Section II, Quantum Mechanics II
Perturbation calculus.
0) Consider the operator ˆA with eigenvalues λ1, λ2, ... and the corresponding eigenvectors ψ1, ψ2.... Furthermore, consider ˆA0 = ˆA + V , where is a small parameter-number and V operator. Find out the general formula for the first correction term for eigenvalues of ˆA0.
a) Assume that eigenvalues of ˆA.
b) Assume that ˆA admits degeneration of the spectrum.
1) A warm up. Consider the following ”small” perturbation of the Pauli matrix σx
σ := 0 1 1 0
+ ε
1 −1
−1 3
, (0.1)
where ε > 0 is a small parameter.
a) Relaying on the perturbation theory calculate eigenvalues as well as eigen- functions of σ up to the first order perturbation terms.
b) Using the results of the previous step calculate the second and the third order perturbation terms for the eigenvalues.
c) Solve the characteristic equation for the matrix σ and show that the result is consistent with perturabation theory, cf. 1a and 1b.
Please be prepared for a discussion on quantum particle with spin, quantum particle in a magnetic field. We will discuss it in details.
2 Suppose we put delta-function bump in the infinite square well defined in interval [0, a]; i.e. the pertubant takes the form
V = αδ(x − a 2) , where R
Rf (x)δ(x − b) = f (b) and α is a real constant. Find the first order correction term to allowed energies.
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3) Zeeman Effect. Consider an electron living in the Coulomb potential VCoulomb= −4πεZe2
0r and a uniform magnetic field with the value B.
a) Show that the stationary Pauli equation for this model can be written as follows
(HA+ HS+ HB1+ HB2)ψ = Eψ , gdzie:
HA= −~2
2m∆ − Ze2 4πε0r, HS = µBσzB , HB1= −~eB
2im
∂
∂φ, and
HB2= e2B2
8m r2sin2θ ;
(r , θ , φ) spherical coordinates, σz – Pauli matrix, m , e denote the mass and the charge of the electron respectively.
b) Assuming that B is small (in the sense defined in the next point) calculate the first order perturbation term to the energies due to the magnetic field.
c) Show that the perturbation method can be applied if B 2m
n~|e||En− En+1| ,
where En denotes the energy level in absence of magnetic field.
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Section II, Quantum Mechanics II (Medical Physics)
Prosz¸e o przygotowanie si¸e do dyskusji: cz¸astka kwantowa ze spinem, cz¸astka kwantowa w polu magnetycznym (r´ownanie Pauliego). B¸edziemy to szczeg´o lowo omawia´c na zaj¸eciach.
0) Rozwa˙zmy operator ˆA. Liczby λ1, λ2, ... stanowi¸a jego warto´sci w lasne, natomiast ψ1, ψ2... okre´slaj¸a wektory w lasne. Nast¸epnie rozwa˙zmy ˆA0 = ˆA +
V , gdzie jest ma lym parametrem a V operatorem. Znale´z´c formu l¸e na pierwsz¸a poprawk¸e na warto´sci w lasne operatora ˆA0.
ps. Za lo˙zy´c, ˙ze widmo operatora ˆA jest niezdegenerowane.
1) Rozwa˙zmy nast¸epuj¸ace, ’ma le’ zaburzenie macierzy Pauliego σx: σ := 0 1
1 0
+ ε
1 −1
−1 3
; (0.2)
gdzie ε > 0 jest ma lym parametrem.
a) Pos luguj¸ac si¸e rachunkiem zaburze´n, obliczy´c warto´sci i wektory w lasne σ z dok ladno´sci¸a do pierwszego rz¸edu rachunku zaburze´n.
b) Korzystaj¸ac z wynik´ow poprzedniego punktu obliczy´c poprawki drugiego i trzeciego rz¸edu do warto´sci w lasnych.
c) Rozwi¸aza´c r´ownanie charakterystyczne dla macierzy σ i pokaza´c, ˙ze wynik jest sp´ojny z rachunkiem zaburze´n, tj. por´ownaj z pkt. a) i b).
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