Prove the following identities that we assumed to calculate [h, x] in [h, p] in class. (a) let a, b and c be arbitrary matrices (possibly infinite dimensional) then prove the following commutator identity: [ab, c] = a[b, c] + [a, c]b . (b) show that the position-momentum commutator [x, p] = i~ generalizes to [x n , p] = i~nxn−1 . (c) assume that f(x) has a convergent taylor expansion, then show that [f(x), p] = i~ df dx (x) .

Which of the following are considered noble gases? a. bromine b. neon c. argon d. chlorine

Aparticle of mass m is placed in a one-dimensional box of length l. the box is so small that the particle’s motion is relativistic, so that e p2/2m is not valid. (a) derive an expression for the energy levels of the particle using the relativistic energy–momentum relation and the quantization of momentum that derives from connement. (b) if the particle is an electron in a box of length l 1.00 1012 m, nd its lowest possible kinetic energy. by what percent is the nonrelativistic formula for the energy in error?

1700 j of energy is lost from 0.14 kg object , the temperature decreases from 50°c to 45°c what is the specific heat of this object, amd what is the material ?