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
Note that the upper limit of the integral is this is because is limited to the region between 0 and Using the identity,
we get
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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