30 lines
621 B
TeX
30 lines
621 B
TeX
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\newpage
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\section*{Example 3}
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For total energy $E$, kinetic energy $K$ and potential energy $V$ we have
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$$E=K+V$$
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The corresponding formula for a quantum harmonic oscillator is
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$$(2n+1)\psi=-{d^2\psi\over dx^2}+x^2\psi$$
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where $n$ is an integer and represents the quantization of energy values.
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The solution to the above equation is
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$$\psi_n(x)=\exp(-x^2/2)H_n(x)$$
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where $H_n(x)$ is the $n$th Hermite polynomial in $x$.
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The following Eigenmath code checks $E=K+V$ for $n=7$.
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\medskip
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\verb$n=7$
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\verb$psi=exp(-x^2/2)*hermite(x,n)$
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\verb$E=(2*n+1)*psi$
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\verb$K=-d(psi,x,x)$
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\verb$V=x^2*psi$
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\verb$E-K-V$
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$$0$$
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