really easy questons that you shouldnt overthink
what color is aiden
white
what is the hair and eye color required to enter agartha
what is blue eyes and blond hair
how many people died in the holocause
what is 271k (lose a billion points if you say 6m)
who is the foid in team 7 in naruto
sakura
who sell big mac
mcdondal
waht is our galaxy
milk way
what is the supreme energy only white people have
what is vril
who is the glorious leader of the place
who is netenyahu
what anime is eastons pfp from
RoR
who sell crunchwrap supreme
taco bell
isreal?
good
who is the mad scientist who created white people
who is yakub
has isreal ever done anything bad
no
which anime uses nen as their power system
HxH
who sell the meat mountain
arby
who owns the evil laser treadmill company
mr beast
what is the alternate name for agartha
hyperborea
what are they
what are jews
Aizen
who sell the sailors seafood boil family meal
red lobster
solve the following:
A particle of mass mmm and electric charge qqq is constrained to move on a 1-dimensional ring of radius RRR (coordinate θ∈[0,2π)\theta\in[0,2\pi)θ∈[0,2π)). A magnetic flux Φ\PhiΦ threads the ring (vector potential can be chosen A=(Φ/2πR) θ^\mathbf{A} = (\Phi/2\pi R)\,\hat{\theta}A=(Φ/2πR)θ^ on the ring).
Write and solve the time-independent Schrödinger equation to find the energy eigenvalues En(Φ)E_n(\Phi)En(Φ) and normalized eigenfunctions ψn(θ;Φ)\psi_n(\theta;\Phi)ψn(θ;Φ).
Compute the persistent current In(Φ)I_n(\Phi)In(Φ) carried by the eigenstate nnn defined as
In(Φ)=−∂En(Φ)∂Φ.I_n(\Phi) = -\frac{\partial E_n(\Phi)}{\partial \Phi}.In(Φ)=−∂Φ∂En(Φ).
State which nnn is the ground-state index at zero temperature (qualitatively) and give the ground-state current.
On the ring the minimal-coupling Hamiltonian (in the θ\thetaθ coordinate) is
H^=12m(p^θ−qAθR)2,p^θ=−iℏ∂∂θ,\hat H = \frac{1}{2m}\left(\frac{\hat p_\theta - qA_\theta}{R}\right)^2, \qquad \hat p_\theta = -i\hbar\frac{\partial}{\partial\theta},H^=2m1(Rp^θ−qAθ)2,p^θ=−iℏ∂θ∂,
and with Aθ=Φ2πRA_\theta=\dfrac{\Phi}{2\pi R}Aθ=2πRΦ (a gauge choice appropriate for the ring) this becomes
H^=12mR2(−iℏ∂∂θ−qΦ2π)2.\hat H = \frac{1}{2mR^2}\left(-i\hbar\frac{\partial}{\partial\theta} - \frac{q\Phi}{2\pi}\right)^2.H^=2mR21(−iℏ∂θ∂−2πqΦ)2.
Look for eigenfunctions of the form ψ(θ)=12πeinθ\psi(\theta)=\dfrac{1}{\sqrt{2\pi}}e^{i n\theta}ψ(θ)=2π1einθ with integer nnn (single-valuedness requires n∈Zn\in\mathbb{Z}n∈Z). Acting with H^\hat HH^ gives
H^ψn=12mR2(ℏn−qΦ2π)2ψn.\hat H \psi_n = \frac{1}{2mR^2}\left(\hbar n - \frac{q\Phi}{2\pi}\right)^2 \psi_n.H^ψn=2mR21(ℏn−2πqΦ)2ψn.
Define the dimensionless flux φ≡ΦΦ0 \displaystyle \varphi\equiv\frac{\Phi}{\Phi_0}φ≡Φ0Φ where Φ0\Phi_0Φ0 is the flux quantum for charge qqq:
Φ0≡hq(so φ=Φ/Φ0).\Phi_0 \equiv \frac{h}{q}\quad(\text{so } \varphi=\Phi/\Phi_0).Φ0≡qh(so φ=Φ/Φ0).
Note h=2πℏh=2\pi\hbarh=2πℏ. Then qΦ2π=ℏφ\dfrac{q\Phi}{2\pi} = \hbar\varphi2πqΦ=ℏφ. Thus the eigenenergies simplify to
En(Φ)=ℏ22mR2(n−φ)2,n∈Z \boxed{\,E_n(\Phi)=\frac{\hbar^2}{2mR^2}\bigl(n-\varphi\bigr)^2,\qquad n\in\mathbb{Z}\,}En(Φ)=2mR2ℏ2(n−φ)2,n∈Z
and the normalized eigenfunctions are
ψn(θ;Φ)=12πeinθ \boxed{\,\psi_n(\theta;\Phi)=\frac{1}{\sqrt{2\pi}}e^{i n\theta}\,}ψn(θ;Φ)=2π1einθ
(you can view the flux as shifting the effective angular momentum by ℏφ\hbar\varphiℏφ).
who invented the agarthan theory
saint-yves d'alveydre
if you were given a billion dollars, how much would you give to isreal
all of it
what anime is this guy from (pullup the image))
who sells the aura
john pork