Phy5645/One dimensional problem

From PhyWiki
Jump to navigation Jump to search

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%.}

Back to One-Dimensional Bound States