Phy5645/One dimensional problem

Let us start with the original box, with its walls 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 x=0\!} 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 x=a,\!} and with the particle in the ground state of this box. The energy and the wave function are

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_{0}=\frac{\pi ^{2}\hbar^{2}}{2ma^{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 \psi_0(x)=\sqrt{\frac{2}{a}}\sin\left (\frac{\pi x}{a}\right ),\, 0<x<a.}

(a) In the new box, with the right-hand wall now located 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 x=4a,\!} the ground state energy and wave function of the electron are

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'_{0}=\frac{\pi ^{2}\hbar^{2}}{32ma^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 \psi'_{0}(x)= \frac{1}{\sqrt{2a}}\sin\left (\frac{\pi x}{4a}\right ),\,0<x<4a.}

The probability of finding the electron in 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_{0}(x)\!} is

$P(E'_{0})=\left|\int _{0}^{a}\psi _{0}^{\ast }(x)\psi '_{0}(x)\,dx\right|^{2}={\frac {1}{a^{2}}}\left|\int _{0}^{a}\sin \left({\frac {\pi x}{a}}\right)\sin \left({\frac {\pi x}{4a}}\right)\,dx\right|^{2}.$

Note that the upper limit of the integral is $a;\!$ this is because $\psi _{0}(x)\!$ is limited to the region between 0 and $a.\!$ Using the identity,

$\sin {a}\sin {b}={\tfrac {1}{2}}[\cos(a+b)-\cos(a-b)],$

we get

$P(E'_{0})={\frac {1}{4a^{2}}}\left|\int _{0}^{a}\cos \left({\frac {5\pi x}{4a}}\right)\,dx-\int _{0}^{a}\cos \left({\frac {3\pi x}{4a}}\right)\,dx\right|^{2}$

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{128}{225\pi ^{2}}=0.058=5.8%.}

(b) The energy and wave function of the first excited state of the new box are

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'_{1}=\frac{\pi ^{2}\hbar^{2}}{8ma^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 \psi'_{1}(x)= \frac{1}{\sqrt{2a}}\sin\left (\frac{\pi x}{2a}\right ),\,0<x<4a.}

The probability of finding the particle in this state 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 P(E'_{2})=\left |\int_{0}^{a}\psi_{0}^{\ast}(x)\psi'_{1}(x)\,dx\right |^2=\frac{1}{a^2}\left |\int_{0}^{a}\sin\left ( \frac{\pi x}{a}\right )\sin(\frac{\pi x}{2a})\right |^{2}dx}

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{16}{9\pi ^{2}}=0.18=18%.}

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