Recent content by Jacob Nie

I'm guessing that there must be some nuance that I do not quite understand in the difference between ##|p\rangle## and ##|E\rangle##? Like, later in the book even ##dk## is used as a variable of integration, where ##k = p/\hbar.## Surely this has effects on the value of the integral - wouldn't...

Thank you for the responses - that makes sense. I forgot to read the sentence of the book that said:

What I don't understand is why ##dS## is expanded in only the two differentials ##dV## and ##dT.## Why doesn't it look more like: $$dS = \left(\dfrac{\partial S}{\partial V}\right)_{T,P,U} \ dV + \left(\dfrac{\partial S}{\partial T}\right)_{V,P,U} \ dT + \left(\dfrac{\partial S}{\partial...

The explicit form would be ##\mathbb{P}_{\omega} = |\omega\rangle \langle \omega | \psi \rangle.## So, ##\langle \psi | \mathbb{P}_{\omega}| \psi \rangle = \langle \psi | \omega \rangle \langle \omega | \psi \rangle = |\langle \omega | \psi \rangle |^2.## Thank you very much for the helpful hint.

I don't see why it is not ##P(\omega)\propto |\langle \psi | \mathbb{P}_{\omega}|\psi\rangle |^2.## After all, the wavefunction ends up collapsing from ##|\psi\rangle## to ##\mathbb{P}_{\omega}|\psi\rangle.##

Sorry - I have solved it. I gave the wrong identities - these are more directly useful: ##\dfrac{d}{dx}\left(x^pJ_p(x)\right) = x^pJ_{p-1}(x)## and ##\dfrac{d}{dx}\left(x^{-p}J_p(x)\right) = -x^{-p}J_{p+1}(x).## (Although, the identities I gave are attained simply by carrying out the...

I tried integration by parts with both ##u = x^2, dv = J_0 dx## and ##u = J_0, du = -J_1 dx, dv = x^2 dx.## But neither gets me in a very good place at all. With the first, I begin to get integrals within integrals, and with the second my powers of ##x## in the integral would keep growing...

The beetle problem: Let $t$ be the number of days since the beetle made his decision to climb. Let ##b(t)## be the height of the beetle, calculated at sunrise. Knowing ##b(0)=0##, we can write the following recursive relation: $$ b(t) = b(t-1) + 0.2\dfrac{b(t-1)}{100} + 0.1.$$ I am assuming...

It's my first time on the forums! Every other problem is like a foreign language to me so I will offer my solution to the clock problem: Let us the define one "ruoh" to be the angular distance between two adjacent numbers on the clock (30 degrees). (The IBWM still hasn't responded to my...