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