Square root of Dirac Delta function

[ismaili]

Homework Statement

I wonder how to deal with the square root of Dirac Delta function, [tex]\sqrt{\delta(x)}[/tex]. Actually, this comes from a problem which asking readers to calculate the wave function of a free particle and of a harmonic oscillator at time [tex]t[/tex], provided that the wave function at time [tex]t=0[/tex] is given, i.e. [tex]\psi(x,t=0) = \sqrt{\delta(x-a)} [/tex].

The way to obtain [tex]\psi(x,t)[/tex] in this problem is by the integral with the propagators.
However, I have no idea how to deal with [tex]\sqrt{\delta(x)}[/tex].

Homework Equations

[tex]\sqrt{\delta(x)}=?[/tex]

The Attempt at a Solution

I tried to differentiate it,
[tex]\frac{d}{dx}\sqrt{\delta(x)} = \frac{1}{2\sqrt{\delta(x)}}\delta'(x)
= -\frac{1}{2\sqrt{\delta(x)}}\frac{\delta(x)}{x} = -\frac{1}{2}\frac{\sqrt{\delta(x)}}{x} [/tex]
[tex] \Rightarrow
\int \sqrt{\delta(x)}f(x)dx = -2\int xf(x)d(\sqrt{\delta(x)})
[/tex]
But this is still no good...

I also tried the other definition of Dirac Delta function,
[tex]\delta(x)\sim\lim_{\epsilon\rightarrow 0}e^{-x^2/\epsilon}[/tex]
[tex]\Rightarrow \sqrt{\delta(x)} = \lim_{\epsilon\rightarrow 0}e^{-x^2/2\epsilon}
=\lim_{\epsilon'\rightarrow 0}e^{-x^2/\epsilon'} = \delta(x) ?? [/tex]

still no good...

I also tried to calculate [tex]\psi(x,t)^2[/tex] to try to get rid of the square root,
but it seems doesn't help.

Is there anyone who has any ideas about [tex]\sqrt{\delta(x)}[/tex] ?
Any help will be appreciated, thanks.

 

[ismaili]

Homework Statement

I wonder how to deal with the square root of Dirac Delta function, [tex]\sqrt{\delta(x)}[/tex]. Actually, this comes from a problem which asking readers to calculate the wave function of a free particle and of a harmonic oscillator at time [tex]t[/tex], provided that the wave function at time [tex]t=0[/tex] is given, i.e. [tex]\psi(x,t=0) = \sqrt{\delta(x-a)} [/tex].

The way to obtain [tex]\psi(x,t)[/tex] in this problem is by the integral with the propagators.
However, I have no idea how to deal with [tex]\sqrt{\delta(x)}[/tex].

Homework Equations

[tex]\sqrt{\delta(x)}=?[/tex]

The Attempt at a Solution

I tried to differentiate it,
[tex]\frac{d}{dx}\sqrt{\delta(x)} = \frac{1}{2\sqrt{\delta(x)}}\delta'(x)
= -\frac{1}{2\sqrt{\delta(x)}}\frac{\delta(x)}{x} = -\frac{1}{2}\frac{\sqrt{\delta(x)}}{x} [/tex]
[tex] \Rightarrow
\int \sqrt{\delta(x)}f(x)dx = -2\int xf(x)d(\sqrt{\delta(x)})
[/tex]
But this is still no good...

I also tried the other definition of Dirac Delta function,
[tex]\delta(x)\sim\lim_{\epsilon\rightarrow 0}e^{-x^2/\epsilon}[/tex]
[tex]\Rightarrow \sqrt{\delta(x)} = \lim_{\epsilon\rightarrow 0}e^{-x^2/2\epsilon}
=\lim_{\epsilon'\rightarrow 0}e^{-x^2/\epsilon'} = \delta(x) ?? [/tex]

still no good...

I also tried to calculate [tex]\psi(x,t)^2[/tex] to try to get rid of the square root,
but it seems doesn't help.

Is there anyone who has any ideas about [tex]\sqrt{\delta(x)}[/tex] ?
Any help will be appreciated, thanks.

I have an idea, but I'm not sure if it's correct.
Consider
[tex] f(0) = \int \delta(x)f(x)dx = \int \sqrt{\delta(x)}\sqrt{\delta(x)} f(x)dx
= \int \sqrt{\delta(x)}\sqrt{\delta(x)} dg(x)[/tex]
where
[tex] \frac{dg}{dx} = f [/tex]
,then integration by part gives,
[tex] f(0) = -2\int g(x)\frac{\delta'(x)}{2\sqrt{\delta(x)}} dx
= \int g(x) \frac{\sqrt{\delta(x)}}{x} dx[/tex]

Now, let [tex] g(x) = xh(x) [/tex], we have,
[tex] f(0) = \int h(x)\sqrt{\delta(x)} dx [/tex]
,i.e.

[tex]
\int h(x)\sqrt{\delta(x)} dx = \left[h(x) + xh'(x)\right]_{x=0}
[/tex]

I'm not sure if I were correct.
Any help would be appreciated!

-----------

Gosh, I found one mistake just now,
I calculated the integration by part wrong,
I omitted one factor of [tex] \sqrt{\delta(x)} [/tex]
So the final formula is incorrect.