Pavel Stránský
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Last updated: 6.9.2018

Onset of chaos

Energy levels and wave functions

Integrable case B=0

Small perturbation B=0.005

Peres lattice - integrable Peres lattice - nonintegrable perturbation
Figure 1: 3000 points of the Peres lattice for the operator L^2 (hbar=0.005)
The intensity of the blue color indicates the level density at given energy.
Note that the nonintegrable perturbation affects the most the states
close to each other both in energy and in the value of the second integral of motion L^2.
Energy levels - integrable Energy levels - nonintegrable perturbation
Figure 2: 150 lowest energy levels (hbar=0.02)
Section γ=0 the potential energy surface is drawed by the yellow curve.
Wave function - integrable, E(0)=-0.23031 Wave function - nonintegrable perturbation, E(0)=-0.23068
Wave function - integrable, E(1)=-0.22649 Wave function - nonintegrable perturbation, E(1)=-0.22619
Wave function - integrable, E(2)=-0.21522 Wave function - nonintegrable perturbation, E(2)=-0.2152
Wave function - integrable, E(3)=-0.19703 Wave function - nonintegrable perturbation, E(3)=-0.19702
Wave function - integrable, E(4)=-0.19159 Wave function - nonintegrable perturbation, E(4)=-0.1919
Wave function - integrable, E(5)=-0.18721 Wave function - nonintegrable perturbation, E(5)=-0.18696
Wave function - integrable, E(6)=-0.17443 Wave function - nonintegrable perturbation, E(6)=-0.17442
Wave function - integrable, E(7)=-0.17255 Wave function - nonintegrable perturbation, E(7)=-0.17255
Wave function - integrable, E(8)=-0.15427 Wave function - nonintegrable perturbation, E(8)=-0.15452
Wave function - integrable, E(9)=-0.15415 Wave function - nonintegrable perturbation, E(9)=-0.15415
Wave function - integrable, E(10)=-0.14911 Wave function - nonintegrable perturbation, E(10)=-0.14891
Wave function - integrable, E(11)=-0.1424 Wave function - nonintegrable perturbation, E(11)=-0.14241
Wave function - integrable, E(12)=-0.13439 Wave function - nonintegrable perturbation, E(12)=-0.13438
Wave function - integrable, E(13)=-0.12732 Wave function - nonintegrable perturbation, E(13)=-0.12732
Wave function - integrable, E(14)=-0.11853 Wave function - nonintegrable perturbation, E(14)=-0.11871
Wave function - integrable, E(15)=-0.11224 Wave function - nonintegrable perturbation, E(15)=-0.11209
Wave function - integrable, E(16)=-0.11162 Wave function - nonintegrable perturbation, E(16)=-0.11162
Wave function - integrable, E(17)=-0.10716 Wave function - nonintegrable perturbation, E(17)=-0.10718
Wave function - integrable, E(18)=-0.09499 Wave function - nonintegrable perturbation, E(18)=-0.09498
Wave function - integrable, E(19)=-0.09474 Wave function - nonintegrable perturbation, E(19)=-0.09475
Wave function - integrable, E(20)=-0.08462 Wave function - nonintegrable perturbation, E(20)=-0.08475
Wave function - integrable, E(21)=-0.08213 Wave function - nonintegrable perturbation, E(21)=-0.08214
Wave function - integrable, E(22)=-0.07659 Wave function - nonintegrable perturbation, E(22)=-0.07649
Wave function - integrable, E(23)=-0.06927 Wave function - nonintegrable perturbation, E(23)=-0.06927
Wave function - integrable, E(24)=-0.0673 Wave function - nonintegrable perturbation, E(24)=-0.06732
Wave function - integrable, E(25)=-0.05709 Wave function - nonintegrable perturbation, E(25)=-0.05711
Wave function - integrable, E(26)=-0.05602 Wave function - nonintegrable perturbation, E(26)=-0.05601
Wave function - integrable, E(27)=-0.053 Wave function - nonintegrable perturbation, E(27)=-0.05307
Wave function - integrable, E(28)=-0.04692 Wave function - nonintegrable perturbation, E(28)=-0.04693
Wave function - integrable, E(29)=-0.04198 Wave function - nonintegrable perturbation, E(29)=-0.04193
Wave function - integrable, E(30)=-0.03683 Wave function - nonintegrable perturbation, E(30)=-0.03684
Wave function - integrable, E(31)=-0.02689 Wave function - nonintegrable perturbation, E(31)=-0.02689
Wave function - integrable, E(32)=-0.02457 Wave function - nonintegrable perturbation, E(32)=-0.02458
Wave function - integrable, E(33)=-0.02323 Wave function - nonintegrable perturbation, E(33)=-0.02325
Wave function - integrable, E(34)=-0.01719 Wave function - nonintegrable perturbation, E(34)=-0.01719
Wave function - integrable, E(35)=-0.01488 Wave function - nonintegrable perturbation, E(35)=-0.0149
Wave function - integrable, E(36)=-0.00798 Wave function - nonintegrable perturbation, E(36)=-0.00799
Wave function - integrable, E(37)=-0.00672 Wave function - nonintegrable perturbation, E(37)=-0.00674
Wave function - integrable, E(38)=-0.00012 Wave function - nonintegrable perturbation, E(38)=-5.0E-5
Wave function - integrable, E(39)=0.0012 Wave function - nonintegrable perturbation, E(39)=0.00118
Wave function - integrable, E(40)=0.00875 Wave function - nonintegrable perturbation, E(40)=0.00873
Wave function - integrable, E(41)=0.01576 Wave function - nonintegrable perturbation, E(41)=0.01574
Wave function - integrable, E(42)=0.02187 Wave function - nonintegrable perturbation, E(42)=0.02182
Wave function - integrable, E(43)=0.02459 Wave function - nonintegrable perturbation, E(43)=0.02418
Wave function - integrable, E(44)=0.02472 Wave function - nonintegrable perturbation, E(44)=0.02469
Wave function - integrable, E(45)=0.02619 Wave function - nonintegrable perturbation, E(45)=0.02673
Wave function - integrable, E(46)=0.03145 Wave function - nonintegrable perturbation, E(46)=0.03142
Wave function - integrable, E(47)=0.03793 Wave function - nonintegrable perturbation, E(47)=0.0379
Wave function - integrable, E(48)=0.04406 Wave function - nonintegrable perturbation, E(48)=0.04403
Wave function - integrable, E(49)=0.04974 Wave function - nonintegrable perturbation, E(49)=0.0497
Wave function - integrable, E(50)=0.05384 Wave function - nonintegrable perturbation, E(50)=0.05371
Wave function - integrable, E(51)=0.05478 Wave function - nonintegrable perturbation, E(51)=0.05474
Wave function - integrable, E(52)=0.05891 Wave function - nonintegrable perturbation, E(52)=0.0588
Wave function - integrable, E(53)=0.06139 Wave function - nonintegrable perturbation, E(53)=0.06087
Wave function - integrable, E(54)=0.06155 Wave function - nonintegrable perturbation, E(54)=0.06247
Wave function - integrable, E(55)=0.07625 Wave function - nonintegrable perturbation, E(55)=0.0762
Wave function - integrable, E(56)=0.08157 Wave function - nonintegrable perturbation, E(56)=0.08153
Wave function - integrable, E(57)=0.08658 Wave function - nonintegrable perturbation, E(57)=0.08654
Wave function - integrable, E(58)=0.08693 Wave function - nonintegrable perturbation, E(58)=0.08684
Wave function - integrable, E(59)=0.09121 Wave function - nonintegrable perturbation, E(59)=0.09116
Wave function - integrable, E(60)=0.09534 Wave function - nonintegrable perturbation, E(60)=0.09527
Wave function - integrable, E(61)=0.09818 Wave function - nonintegrable perturbation, E(61)=0.09806
Wave function - integrable, E(62)=0.09882 Wave function - nonintegrable perturbation, E(62)=0.09872
Wave function - integrable, E(63)=0.10144 Wave function - nonintegrable perturbation, E(63)=0.10105
Wave function - integrable, E(64)=0.1022 Wave function - nonintegrable perturbation, E(64)=0.1021
Wave function - integrable, E(65)=0.10279 Wave function - nonintegrable perturbation, E(65)=0.10379
Wave function - integrable, E(66)=0.12298 Wave function - nonintegrable perturbation, E(66)=0.12291
Wave function - integrable, E(67)=0.13111 Wave function - nonintegrable perturbation, E(67)=0.13105
Wave function - integrable, E(68)=0.13517 Wave function - nonintegrable perturbation, E(68)=0.13511
Wave function - integrable, E(69)=0.13684 Wave function - nonintegrable perturbation, E(69)=0.13678
Wave function - integrable, E(70)=0.1389 Wave function - nonintegrable perturbation, E(70)=0.13883
Wave function - integrable, E(71)=0.14222 Wave function - nonintegrable perturbation, E(71)=0.14213
Wave function - integrable, E(72)=0.14407 Wave function - nonintegrable perturbation, E(72)=0.14391
Wave function - integrable, E(73)=0.14503 Wave function - nonintegrable perturbation, E(73)=0.14489
Wave function - integrable, E(74)=0.14721 Wave function - nonintegrable perturbation, E(74)=0.14677
Wave function - integrable, E(75)=0.1476 Wave function - nonintegrable perturbation, E(75)=0.14735
Wave function - integrable, E(76)=0.14857 Wave function - nonintegrable perturbation, E(76)=0.14865
Wave function - integrable, E(77)=0.14885 Wave function - nonintegrable perturbation, E(77)=0.15011
Wave function - integrable, E(78)=0.16153 Wave function - nonintegrable perturbation, E(78)=0.16147
Wave function - integrable, E(79)=0.17741 Wave function - nonintegrable perturbation, E(79)=0.17737
Wave function - integrable, E(80)=0.18732 Wave function - nonintegrable perturbation, E(80)=0.18723
Wave function - integrable, E(81)=0.18908 Wave function - nonintegrable perturbation, E(81)=0.18899
Wave function - integrable, E(82)=0.19202 Wave function - nonintegrable perturbation, E(82)=0.19192
Wave function - integrable, E(83)=0.19347 Wave function - nonintegrable perturbation, E(83)=0.19332
Wave function - integrable, E(84)=0.1946 Wave function - nonintegrable perturbation, E(84)=0.19448
Wave function - integrable, E(85)=0.19677 Wave function - nonintegrable perturbation, E(85)=0.19656
Wave function - integrable, E(86)=0.19713 Wave function - nonintegrable perturbation, E(86)=0.19684
Wave function - integrable, E(87)=0.19842 Wave function - nonintegrable perturbation, E(87)=0.19797
Wave function - integrable, E(88)=0.19908 Wave function - nonintegrable perturbation, E(88)=0.19879
Wave function - integrable, E(89)=0.19947 Wave function - nonintegrable perturbation, E(89)=0.19989
Wave function - integrable, E(90)=0.19976 Wave function - nonintegrable perturbation, E(90)=0.20117
Wave function - integrable, E(91)=0.20232 Wave function - nonintegrable perturbation, E(91)=0.20227
Wave function - integrable, E(92)=0.21986 Wave function - nonintegrable perturbation, E(92)=0.21983
Wave function - integrable, E(93)=0.23203 Wave function - nonintegrable perturbation, E(93)=0.23196
Wave function - integrable, E(94)=0.2405 Wave function - nonintegrable perturbation, E(94)=0.24042
Wave function - integrable, E(95)=0.24515 Wave function - nonintegrable perturbation, E(95)=0.2451
Wave function - integrable, E(96)=0.2464 Wave function - nonintegrable perturbation, E(96)=0.24628
Wave function - integrable, E(97)=0.24999 Wave function - nonintegrable perturbation, E(97)=0.24985
Wave function - integrable, E(98)=0.25043 Wave function - nonintegrable perturbation, E(98)=0.25025
Wave function - integrable, E(99)=0.2519 Wave function - nonintegrable perturbation, E(99)=0.25171
Wave function - integrable, E(100)=0.25305 Wave function - nonintegrable perturbation, E(100)=0.25271
Figure 3: Wave functions for the lowest 100 energy levels from Figure 2