Quantum Structure of Matter
Q1: Write downthe time-dependent and the stationary Schrödinger equation (in position
representation) for a particle of mass moning in a scalar potential V(r).
Q2: Which values do the following commutators
Q3: What is the general expression of the uncertainty relation for two quantum mechanical
observables A and B? In which case are therefore two quantum mechanical observables
simultaneously sharply measurable?
Q4: What is the dynamical equation for the expectation value of an observable o (Ehrenfest
theorem)? What are the Ehrenfest equations for and
for a free particle?
Q5: What are the simultaneous eigenfunctions of the momentum and the energy operator for a free
particle (3 dimensional)? What is the corresponding dispersion relation?
Q6: Express the Hamiltonian of a particle moving in a one dimensional harmonic oscillator potential
by means of the creation and annihilation operator a and d What is the fundamental commutator
relation of & and a What are the energy eigenvalues?
Q7: Which form do the energy eigenfunctions of a particle possess if it is moving in a one
dimensional periodic potential with lattice constant a? Which values can the Bloch wavevector have?
Q8: Assume that the Hamiltonian Ho has a non-degenerate discrete spectrum of energy eigenvalues
What is the first-order correctionto the energy eigenvalues in Rayleigh-Schrödinger
perturbation if we add the perturbation # to Ho so that the full Hamiltonian of the systemis
A2 Particle in a potential well
A particle of mass mmove4s in the one-dimensional potential
00,if x < 0
V(x) 0,if0Where a determines the width of the potential. Sketch the potential and find the transcendental
equation which determines the energy eigenvalues of the bound states for energies 0 < E < Vo
Determine the spectrum of the energy eigenvalues in the limit Vo - 00. Therefore, to the spectrum
of which potential do the energy values converge, if vo - 00?
A3 Tunneling of a particle through a 6 potential
A particle of mass mis incident from the left on a delta potential V(x) - Wõ(x) withthe strength
i) Make an ansatz for the incident, reflected and transmitted wave. By imposing the correct
boundary conditions determine the reflection and transmission coefficients, where andt
are the amplitude reflection and transmission coefficients.
i) What values do you obtain for |r|² and in the limit W - 0 and W - 00?
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