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Consider the second order ODE (actually it is time-independent Schrödinger equation)

$$\frac{d^2\phi}{dx^2} + V(x)\phi = E\phi$$

We know that $\phi$ is a bound solution. This means the following two things:

(i)$$\lim_{x\to \pm\infty}\phi(x)=0 .$$

(ii) $0<E<\max[V(x)]$.

Using this information, can we prove

$$\int_{-\infty}^{\infty}\vert\phi\vert^2\frac{dV}{dx} dx=0\,?$$

You are free to make any reasonable assumptions about the nature of $V(x)$ and $\phi(x)$ along your way.

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Try doing by-parts integration twice. And nice name. –  anon Nov 17 '11 at 19:40

1 Answer 1

up vote 2 down vote accepted

If you assume that the potential $V(x)$ is bounded, then we can prove it by integration by parts as suggested by anon: $$ \int^\infty_{-\infty}\phi^2\frac{dV}{dx}{dx}=\int^\infty_{-\infty}\phi^2dV =\phi^2V\,\big|^\infty_{-\infty}-2\int^\infty_{-\infty}V\phi\frac{d\phi}{dx}dx $$ where the last equality follows from integration by parts. Since $\lim_{x\to \pm\infty}\phi(x)=0$ by your assumption and $V(x)$ is bounded by my assumption, $\phi^2V\,\big|^\infty_{-\infty}=0$. Now using the equation $\frac{d^2\phi}{dx^2} + V(x)\phi = E\phi$, we can rewrite the above equation as $$ \int^\infty_{-\infty}\phi^2\frac{dV}{dx}{dx}=-2\int^\infty_{-\infty}V\phi\frac{d\phi}{dx}dx=2\int^\infty_{-\infty}\frac{d^2\phi}{dx^2}\frac{d\phi}{dx}dx-2E\int^\infty_{-\infty}\frac{d\phi}{dx}dx:=I+II. $$ Note that $$I=2\int^\infty_{-\infty}\frac{d^2\phi}{dx^2}\frac{d\phi}{dx}dx=2\int^\infty_{-\infty}\frac{d\phi}{dx}d\big(\frac{d\phi}{dx}\big)=\big(\frac{d\phi}{dx}\big)^2\Big|^\infty_{-\infty}=0$$ since $\lim_{x\to \pm\infty}\phi(x)=0$ implies that $\lim_{x\to \pm\infty}\frac{d\phi}{dx}(x)=0$. Note also that $$II=-2E\int^\infty_{-\infty}\frac{d\phi}{dx}dx=-2E\phi\,\big|^\infty_{-\infty}=0.$$ Now the result follows.

Note added: To see that $\lim_{x\to \pm\infty}\frac{d\phi}{dx}(x)=0$, first note that $\frac{d\phi}{dx}$ is bounded. To see this, since $\frac{d^2\phi}{dx^2} + V(x)\phi = E\phi$, $\lim_{x\to \pm\infty}\phi(x)=0$ and $V(x)$ is bounded, we have $\lim_{x\to \pm\infty}\frac{d^2\phi}{dx^2}=0$. This implies that $\frac{d\phi}{dx}$ is bounded. Next, if we multiply $\phi$ to $\frac{d^2\phi}{dx^2} + V(x)\phi = E\phi$, we get $$\phi\frac{d^2\phi}{dx^2}=(E-V(x))\phi^2$$ Now integrate it from $M$ to $\infty$ and we get $$\int_M^\infty\phi\frac{d^2\phi}{dx^2}dx=\int_M^\infty(E-V(x))\phi^2dx.$$ The right hand side is zero when $M$ tends to infinity since $V(x)$ is bounded and the eigenfunction $\phi$ is $L^2$. On the other hand, by integration by parts, the left hand side is $$\int_M^\infty\phi\frac{d^2\phi}{dx^2}dx=(\phi\frac{d\phi}{dx})\Big|^\infty_M-\int_M^\infty(\frac{d\phi}{dx})^2dx=-\phi(M)\frac{d\phi}{dx}(M)-\int_M^\infty(\frac{d\phi}{dx})^2dx.$$ The first term goes to zero since $\lim_{x\to \pm\infty}\phi(x)=0$ and $\frac{d\phi}{dx}$ is bounded. Combining all these, we have $\lim_{x\to \infty}\frac{d\phi}{dx}(x)=0$. Similarily we can prove that $\lim_{x\to -\infty}\frac{d\phi}{dx}(x)=0$.

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Paul, thank you so much. Just a follow up question: How does $$\lim_{x\to\pm\infty}\phi(x)=0$$ imply $$\lim_{x\to\pm\infty}\frac{d\phi(x)}{dx}=0?$$ –  anon Nov 21 '11 at 21:14
    
see my edited answer. –  Paul Nov 26 '11 at 7:21
    
Paul thanks for the rigorous explanation –  anon Nov 28 '11 at 22:02
    
You are welcome. You may want to accept the answer. –  Paul Nov 29 '11 at 13:27

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