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, where . With a trial wave function of your own devising, estimate the binding energy of a "hydrogen" atom with this potential. Assume , and give your answer correct to order
Solution:
The simplest trial function looks exactly like the hydrogen atom ground wavefunction 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, 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.