16.2: Exercises - Perturbations
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An asterisk denotes a harder problem, which you are nevertheless encouraged to try!
The following trigonometric identities may prove useful in the first two questions:
cos2A≡1−sin2A
sin2A≡2sinAcosA
sinAsinB≡12[cos(A−B)−cos(A+B)]
cosAcosB≡12[cos(A−B)+cos(A+B)]
1. A quantum dot is a self assembled nanoparticle in which a single electron state can be confined. A model for such an object is a particle moving in one dimension in the potential
V(x)=∞,|x|>a,V(x)=V0cos(πx/2a),|x|≤a
Identify an appropriate unperturbed system and perturbation term.
Calculate the energies of the two lowest states to first order in perturbation theory.
What is the sign of V0?
State two ways in which the colour of a material containing dots can be shifted towards the red.
2. A particle moves in one dimension in the potential
V(x)=∞,|x|>a,V(x)=V0sin(πx/a),|x|≤a
- show that the first order energy shift is zero;
- *obtain the expression for the second order correction to the energy of the ground state,
ΔE(2)1=−(32V015π)28ma23π2ℏ2−(64V0105π)28ma215π2ℏ2−…
3. The 1-d anharmonic oscillator: a particle of mass m is described by the Hamiltonian
ˆH=ˆp22m+12mω2ˆx2+γˆx4
- Assuming that γ is small, use first-order perturbation theory to calculate the ground state energy;
- *show more generally that the energy eigenvalues are approximately
En≃(n+12)ℏω+3γ(ℏ2mω)2(2n2+2n+1)
Hint: to evaluate matrix elements of powers of ˆx, write ˆx in terms of the harmonic oscillator raising and lowering operators ˆa and ˆa†. Recall that the raising and lowering operators are defined by
ˆa≡√mω2ℏˆx+i√2mωℏˆp and ˆa†≡√mω2ℏˆx−i√2mωℏˆp
with the properties that
ˆa|n⟩=√n|n−1⟩ and ˆa†|n⟩=√n+1|n+1⟩
4. A 1-dimensional harmonic oscillator of mass m carries an electric charge, q. A weak, uniform, static electric field of magnitude E is applied in the x-direction. Write down an expression for the classical electrostatic potential energy for a point particle at x.
The quantum operator is given by the same expression with x→ˆx. By considering the symmetry of the integrals, or otherwise, how that, to first order in perturbation theory, the oscillator energy levels are unchanged, and calculate the second-order shift. Can you show that the second-order result is in fact exact?
Hint: to evaluate matrix elements of ˆx, write ˆx in terms of the harmonic oscillator raising and lowering operators ˆa and ˆa† and use the results ˆa|n⟩=√n|n−1⟩ and ˆa†|n⟩=√n+1|n+1⟩. To obtain an exact solution, change variables to complete the square in the potential and show it remains a harmonic oscillator.
5. Starting from the relativistic expression for the total energy of a single particle, E=(m2c4+p2c2)1/2, and expanding in powers of p2, obtain the leading relativistic correction to the kinetic energy, for a plane wavefunction Φ(x)=Acos(kx), and determine whether Φ(x) is an eigenstate for a relativistic free particle. Hint For normalisation of the wavefunction, it helps to keep the integral form for |A|−2=∫cos2kxdx.