QM Fourier Transform and Integral rules

[katkatkat]

Show G(k)=$\sqrt{2π}$g₁(k)g₂(k)

Given that G(k) is the Fourier transform of F(x), g₁(k) is Fourier trans of f₁(x), g₂(k) is Fourier trans of f₂(X) and

F(x)=$^{+∞}_{-∞}$∫dyf₁(y)f₂(x-y)

SO FAR

G(k)=$1/\sqrt{2π}$$^{+∞}_{-∞}$∫F(x)e^-ikxdx <-def'n of Fourier transform

=$1/\sqrt{2π}$$^{+∞}_{-∞}$∫$^{+∞}_{-∞}$∫dyf₁(y)f₂(x-y)e^-ikxdx

mult by e^-ikye^iky

=$1/\sqrt{2π}$$^{+∞}_{-∞}$∫dyf₁(y)e^-iky$^{+∞}_{-∞}$∫f₂(x-y)e^-ik(x-y)dx

=g₁(k)$^{+∞}_{-∞}$∫f₂(x-y)e^-ik(x-y)dx

Problem starts here... I have arg of x-y in the remaining part, with dx...
Thought was that sub dx=dx-dy+dx=d(x-y)+dx <-Pretty sure I'm allowed to do this

=g₁(k)$^{+∞}_{-∞}$∫f₂(x-y)e^-ik(x-y)d(x-y)+g₁(k)$^{+∞}_{-∞}$∫f₂(x-y)e^-ik(x-y)d(y)

=$\sqrt{2π}$g₁(k)g₂(k)+g₁(k)$^{+∞}_{-∞}$∫f₂(x-y)e^-ik(x-y)d(y)

I'm definitely on the right track up to the problems part, do I say that ∫arg*dy is equal to zero because F(x) is a function of x... or a different reasoning... not sure... If I can make the right half of the equation = 0, I'll have the desired result

=$\sqrt{2π}$g₁(k)g₂(k)

=G(k)

 

[gabbagabbahey]

=$1/\sqrt{2π}$$^{+∞}_{-∞}$∫dyf₁(y)e^-iky$^{+∞}_{-∞}$∫f₂(x-y)e^-ik(x-y)dx

=g₁(k)$^{+∞}_{-∞}$∫f₂(x-y)e^-ik(x-y)dx

This is not correct, you cannot claim that

$$\int_{ -\infty }^{ \infty }\int_{ -\infty }^{ \infty }dydx f_1(y)f_2(x-y) e^{-iky} e^{-ik(x-y)} = \left( \int_{ -\infty }^{ \infty }dx f_2(x-y) e^{-ik(x-y)} \right) \left( \int_{ -\infty }^{ \infty }dy f_1(y)e^{-iky} \right)$$

The first integral contains the variable $y$, so is not constant when integrating over $y$ and cannot be pulled out of the $y$ integral. Instead, try integrating over $x$ first while treating $y$ as a constant (which you can do since they are independent variables over the entire region of integration).

 

[katkatkat]

I can't actually solve the integral as I'm supposed to be subbing in the symbol for the Fourier transform. I have no choice but to separate them somehow.

Here's the prof's solution. I'm not sure how he got to the 3rd step though. How did he get d(x-y) out of dx.

G(k)=$1/\sqrt{2π}$$^{+∞}_{-∞}$∫F(x)e^ikxdx

=$1/\sqrt{2π}$$^{+∞}_{-∞}$∫dx*e^ikx$^{+∞}_{-∞}$∫f₁(y)f₂(x-y)dy

=$1/\sqrt{2π}$$^{+∞}_{-∞}$∫d(x-y)f₂(x-y)e^ik(x-y)$^{+∞}_{-∞}$∫f₁(y)e^ikydy

=$\sqrt{2π}$g₁(k)g₂(k)

I know his solution is right, this is coming from someone who's specialty is Fourier transforms. I'm just not sure how he did the d(x-y) step from the dx step (line 2 to line 3)

 

[gabbagabbahey]

Just use a substitution: $u(x)=x-y$, when integrating over x, y is constant, so $du=?$...

 

[rude man]

Show G(k)=$\sqrt{2π}$g₁(k)g₂(k)

Given that G(k) is the Fourier transform of F(x), g₁(k) is Fourier trans of f₁(x), g₂(k) is Fourier trans of f₂(X) and

F(x)=$^{+∞}_{-∞}$∫dyf₁(y)f₂(x-y)

SO FAR

G(k)=$1/\sqrt{2π}$$^{+∞}_{-∞}$∫F(x)e^-ikxdx <-def'n of Fourier transform

=$1/\sqrt{2π}$$^{+∞}_{-∞}$∫$^{+∞}_{-∞}$∫dyf₁(y)f₂(x-y)e^-ikxdx

mult by e^-ikye^iky

=$1/\sqrt{2π}$$^{+∞}_{-∞}$∫dyf₁(y)e^-iky$^{+∞}_{-∞}$∫f₂(x-y)e^-ik(x-y)dx

No, the problem starts here.

The second integral, which, when evaluated (integrated over x), being a function of y, therefore belongs under the 1st integral. You put the second integral outside the first.

For the last step, hint: ∫f₂(x-y)exp{-ik(x-y)}dx = same expression with dx → d(x-y), since y is constant in the x integration.

 

[madhatter42]

Hi,

I just wanted to note that you are essentially trying to prove the convolution theorem for the Fourier transform. Knowing that may help you out by looking at its proof.

 

[katkatkat]

The second integral, which, when evaluated (integrated over x), being a function of y, therefore belongs under the 1st integral. You put the second integral outside the first.

I don't think that matters. It's a function of the variable (x-y) not y. Can you not sub in z=x-y? Then you no longer have that problem. I don't think it'll fall under the dy integral for that reason since it's not a function of the y variable.

 

[katkatkat]

I'm going to sub in z(x)=x-y. Then dx=dz and f(x-y)=f(z) etc. Thanks for the help guys :) This looks on the right track according to some of the proofs I've seen.

 

[vela]

You could use the change of variables u=y, v=x-y. It's straightforward to show that dx dy = du dv so that
$$\int_{-\infty}^\infty \int_{-\infty}^\infty dy\,dx\,f_1(y)\,f_2(x-y) e^{-iky} e^{-ik(x-y)} = \int_{-\infty}^\infty \int_{-\infty}^\infty du\,dv\,f_1(u)\,f_2(v) e^{-iku} e^{-ikv}.$$

 

[rude man]

I don't think that matters. It's a function of the variable (x-y) not y. Can you not sub in z=x-y? Then you no longer have that problem. I don't think it'll fall under the dy integral for that reason since it's not a function of the y variable.

Ahem - if it' a function of (x-y) it's ipso facto a function of x ... the second integral, when integrated wrt x, is now a function only of y.

You don't need to substitute anything, just realize that dx and d(x-y) are the same thing if y is a constant, which it is in the second integral (the one ending in dx). And you can't just say, "well, I'll get rid of y by substituting z = x-y." That's actually quite amusing ...