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In the textbook titled Mathematical Methods for Physics and Engineering by Riley, Hobson and Bence, 3rd edition, there is a section on page 664 on the derivation of the uncertainty principle.

It starts of by considering two state vectors $ \left|u \right>$ and $ \left|v\right>$, and a real scalar $\lambda$. Letting $ \left|w\right> = \left|u \right> + \lambda \left|v \right> $, we take the inner product of $ \left|w\right> $ with itself and remember that the norm squared is always greater or equal to $0$.

$$ 0 \leq \left<w|w\right> = \left<u|u\right> + \lambda(\left<u|v\right> + \left<v|u\right>) + \lambda^2 \left<v|v\right> $$

It then says that this is a quadratic inequality in $\lambda$ and therefore the quadratic equation formed by equating the RHS to zero must have no real roots. What I don't understand is why mustn't it have real roots? We defined that $\lambda$ was real in the first place!

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  • $\begingroup$ Hint : calculate $b^2 - 4ac$ and apply Schwarz inequality $\endgroup$ – Oswald Aug 7 '16 at 14:32
  • $\begingroup$ Varying $\lambda \in \mathbb R$ you obtain a parabola with axis parallel to the $y$-axis. You already know that this curve completely stays in the half-plane $y\geq 0$. Do you think that this curve can intersect the $x$ ($\lambda$) axis in a couple of solutions? $\endgroup$ – Valter Moretti Aug 7 '16 at 15:08
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Let me put it a little differently. Let $f(\lambda) = \langle u|u \rangle + \lambda(\left<u|v\right> + \left<v|u\right>) + \lambda^2 \left<v|v\right>$; we can prove that $f(\lambda) \ge 0$ for all real $\lambda$. Therefore, if there's some real number $\lambda_0$ such that $f(\lambda_0)=0$ (i.e., if $f$ has a real root), it must be a double root. This is because if $f$ had two distinct real roots, it would have to go negative at some point.

You can see this graphically: if you have a quadratic that's always non-negative, then it either has a double root (that is, it just barely touches the x-axis) or it has no real roots. In either case, the discriminant is $\le 0$.

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Because the quadratic is never negative, any real root must be a double root. Therefore there are two cases: No real roots, in which case the discriminant $ b^2−4ac$ is negative, or one double root, in which case the discriminant is zero. Either way you get that the discriminant is non-negative, which is the uncertainty principle.

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