Recent content by Hakkinen

Homework Statement Consider a system of two spin 1/2 particles, labeled 1 and 2. The Pauli spin matrices associated with each particle may then be written as \vec{\hat{\sigma _{1}}} , \vec{\hat{\sigma _{2}}} a)Prove that the operator \hat{A]}\equiv \vec{\hat{\sigma _{1}}}\cdot...

My E&M professor brought up this problem of considering a uniform charge density, rho, that is infinite in volume and then using Gauss's Law to find the electric field at a point. It's resulted in a lot of head scratching and I'd appreciate some help/discussion to guide me towards a resolution...

I don't know if I'm setting up the integral correctly in a form where a straightforward application of CIF could be done. I know about making arcs to form a closed contour but would I just need to evaluate one of these arcs? In this problem for example can I just use the "lower" arc that...

Thanks for the reply, but I had already solved that

Homework Statement Given a linear operator L=\frac{d^3}{dx^3}-1, show that the Fourier transform of the Green's function is \tilde{G}(k)=\frac{i}{k^3-i} and find the three complex poles. Use the Cauchy integral theorem to compute G(x) for x < 0 and x > 0. Homework Equations The...

Thanks for the reply. So the eigenvalue is 0 and the eigenfunction is just ax, with a determined by some IV?

I just have a question about the problem for when the eigenvalue = 0 Homework Statement for y_{xx}=-\lambda y with BC y(0)=0 , y'(0)=y'(1) Homework Equations The Attempt at a Solution y for lamda = 0 is ax+b so from BC: y(0)=b=0 and a=a What is the conclusion to...

Thanks for the reply! sorry for the simple mistakes So is this not the correct approach to show this? If it is, should I start with a|0> = k|0> and then conjugate, leaving the annihilation operator acting on the ground state bra? so 0 = <0|\bar{k} but this doesn't necessarily show...

Homework Statement Show there are no eigenvectors of a^{\dagger} assuming the ground state |0> is the lowest energy state of the system. Homework Equations Coherent states of the SHO satisfy: a|z> = z|z> The Attempt at a Solution Based on the hint that was given (assume there...

I think I've got a solution for this but I'm not sure if it's general enough. So since each operators eigenbasis forms an orthonormal set you can choose any convenient orthonormal basis. If I pick the standard cartesian basis (i,j,k) then each operator can be represented by the identity...

Homework Statement Assume A and B are normal linear operators [A,A^{t}]=0 (where A^t is the adjoint) show that det AB = detAdetB Homework Equations The Attempt at a Solution Well I know that since the operators commute with their adjoint the eigenbases form orthonormal sets...

So for the general solution of u I have u(x,t)=\sum_{n=1}^{\infty}A_{n}\sin [\frac{\pi}{2}(1+2n)x]\exp -2t[\frac{\pi}{2}(1+2n)]^2 and the coefficient A_n given by A_{n}=\frac{12}{\pi(1+2n)}\cos \pi(1+2n) There was another cosine term with pi/2 in the argument that was always zero for...

Homework Statement solve the heat equation over the interval [0,1] with the following initial data and mixed boundary conditions.Homework Equations \partial _{t}u=2\partial _{x}^{2}u u(0,t)=0, \frac{\partial u}{\partial x}(1,t)=0 with B.C u(x,0)=f(x) where f is piecewise with values: 0...

however, could there be a typo in part c? I wrote it exactly as typed from the professor but if the divergence of that 2nd region is zero than the whole volume integral should be zero too?

Thanks! So it's much simpler than I thought, just a straightforward application of the divergence theorem in the context of gauss' law.