Consider the position and momentum operators $\hat q$ and $\hat p$, defined respectively in terms of creation and destruction operators in the usual way: $$ \hat q = c (\hat a + \hat a^\dagger), \qquad \hat p = \frac{i\hbar}{2c} ( \hat a^\dagger - \hat a), \tag 1 $$ for some constant $c$, where $\hat a$ and $\hat a^\dagger$ satisfy the canonical commutation relations: $$ [\hat a, \hat a^\dagger] = 1, \qquad [\hat a,\hat a] = [\hat a^\dagger, \hat a^\dagger] = 0. \tag 2$$ If follows immediately that, as expected, $[\hat q, \hat p] = i \hbar$.  My question is now: starting from these definitions of $\hat p$ and $\hat q$, how can I derive the common relation $\hat p = -i \hbar \frac{\partial}{\partial q}$, or some equivalent version of this relation? In other words, how can I compute the value of $\langle q | \hat p| \psi \rangle$, or equivalently of $\langle q| \hat p | q' \rangle$?  If I try to simply compute this using the definition of $\hat p$ given in (1) I get nonsensical results, I'm gessing because of $|q \rangle$ being not normalized eigenstates. What method should then be used in this context?
I have a question about deriving the coordinate representation of momentum operator from the commutation relation, $[x,p]= i$.  One derivation (ref W. Greiner's Quantum Mechanics: An Introduction, 4th edi, p442) is as following: $$ \langle x|[x,p]|y \rangle = \langle x|xp-px|y \rangle = (x-y) \langle x|p|y \rangle. $$ On the other hand,  $ \langle x|[x,p]|y \rangle = i \langle x|y \rangle = i \delta (x-y)$. Thus $$ (x-y) \langle x|p|y \rangle = i \delta(x-y). \tag{1} $$  We use $(x-y) \delta(x-y) = 0$. Take the derivative with respect to $x$; we have $\delta(x-y) + (x-y) \delta'(x-y) = 0$. Thus $$ (x-y) \delta'(x-y) = - \delta(x-y). \tag{2} $$  Comparing Eqs. (1) and (2), we identify $$ \langle x|p|y \rangle = -i \delta'(x-y). \tag{3} $$  In addition, we can add $\alpha \delta(x-y)$ on the right-hand side of Eq. (3), i.e. $$ \langle x|p|y \rangle = -i \delta'(x-y)  + \alpha \delta(x-y), $$ and $[x,p] = i$ is still satisfied. We can also add $$ \frac{\beta}{\sqrt{|x-y|}}\delta(x-y)$$ on the RHS of Eq. (3).  Here $\alpha$ and $\beta$ are two real numbers.  My question is, what is the most general expression of $\langle x|p|y \rangle$? Can we always absorb the additional term into a phase factor like Dirac's quantum mechanics book did?  Thank you very much in advance.
Benefits and Drawbacks of Swapping Associations of Positivity and Negativity of the Proton and Electron For historical reasons, I understand that a proton's charge was assigned the label of "positive", the electron's charge was assigned the label of negative, and, crucially, equations of current in E &amp; M are written with the notion that current is the flow of positive charge despite the rarity of actual flows of protons in real systems (which I understand are mostly biological systems).   If the possibility existed to instantaneously and at no cost, retcon these associations such that protons have a negative charge, electrons have a positive charge, and current remains the flow of positive charge, would there be any beneficial or detrimental outcomes?  For instance, would there be:   new interpretations of physical laws that would provide different or better insights into any relevant subfield of Physics significant simplifications to notation in any field of Physics or Engineering significant increases in understanding in the fields of electricity, electronics, electrical engineering, radiation/radiative transport, etc. a noticeably easier learning curve for any subfield of physics dealing with the concept of charge   Additionally, is there a permutation of positive charge, negative charge, and flow of current definition that would improve upon the one I suggested? Reference: Why are my "special" Unicode characters encoded weird using json_encode? I've just created an account to submit an application to the Ubuntu Software Centre and the agreement says: "You are responsible for determining applicable taxes in connection with distributing your App, and you shall pay applicable taxes to the applicable tax authorities.".  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When I tried to change it using xrandr --output XWAYLAND0 --mode 1280x1024_60.00 --rate 60 But an error occurred:  $ xrandr --output XWAYLAND0 --mode 1280x1024  --rate 60 X Error of failed request:  BadValue (integer parameter out of range for operation)   Major opcode of failed request:  140 (RANDR)   Minor opcode of failed request:  7 (RRSetScreenSize)   Value in failed request:  0x0   Serial number of failed request:  21   Current serial number in output stream:  22   I don't know what did I do wrong. Or how to change resolution to the correct one: 1280x1024 How can I draw outlines around 3D models? How can I draw outlines around 3D models? I'm referring to something like the effects in a recent Pokemon game, which appear to have a single-pixel outline around them: Possible Duplicate:          I have recently seen this image:    Should "a" have been used instead of "an" in the "...an $100,000 apartment" part? 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