Chapter4problem

(Problem submitted by team 9, based on problem 7.11 of Griffiths)

(a) Using the wave function $\psi ={\begin{cases}A*cos({\frac {\pi *x}{a}})&{\frac {-a}{2}}<x<{\frac {a}{2}}\\0&otherwise\end{cases}}$

obtain a bound on the ground state energy of the one-dimensional harmonic oscillator. Compare $<H>_{min}$ with the exact energy. Note: This trial wave function has a discontinuous derivative at ${\frac {\pm a}{2}}$.

(b) Use $\Psi =B*sin({\frac {\pi *x}{a}})$ on the interval (-a,a) to obtain a bound on the first excited state. Compare to the exact answer.

Solution

(a)

$1=\int |\Psi |^{2}dx=\int \limits _{-a/2}^{a/2}cos^{2}({\frac {\pi *x}{a}})\,dx=|A|^{2}*{\frac {a}{2}}\Rightarrow A={\sqrt {(}}{\frac {2}{a}})$

$<T>=-{\frac {\hbar ^{2}}{2m}}\int \Psi {\frac {d^{2}\Psi }{dx^{2}}}={\frac {\hbar ^{2}}{2m}}({\frac {\pi }{a}})^{2}\int \Psi ^{2}dx={\frac {\pi ^{2}\hbar ^{2}}{2ma^{2}}}$

$<V>=.5m\omega ^{2}\int x^{2}\Psi ^{2}dx=.5m\omega ^{2}{\frac {2}{a}}\int \limits _{-a/2}^{a/2}x^{2}cos^{2}({\frac {\pi x}{a}}dx$ $={\frac {m\omega ^{2}}{a}}({\frac {a}{\pi }})^{2}\int \limits _{-\pi /2}^{\pi /2}y^{2}cos^{2}(y)dy$ $={\frac {m\omega ^{2}a^{2}}{\pi ^{3}}}[{\frac {y^{3}}{6}}+({\frac {y^{2}}{4}}-{\frac {1}{8}})sin(2y)+{\frac {ycos(2y)}{4}}]_{-a/2}^{a/2}$ $={\frac {m\omega ^{2}a^{2}}{4\pi ^{2}}}({\frac {\pi ^{2}}{6}}-1)$

$<H>={\frac {\pi ^{2}\hbar ^{2}}{2ma^{2}}}+{\frac {m\omega ^{2}a^{2}}{4\pi ^{2}}}({\frac {\pi ^{2}}{6}}-1)$

${\frac {\partial <H>}{\partial a}}=-{\frac {\pi ^{2}\hbar ^{2}}{ma^{3}}}+{\frac {m\omega ^{2}a}{2\pi ^{2}}}({\frac {\pi ^{2}}{6}}-1)=0$

$\Rightarrow a=\pi {\sqrt {\frac {\hbar }{m\omega }}}({\frac {2}{\pi ^{2}/6-1}})^{1/4}$

$<H>_{min}={\frac {\pi ^{2}\hbar ^{2}}{2m\pi ^{2}}}{\frac {m\omega }{\hbar }}{\sqrt {\frac {\pi ^{2}/6-1}{2}}}+{\frac {m\omega ^{2}}{4\pi ^{2}}}(\pi ^{2}/6-1)\pi ^{2}{\frac {\hbar }{m\omega }}{\sqrt {\frac {2}{\pi ^{2}/6-1}}}$

$=.5\hbar \omega {\sqrt {\pi ^{2}/3-2}}=.5\hbar \omega (1.136)>.5\hbar \omega$

We do not need to worry about the discontinuity at ${\frac {\pm a}{2}}$. It is true that 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{d^2 \Psi}{dx^2} } has delta functions there, but 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 \Psi(\frac{\pm a}{2})=0} no extra contribution comes from these points.

(b) Because this trial function is odd, it is orthogonal to the ground state. 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 <\Psi|\Psi_{gs}>=0} . 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 <H> \ge E_{fe} } 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 E_{fe} } is the energy of the first excited state.

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 1=\int|\Psi|^2dx=|B|^2 \int_{-a}^a sin^2(\frac{\pi x}{a})dx = |B|^2a \Rightarrow B = \frac{1}{\sqrt(a)} }

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 <T> = \frac{-\hbar^2}{2m} \int \Psi \frac{d^2 \Psi}{dx^2}dx = \frac{\hbar^2}{2m}\frac{\pi^2}{a^2} \int \Psi^2 dx = \frac{\pi^2 \hbar^2}{2ma^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 <V> = .5m\omega^2 \int x^2 \Psi^2 dx = .5m\omega^2a^{-1} \int_{-a}^{a} x^2sin^2(\frac{\pi x}{a}) dx = \frac{m\omega^2}{2a}\frac{a^3}{\pi^3} \int_{\pi}^{\pi} y^2 sin^2(y)dy } 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{m \omega^2 a^2}{2ma^2} + \frac{m \omega^2 a^2}{4\pi^2}(\frac{2\pi^2}{3}-1)}  ;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 <H>}{\partial a} = \frac{-\pi^2 \hbar^2}{ma^3} + \frac{m \omega^2a}{2\pi^2}(\frac{2 \pi^2}{3}-1)=0 }

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 \rightarrow a=\pi \sqrt(\frac{\hbar}{m\omega})(\frac{2}{2\pi^2/3-1})^{1/4}}

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 <H_{min}> = \frac{\pi^2 \hbar^2}{2m\pi^2} \frac{m\omega}{\hbar} \sqrt(\frac{2\pi^2 /3-1}{2}) + \frac{m \omega^2}{4\pi^2}(\frac{2\pi^2}{3}-1) \pi^2 \frac{\hbar}{m\omega} \sqrt(\frac{2}{2\pi^2/3-1})}