Quantum Physics

An electron in a "double-well" structure can be in one of two positions: "left" corre- sponding to the quantum state |L) and "right" corresponding to R). The system is subject to external fields and the energy eigenstates are Eg) = A (i|L) - 2|R)) with energy Eg = 0 and |Ee) = A (2|L) - i|R)) with energy Ee =A > 0. A is a real, positive constant.

Consider a particle of mass m and charge e confined to the xy-plane interacting with a magnectic field B = BK with vector potential A= ax i. construct a

Consider a one-dimensional harmonic oscillator, find the state [z) that satisfies a|z) = 2|2) where a is the annihilation operator. Hint: Try |2) = a,n) where H❘n) = hw(n + 1)|n).

Coherent States A harmonic oscillator (mass m, frequency w) is prepared in the state (wavepacket)

Coherent states A harmonic oscillator (mass m, frequency w) is prepared in the state (wavepacket) [4) = e l1/2 Σ ýntln) n=0 What is the probability mass distribution for the outcome of energy measurements made of this system?

Three-D oscillator Consider a particle in a 3 -D oscillator potential Determine its ground state wavefunction and energy in iterms of the three oscillatorquantum

Question 1

Electric fields The one dimensional Harmonic oscillator is subjected to a constant electric field Ex=E0. find the exact wavefunctions

A particle of mass m is moving in a one-dimensional potential. (x) are the orthonormal energy eigenstates (their explicit form is not needed) and the energy eigenvalues are En nE with n= 1, 2, 3, ... and E some real constant. In this problem no integrations are needed; use orthonormality of the Un and/or any results derived in class. At t=0 the system is prepared in the state: (a) Find a by requiring that (x, 0) is correctly normalised. You may assume that a is a real and positive number. [2] (b) At t = 0 a measurement of energy is performed and the most likely energy eigenvalue is found. Which eigenvalue is this? Find the corresponding probability and the wave function after the measurement at t > 0. [3] [3] (c) Calculate the expectation values (E) = (Î) and (E²) = (Ĥ²), and the energy uncertainty AE.

Consider a quantum system with Hamiltonian and let us denote by |p] the eigenstates of the