[Lz, |Îz, Îy] = iħÎz [Îy, Î.] = iħÎ₂ [Îz,, Îx] = iħÎy

What do we get if we multiply the angular momentum raising and lowering operators together?

5. Obtain the the expression for the density of states for a system of free electrons in a three-dimensional infinite square well.

Explain what spin up and spin down is in quantum entanglement.

Physics When we use the notation |k,m,n> in angular momentum, for the case of spin we write, for example |k,1/2,1/2>. In that case, what is the meaning of the k?

At what temperature would we find about half as many hydrogen atoms in first excited state as in the groundstate?..

Consider a hydrogen-like atom such as He+ or Li++ that has a single electron outside a nucleus of charge Ze. (a) Rewrite the Schrödinger equation with the new Coulomb potential. (b) What change does this new potential have on the separation of variables? (c) Will the radial wave functions be affected? Explain. (d) Will the spherical harmonics be affected? Explain.

If a very small uniform-density sphere of charge is in an clectrostatic potential V(r), its potential cnergy is U(r) = V(r) + rởV²V(r) + -…, where r is the position of the center of the charge and ro is its very small radius. The “Lamb shift" can be thought of as the sinall correction to the energy levels of the hydrogen atom because the physical electron does have this property. If the r term of U is treated as a very small perturbation compared to the Coulomb interaction V (r) = -c2/r., what are the Lanb shifts for the 1s and 2p levels of the hydrogen atom? Express your result in terms of ro and fundameutal constants. The umperturbed wave functions are P16(r) = 2a,2. -3/2. e-r/anY; bzpm(r) : 1 -5/2 re-r/20BYm %3D V24 where as = h? /mee?.

Explain the difference between the group velocity and the phase velocity. Tell which one is associated with the probabilistic center of mass of the electron.

Find Ly, Ly Lz then estimated [12, Lx] when L= F xp (the angular momentum)