct to order $(\mu a)^{2}$

Solution:

The simplest trial function looks exactly like the hydrogen atom ground wavefunction $(\psi ={\frac {1}{\sqrt {\pi a^{3}}}}e^{-r/a}.$ but with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} as a variational parameter. The hydrogen atom Hamiltonian is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal H = \frac{-\hbar^{2}}{2m} - \frac{e^{2}}{4\pi \epsilon_{0}} \frac{1}{r} = T+V}

For hydrogen atom with standard Coulomb potential (massless photons), we have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle T\rangle = \langle V\rangle,} with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle T\rangle = \frac{\hbar^{2}}{2ma^2}}

For the Yukawa potential, $\langle T\rangle$ remains he same, but Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle V \rangle} gets modified.

$V={\frac {-e^{2}}{4\pi \epsilon _{0}}}{\frac {1}{\pi a^{3}}}\int _{0}^{\infty }{\frac {e^{-2r/b}e^{-\mu r}}{r}}r^{2}drd\Omega ={\frac {-e^{2}}{4\pi \epsilon _{0}}}{\frac {4}{a^{3}}}\int _{0}^{\infty }e^{-(\mu +2/a)r}rdr={\frac {-e^{2}}{4\pi \epsilon _{0}}}{\frac {4}{a^{3}}}{\frac {1}{(\mu +2/a)^{2}}}=$