Chapter4problem

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

(a) Using the wave function 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= \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)

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

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

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

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

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

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

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{\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}}}

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 = .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 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{\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.