Phy5646/soham1

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(Introduction to Quantum 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, 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 V(r)= \frac{e^{2}}{4\pi\epsilon_{0}} \frac{e^{-\mu r}}{r}} where 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 \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 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 \mu a \ll 1} , where 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 = \frac{4\pi\epsilon_{0}\hbar^{2}}{me_{2}}} , and give your answer correct to order 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 (\mu a)^{2}}

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} changed to 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 b} . 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 (\psi = \frac{1}{\sqrt{\pi b^{3}}}e^{-r/b}.} . 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 b} acts 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 = -E_{1}\frac{\hbar^{2}}{2ma^2}}

For the Yukawa potential, 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}}{2mb^2} } and

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 V = \frac{-e^2}{4\pi \epsilon_{0}}\frac{1}{\pi b^{3}} \int_{0}^{\infty}\frac{e^{-2r/b} e^{-\mu r}}{r} r^{2}dr d\Omega = \frac{-e^2}{4\pi \epsilon_{0}}\frac{4}{b^{3}} \int_{0}^{\infty}e^{-(\mu + 2/b)r} r dr = \frac{-e^2}{4\pi \epsilon_{0}}\frac{4}{b^{3}}\frac{1}{(\mu+2/b)^2} = \frac{-e^2}{4\pi \epsilon_{0}}\frac{1}{b(1+\mu b/2)^2}}

Or,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 = \langle T\rangle + \langle V \rangle = \frac{\hbar^{2}}{2mb^2} - \frac{-e^2}{4\pi \epsilon_{0}}\frac{1}{b(1+\mu b/2)^2}}

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 \frac{\partial \mathcal H}{\partial b} = 0 = \frac{\hbar^{2}}{mb^{3}} + \frac{-e^2}{4\pi \epsilon_{0}}\left [\frac{1}{b(1 + \mu b/2)^{2}} + \frac{\mu}{b(1+\mu b/2)^{3}} \right ] = frac{\hbar^{2}}{mb^{3}} + \frac{-e^2}{4\pi \epsilon_{0}b^{2}}\left [\frac{1 + 3\mu b/2)}{(1 + \mu b/2)^3} \right ]}

Or, 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 frac{4\pi\epsilon_{0}\hbar^{2}}{me_{2}} = b\frac{1 + 3\mu b/2)}{(1 + \mu b/2)^3}} Or, 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 b\frac{1 + 3\mu b/2)}{(1 + \mu b/2)^3} = a} Now since 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 \mu a \ll 1, <math>\mu b \ll 1} </math>. So

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 b(1 + 3\mu b/2)\left [1 - \frac{3\mu b}{2} + 6 \frac{\mu^{2} b^{2}}{4} \right ] \approx a} Or, 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 \approx = b\left [ 1-3(\mu b)^{2}/4 \right ]}

In the second order term, we can replace 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 b} by 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} So

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 b \approx a\left [1+3(\mu a)^{2}/4 \right ]}

With this optimized value of 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 b} , we can find out 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 \mathcal H\rangle}

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 \mathcal H\rangle = \frac{\hbar^{2}}{2ma^{2}\left [1+3(\mu a)^{2}/4 \right ]} - \frac{e^2}{4\pi \epsilon_{0}}\frac{1}{a\left [1+3(\mu a)^{2}/4 \right ]\left [1+\mu a/2 \right ]^2}} where we have approximated 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 b} by 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} in the last bracketed term in the denominator.

Or,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 \mathcal H\rangle = \frac{\hbar^{2}}{2ma^2} \left [1-6(\mu a)^{2}/4 \right ] - \frac{-e^2}{4\pi \epsilon_{0}a}\left [1-3(\mu a)^{2}/4 \right ]\left [1-2(\mu a)/2 +3(\mu a/2)^{2} \right ] = -E_{1}\left [1-3(\mu a)^{2}/2 \right ] +2E_{1}\left [1 -\mu a - \frac{3}{4}(\mu a)^2 + \frac{3}{4}(\mu a)^2 \right ]} Or, 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 \mathcal H\rangle = E_{1}\left [1-2\mu a +\frac{3}{2}(\mu a)^{2} \right ]}

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