Validity of Mathematical Proof of Uncertainty Principle

[phreak]

I saw a rather easy proof of the Heisenberg Uncertainty Principle in a PDE textbook the other day, but I'm not sure if it's correct. The proof goes as following:

Note that $$\left| \int xf(x) f'(x) \right| \le \left[ \int |xf(x)|^2 dx \right]^{1/2} \left[ \int |f'(x)|^2 dx \right]^{1/2}$$, by the Cauchy-Schwartz inequality. Now, the Fourier transform of df/dx is ipF(p), so along with Parseval's Equality, the right side of the above equation equals:

$$\overline{x} \cdot \left[ \int |ipF(p)|^2 \frac{dp}{2\pi} \right]^{1/2} = \overline{x} \cdot \overline{p}.$$

Integrating the left side by parts, we then get that it is 1/2, so that $$\overline{x} \cdot \overline{p} \ge 1/2$$.

Now, here is my problem with this proof. I don't understand why $$\int |ip F(p)|^2 \frac{dp}{2\pi} = \overline{p}^2$$. Using Parseval's equality, this would be fine if the Fourier transform of xf(x) is kF(k), but as far as I know this isn't true. Can anyone point me in the right direction?

 

[malawi_glenn]

here is one version of the proof, I don't know at the moment the version you are referring to.

https://www.physicsforums.com/library.php?do=view_item&itemid=207

 

[Ilja]

I don't understand why $$\int |ip F(p)|^2 \frac{dp}{2\pi} = \overline{p}^2$$.

Whatever the observable O, if $\psi(o)$ is the wave function in the representation defined by O (that means, $\psi(o)$ is the amplitude of O having the value o), and whatever the function f(o), the expectation value of f(o) is


$$\int f(o)|\psi(o)|^2 do = \overline{f(o)}$$


Apply this to o=p and f(o)=o².

 

[Josyulasharma]

can anybody here please suggest me a book for quantum mechanics.(i'm good at math)

 

[Josyulasharma]

@phreak
can please give the name of the book you reffered.

 

[phreak]

Thank you for your comments.

Josyulasharma: The book this is from is 'Partial Differential Equations' by Walter Strauss.

 

[malawi_glenn]

can anybody here please suggest me a book for quantum mechanics.(i'm good at math)

i) don't ask unrelated questions to the thread, i.e don't get off topic.

ii) We have science book sub forum here, look around an you'll find it.

iii) when you do find it, please search for old threads, this question has been asked 100000 times before,