How is the pitch of a sound defined quantitatively when it is a mixture of many frequencies? For example, the sound emitted by a plucked guitar string, or say, the pitch of somebody's (normal) voice. I know that female voices are generally of higher pitch. But can we quantify the notion of pitch?...

It is said in wikipedia that Minkowski spacetime isometries, i.e. the transformation that preserves $$ (x_1-x_2)^2+(y_1-y_2)^2+(z_1-z_2)^2-(t_1-t_2)^2 $$ between points, can be represented as $\mathbb{R}^{1,3}\rtimes O(1,3)$, meaning that it consists of transformations with the form $$ x \mapsto ...

In the quantum textbook I'm currently working from, the completeness relation is written as: $$ \sum_i |\psi_i \rangle \langle \psi_i| = \mathbb{1}. $$ But this seems to specifically require knowledge of individual bra and ket vectors. I know wavefunctions are supposed to satisfy both orthogonal ...

We have two tires with radius r and R respectively. They are connected to two almost identical vehicles with same net mass. "Almost", because Gear ratios are adjusted such that At equal Engine RPMs, both vehicles have same road speed, on paper at least. So, does this have any effect on the offroa...

Let $\rho$ and $\rho'$ be $n\times n$ density matrices, and suppose that for every observable $A$ and every $\lambda$ in the spectrum of $A$ we have $$ \text{tr}(\rho P_{\lambda})=\text{tr}(\rho' P_{\lambda}), $$ where $P_{\lambda}$ is the orthogonal projection onto the $\lambda$-eigenspace of $A...

De Broglie's equation regarding the wave particle duality of matter $\lambda =\frac{h}{mv}$ depends on velocity. Now, velocity is relative and depends on the frame of reference of the observer. But does this mean that even the wavelength is relative and will be different for different observers a...