Post on 22-Jul-2020
2013 Student Practice in JINR Fields of Research (2nd Stage)
Student: Andrej Babič1
Supervisor: Prof. Vladimir B. Belyaev2
1 Comenius University in Bratislava, Slovakia 2 Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear
Research, Dubna, Russia
Dubna, 26.07.2013
Finite-rank approximation (FRA):
Non-standard method for solving QM problems.
Applicable beyond limitations of perturbation theory.
Successfully applied to:
3-body bound-state problems (H−, He, H2+, 𝑝𝑝𝜇, 𝑑𝑑𝜇 etc.).
Scattering with short-range interactions (𝜋 He3 → 𝜋 He3 ).
Bound-state problem:
3D isotropic quantum anharmonic oscillator.
Interactions of power type: 𝑉 𝑟 ∝ 𝑟3, 𝑟4.
Calculation of energy spectrum.
Comparison with perturbation theory.
r2 + r4 r2 + r3
r2
Calculation of energy spectrum from 𝐻 𝜓 = 𝐸 𝜓 , where:
𝐻 = 𝐻0 + 𝑉 = −ℏ2
2𝜇Δ +
1
2𝜇𝜔2𝑟2 + Λ𝑟3 4 .
For this choice of 𝐻0 and 𝑉, solutions of 𝐻0 𝜙𝑛 = 휀𝑛 𝜙𝑛 are:
𝒓 𝜙𝑘𝑙𝑚 = 𝑁𝑘𝑙𝑟𝑙𝐿𝑘
𝑙+1
2 𝜌2 𝑒−𝜌2
2 𝑌𝑙𝑚 𝜗,𝜑 , 𝜌 =𝜇𝜔
ℏ𝑟,
휀𝑘𝑙 = ℏ𝜔 2𝑘 + 𝑙 + 3/2 .
Introducing QHO Green function:
𝐺0 𝐸 = 𝐻0 − 𝐸 −1,
𝐺0 𝜙𝑛 = 휀𝑛 − 𝐸 −1 𝜙𝑛 ,
𝜓 = −𝐺0𝑉 𝜓 .
2k + l (k, l)n
0 (0, 0)1
1 (0, 1)2
2 (0, 2)3; (1, 0)4
3 (0, 3)5; (1, 1)6
4 (0, 4)7; (1, 2)8; (2, 0)9
5 (0, 5)10; (1, 3)11; (2, 1)12
6 (0, 6)13; (1, 4)14; (2, 2)15; (3, 0)16
Approximation by finite-rank operator:
𝑉 ∼ 𝑉𝑁 = 𝑉 𝜙𝑚 𝑑𝑚𝑛−1 𝜙𝑛 𝑉𝑁
𝑚,𝑛 , 𝑑𝑚𝑛 = 𝜙𝑚 𝑉 𝜙𝑛 .
𝑉 and 𝑉𝑁 have common action on 𝜙𝑛 :
𝑉 𝜙𝑛 = 𝑉𝑁 𝜙𝑛 .
Reduction to system of homogeneous linear algebraic equations:
𝜓 ~ − 𝐺0𝑉𝑁 𝜓 = − 𝐺0𝑉 𝜙𝑚 𝑑𝑚𝑛
−1 𝜙𝑛 𝑉 𝜓 ,𝑁𝑚,𝑛
𝐵𝑝 = − 𝜙𝑝 𝑉𝐺0𝑉 𝜙𝑚 𝑑𝑚𝑛−1 𝐵𝑛
𝑁𝑚,𝑛 , 𝐵𝑛 = 𝜙𝑛 𝑉 𝜓 ,
𝐵𝑝 = − 𝜙𝑝 𝑉𝐺0 𝜙𝑛 𝐵𝑛𝑁𝑛 ,
𝐴𝑝𝑛𝐵𝑛𝑁𝑛 = 0,
𝐴𝑝𝑛 𝐸 = 𝛿𝑝𝑛 + 𝜙𝑝 𝑉𝐺0 𝐸 𝜙𝑛 ,
det 𝐴 = 0.
Perturbation theory (2nd order):
𝐸𝑛 ≈ 휀𝑛 + 𝜙𝑛 𝑉 𝜙𝑛 + 𝜙𝑚 𝑉 𝜙𝑛
2
𝜀𝑛−𝜀𝑚
𝑁𝑚,𝜀𝑚≠𝜀𝑛
.
𝑉 𝑟 ∝ 𝑟3 𝑉 𝑟 ∝ 𝑟3
𝑉 𝑟 ∝ 𝑟4 𝑉 𝑟 ∝ 𝑟4
V (r ) ∝ r 3 Λ = 0 Λ = 0.1 Λ = 0.1 Λ = 1 Λ = 10
E1 3 3.20658 3.20900 4.44642 11.3979
E2 5 5.45135 5.45135 7.77366 22.8174
E3 7 7.72216 7.72216 11.84610 49.1358
E4 7 7.80896 7.80654 12.44290 59.1604
V (r ) ∝ r 4 Λ = 0 Λ = 0.1 Λ = 0.1 Λ = 1 Λ = 10
E1 3 3.28125 3.30764 4.67628 12.8285
E2 5 5.87500 5.87500 8.60390 29.9121
E3 7 8.57500 8.57500 15.36440 84.4309
E4 7 8.96875 8.94236 17.47800 110.512
FRA – non-standard method with large potential.
Outstanding agreement with perturbation theory (small Λ).
Works for any system of 𝜙𝑛 for which 𝑑𝑚𝑛−1 exists.
Key property – negligible dependence on N:
Numerous properties yet to be revealed…
Thank you for your attention!
𝑉 𝑟 ∝ 𝑟3
Λ = 0.1