Download Commutators Consider two operators A and B, such that their

HAND-INS
Week 2: Commutators
Consider two operators A and B, such that their commutator [A, B] = 1. Express the commutators [A, B 2 ], [A, B 3 ] and [A, B 4 ] only in terms of B. (use the
Jacobi identity [A, BC] = [A, B]C +B[A, C] ). On the basis of the previous results,
what would be the value of [A, B n ], n > 0 ? Prove your intuition (for example using
the induction principle). Use the general formula for [A, B n ] to compute [A, eB ].
Week 3: Uncertanties
Problem 14 in chapter 5
Week 4: Band structure
In the book is considered a system of N potential wells placed side by side, as
a model for band structure in a solid. If there is no interaction between the wells
the lowest state is the same in all wells, say E1 . That is, the ground state consists
of N degenerate states. If we (as is done in the book) assume that an electron
placed in one well is affected only by the two neighboring wells with a coupling
γ, the hamiltonian can be written as a quite simple NxN matrix: All diagonal
matrix elements are the same and equals E1 , and all non-diagonal matrix elements
are zero, except two bands parallel to the diagonal, where the matrix elements are
the same and equals γ (see page 101 in the book).
a) Calculate the N energy eigen values by diagonalizing the matrix for system
sizes N=2, 10, 20 and 100. Plot the N energies as is done in figure 4.34 in the
book. Choose γ = 0.1 and E1 = 1.
b) Plot in the same figure the N states around the first excited states in the well,
E2 = 4E1 . Only for N = 100. The interaction between the different wells with an
electron in the first excited state is also assumed to be γ. Assume no interaction
between the first excited state and ground state. What kind of system do you
obtain if you fill the system with N/2, N or 1.5N electrons?
Week 5: Spherical harmonics
Problem 9 in chapter 7
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HAND-INS
Week 6: Helium atom
The properties of the Helium atom is discussed on pages 171-174 in the book.
It is shown how to treat the interaction between the two electrons by means of
perturbation theory as well as with the variational approach. Another approach
is the so called ”mean field approach”. The electric potential caused by the average charge density of the first electron, screening the nucleus charge, is taken into
account when calculating the energy of the second electron and vice versa. Here
you are going to study this screening effect by treating the average, or mean field,
potential to first order in perturbation theory.
(a) Calculate the average charge density of an electron in the lowest energy level
in the potential of a Helium nucleus,
He
ρ(r) = ehφHe
100 |φ100 i
where |φHe
100 i =
p
23 /(πa30 ) exp(−2r/a0 ).
(b) Solve Poissons equation
1 d2
(rU(r)) = ρ(r)/ε0
r dr 2
to determine the potential U(r) from the electron. Make sure the solution for U(r)
is bounded at r → 0 and goes to zero at r → ∞.
∇2 U(r) = ρ(r)/ε0
⇒
(c) Calculate the energy shift ∆E of the ground state to first order in perturbation
theory
He
∆E = hφHe
100 |V (r)φ100 i
where V (r) = eU(r).
(d) Compare the ground state energy
E = 2 × 22 × (−13.6eV ) + 2∆E = −108.8eV + 2∆E
with the different approximate results in the book. Is the mean field approach
better or worse? Note: the factor of 2 in front of ∆E comes from that electron 1
screens electron 2 and vice versa.