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noblegas
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Homework Statement
b)Use the equations [tex]
a_{-}|\varphi_0\rangle=0 \implies \left(\left(\frac{k}{2}\right)^{1/2}x-\left(\frac{1}{2k/(2m)^(1/2)}\right)^{1/2}\frac{d}{d x}\right)\varphi_0(x)=0
[/tex] and [tex]
a_{+}|\varphi_0\rangle=0 \implies \left(\left(\frac{k}{2}\right)^{1/2}x+\left(\frac{1}{2k/(2m)^(1/2)}\right)^{1/2}\frac{d}{d x}\right)\varphi_0(x)=0
[/tex] to express as a linear combination of the lowering and raising operators a_ and a+, and then use the operator methods to show that x[itex]
\varphi_0(x)
[/itex] is the(not normalized) wave function for the first excited state;
c) Use the properties of the raising operator to argue that the wave function for the nth energy level, with E_n=(n+1/2)*omega*[tex]\hbar[/tex], for the oscillator is a polynomial multiplied by a gaussian exp(-c*x^2), and find the order of the polynomial
Homework Equations
The Attempt at a Solution
b) I don't think it is diffiicult to find the linear combination; c*(a_+ a_+)=c*2*(K/2)^(1/2); Not sure how I would show that it does not normalized for the first excited state but would I start the problem in this manner? a* [itex]
\varphi_1(x)
[/itex]=0, to find [itex]
\varphi_1(x)
[/itex] and plug
[tex]
\int_{-\infty}^{\infty} |\varphi_1(x)|^2 dx=1
[/tex] ; How would I not know that x[itex]
\varphi_0(x)
[/itex] is not normalizable?
c) [tex]
a_{+}|\varphi_0\rangle=0 \implies \left(\left(\frac{k}{2}\right)^{1/2}x+\left(\frac{1}{2k/(2m)^(1/2)}\right)^{1/2}\frac{d}{d x}\right)\varphi_0(x)=0
[/tex]
Not sure what they are asking for.
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