Translation operator problems.

The translation operator S(a), for arbitrary constant real vector a, is defined such that... a) Show, using the Taylor expansion of Φ(x-a) about x, that S(a) may be expressed in terms of the momentum operator p=.. Sum the expansion to give a simple closed-form expression for S(a). Please see attached for full question.

Ehrenfest's theorem

Consider a 1-D free particle, describable as a wave packet at initial time t0. a) Show, applying Ehrenfest's theorem, that is a linear function of time and

is a constant. b) Write the equations of motion for the mean values and . Integrate these equations. c) Show that, with a suitable choice of the time origin, the rms deviation <(X-)^2> is given by... Please see attached for full question.

Accuracy of first order theory

Problem 20 The finite size of the proton perturbs the energy of the n=1 orbit in a hydrogen atom. (i) Given the proton to be a uniformly charged sphere... (see attachment)

Heisenberg Uncertainty Principle

Calculate the uncertainty in position of an electron that has been accelerated across a potential difference of 100 kV +/- 100 V. Hints: Derive an expression for the momentum as a function of the voltage (potential difference) and differentiate. Also dV may be represented by the uncertainty in the voltage (i.e. +/- 100 volts = 200 V) Relationship between kinetic energy and voltage : E_k = V*q

General Quantum Mechanics

a) Briefly describe the field of physics known as quantum mechanics. How are quantum mechanical principles related to radiation science? b) Briefly describe the significance of Schroedinger's Cat to quantum mechanics.