A particle of mass (an electron) sits in a one-dimensional harmonic trap of angular…
PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Wave mechanics in one dimension · Quantum dynamics and its formulations
The Structure of Quantum Mechanics, Ch The oscillator as an algebra
Problem
A particle of mass m=9.11×10−31kg (an electron) sits in a one-dimensional harmonic trap of angular frequency ω=2.00×1014rad s−1, so that with no extra force H^0=ℏω(a^†a^+21) with the usual x^=ℏ/2mω(a^+a^†). A uniform force F=5.00×10−12N (for instance from a uniform electric field acting on the electron's charge) is then switched on, so the full Hamiltonian is H^=ℏω(a^†a^+21)−Fx^.
((a)) Show that defining b^=a^−λ for a suitably chosen real number λ turns H^ into ℏω(b^†b^+21)+constant, with [b^,b^†]=1 unchanged. Give λ and the resulting energy levels En in terms of F, m, ω, ℏ and the integer n=0,1,2,…
((b)) Evaluate E0 and E1 numerically, in electronvolts.
((c)) Write x^ as ℏ/2mω(b^+b^†) plus a constant, and find the constant. The ladder part b^+b^† obeys the same selection rule as a^+a^†: its matrix elements between the number states ∣n⟩b of b^†b^ are nonzero only when the b^-numbers differ by exactly one. Use this to show that ⟨n∣bx^∣n⟩b is the same for every n, evaluate it numerically in nanometres, and compare it with the classical equilibrium position where d/dx[21mω2x2−Fx]=0.