cx%3D+00 Using only the property f(a) = o f(x)8(x – a)dx as a definition of the Dirac delta, prove the x=-00 following properties of the Dirac Delta rx=+00 i) L 8(x – a)dx = 1 x=-00 = (x) ii) 8-x) ii) а8(tax) —D &х); а > 0 in) 8(x? — а?) %3D - 16(х + а) + 6(х - а)] 2a

Find the gradient field F = Vo for the potential function o below. P(x,y.z) = In (3x +y° +z²) Vøxy.z) = (O Vo(x.y.z z)3D

The distance from the point (-8,-7,-2) to the yz plane is: ?

View of a one-dimensional harmonic oscillator system on the x axis of a charged particle q with an energy spectrum of En = (n + ½)ħw. The system is then disturbed by an oscillating electric field as a wo• function of time t such that the disturbance potential energy can be expressed as untuk t o 0 v (t) = {_q {-9€x sin wt e-t With & electric field amplitude. (notes : untuk means for) a. Calculate the transition probability from state n to state m. b. State which transitions n → m are allowed and which are not allowed to occur. c. Explain what happens if w → wo and/or → 0. [Hint: Use the create and annihilate operators to enumerate Matrix elements.]