diff --git a/lecture_notes/3-25/central potential states b/lecture_notes/3-25/central potential states new file mode 100644 index 0000000..b1d0eee --- /dev/null +++ b/lecture_notes/3-25/central potential states @@ -0,0 +1,131 @@ +Bound states of a central potential +━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ + + For any central potential V(r) = V(│r│) the eigenfunctions of H can be + separated as + + ❬r❙E,l,mₗ❭ = Rₑ﹐ₗ(r) Yₗ﹐ₘ(θ,φ) + + (pic) The radial S.E. is + + ⎡−͟ħ͟²⎛ ∂͟²͟ + 2͟ ∂͟ ⎞ + l͟(l͟+͟1͟)ħ͟² + V(│r│) ⎤ Rₑ﹐ₗ(r) = E Rₑ﹐ₗ(r) + ⎣2m ⎝ ∂r² r ∂r ⎠ 2 m r² ⎦ + + (pic) Developed radial schrodinger equation using U(r) replacement + + Rₑ﹐ₗ(r) = Uₑ﹐ₗ(r) /r + + ⎡−͟ħ͟²⎛ ∂͟²͟ + 2͟ ∂͟ ⎞⎤ U͟ = E ∂͟²͟ U + ⎣2m ⎝ ∂r² r ∂r ⎠⎦ r ∂r² + + ∂͟ U͟ = 1͟ ∂͟ U - U͟ + ∂r r r ∂r r² + + ∂͟²͟ U͟ = 1͟ ∂͟²͟ U + 2͟U͟ - 2͟ ∂͟ U + ∂r² r r ∂r² r³ r²∂r + + ⎡−͟ħ͟²⎛ ∂͟²͟ U͟ + 2͟ ∂͟ U͟ ⎞⎤ = E ∂͟²͟ U + ⎣2m ⎝ ∂r² r r ∂r r ⎠⎦ ∂r² +──────────────────────────────────────────────────────────────────────────────── + ⎡−͟ħ͟²⎛ 1͟ ∂͟²͟ U + 2͟U͟ - 2͟ ∂͟ U + 2͟ 1͟ ∂͟ U - 2͟ U͟ ⎞⎤ + ⎣2m ⎝ r ∂r² r³ r²∂r r r ∂r r r² ⎠⎦ + + = E 1͟ ∂͟²͟ U + E 2͟U͟ - E 2͟ ∂͟U + r ∂r² r³ r²∂r +──────────────────────────────────────────────────────────────────────────────── + ⎡−͟ħ͟²⎛ 1͟ ∂͟²͟ + 2͟ - 2͟ ∂͟ + 2͟ ∂͟ - 2͟ ⎞U⎤ + ⎣2m ⎝ r ∂r² r³ r²∂r r² ∂r r³⎠ ⎦ + + = E ⎛1͟ ∂͟²͟ + 2͟ - 2͟ ∂ ⎞U + ⎝r ∂r² r³ r²∂r⎠ +──────────────────────────────────────────────────────────────────────────────── + ⎡⎛−͟ħ͟² 1͟ ∂͟²͟ + −͟ħ͟² 2͟ - −͟ħ͟² 2͟ ∂͟ + −͟ħ͟² 2͟ ∂͟ - −͟ħ͟² 2͟ ⎞U⎤ + ⎣⎝2m r ∂r² 2m r³ 2m r²∂r 2m r² ∂r 2m r³⎠ ⎦ + + = E ⎛1͟ ∂͟²͟ + 2͟ - 2͟ ∂ ⎞U + ⎝r ∂r² r³ r²∂r⎠ +──────────────────────────────────────────────────────────────────────────────── + ⎡⎛−͟ħ͟² 1͟ ∂͟²͟ - −͟ħ͟² 2͟ ∂͟ + −͟ħ͟² 2͟ ∂͟ + −͟ħ͟² 2͟ - −͟ħ͟² 2͟ ⎞U⎤ + ⎣⎝2m r ∂r² 2m r²∂r 2m r² ∂r 2m r³ 2m r³⎠ ⎦ + + = E ⎛1͟ ∂͟²͟ + 2͟ - 2͟ ∂ ⎞U + ⎝r ∂r² r³ r²∂r⎠ +──────────────────────────────────────────────────────────────────────────────── + ↓ + Not sure this makes sense, but the final result is + ↓ + + ⎡−͟ħ͟²∂͟²͟ - ħ͟² l(l+1) + V(│r│)⎤ Uₑ﹐ₗ(r) = E Uₑ﹐ₗ(r) + ⎣2m ∂r² 2mr² ⎦ + + + Normalization Condition + + ∞ ∞ + ∫ │Rₑ﹐ₗ│² r² dr = ∫ │Uₑ﹐ₗ│² dr + 0 0 + + + If V(r) is not more singular at the origin than 1/r^2 then the SE has power + series solutions. + + Thus for small r we take U(r) → rˢ + + (pic) substitute U(r) = rˢ into S.E. + + Uₑ﹐ₗ(r) ≈ rˢ + + ⎡−͟ħ͟² ∂͟²͟ + ħ͟² l(l+1) + V(│r│)⎤ rˢ = E rˢ + ⎣2m ∂r² 2mr² ⎦ + + −͟ħ͟² s(s-1)∂͟ rˢ⁻² + l͟ (l+1)ħ²rˢ⁻² + V(│r│)rˢ = E rˢ + 2m ∂r 2m + + −͟ħ͟² ⎛s(s-1)∂͟ + l͟ (l+1)ħ²⎞rˢ⁻² + V(│r│)rˢ = E rˢ + 2m ⎝ ∂r 2m ⎠ + + + −͟ħ͟² ⎛s(s-1)∂͟ + l͟ (l+1)ħ²⎞rˢ⁻² + V(│r│)rˢ = E rˢ + 2m ⎝ ∂r 2m ⎠ + + −͟ħ͟² ⎛s(s-1)∂͟ + l͟ (l+1)ħ²⎞ + V(│r│)r² = E r² + 2m ⎝ ∂r 2m ⎠ + + + For r → 0, + r² → 0, + V(r) r² → 0. + ⇓ + s(s-1) + l(l+1) = 0 + ⇓ + s = l+1 or s = -l + + If s = -l, the normalization conditions + + ∞ │∞ + ∫ r⁻²ˡ dr = 1/(2l-1) 1/(r²ˡ⁻¹) │ → diverges + 0 │0 + + + So, for small r, + + Uₑ﹐ₗ(r) → (r→0) → rˡ⁺¹; + + Rₑ﹐ₗ(r) → (r→0) → rˡ. + +Eigenfunctions +━━━━━━━━━━━━━━ + + Ψₑ﹐ₗ﹐ₘ(r,θ,φ) = Rₑ﹐ₗ(r) Yₗ﹐ₘ(θ,φ) = Uₑ﹐ₗ /r Yₗ﹐ₘ(θ,φ) + + ⇒ d͟²͟ U - 1͟ l(l+1) + ⎛λ͟ - 1͟⎞U = 0 + dρ² ρ² ⎝ρ 4⎠ + + ⎡−͟ħ͟² d² + l͟ (l+1)ħ² - Z͟e͟²⎤Uₑ﹐ₗ(r) = E Uₑ﹐ₗ(r) + ⎣2m dr² 2mr² r ⎦ + + ρ = √⎛8͟m͟ │E│⎞r + ⎝ ħ² ⎠ + + λ = Z͟e͟² √⎛_͟m͟_͟ ⎞ + ħ ⎝2│E│⎠ diff --git a/lecture_notes/3-25/hydrogen atom b/lecture_notes/3-25/hydrogen atom new file mode 120000 index 0000000..7a70ff1 --- /dev/null +++ b/lecture_notes/3-25/hydrogen atom @@ -0,0 +1 @@ +../3-28/hydrogen atom \ No newline at end of file diff --git a/lecture_notes/3-25/overview b/lecture_notes/3-25/overview index 4ac7100..23f6ff7 100644 --- a/lecture_notes/3-25/overview +++ b/lecture_notes/3-25/overview @@ -1,65 +1,10 @@ -Bound states of a central potential -━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ +Computed the bound states of a central potential with eigenfunctions of H + ❬r❙E,l,mₗ❭ = Rₑ﹐ₗ(r) Yₗ﹐ₘ(θ,ϕ). -For any central potential V(r) = V(│r│) the eigenfunctions of H can be separated as - - ❬r❙E,l,mₗ❭ = Rₑ﹐ₗ(r) Yₗ﹐ₘ(θ,ϕ) - -The radial S.E. is - - ⎡−͟ħ͟² ⎛ ∂͟²͟ + 2͟∂͟ ⎞ + l͟(l͟+͟1͟)ħ͟² + V(│r│) ⎤ Rₑ﹐ₗ(r) = E Rₑ﹐ₗ(r) - ⎣2m ⎝ ∂r² r∂r ⎠ 2 m r² ⎦ - - - Rₑ﹐ₗ(r) = U͟ₑ͟﹐͟ₗ͟(r) - r - - (pic) ... - -(pic) Developed radial schrodinger equation using U(r) replacement - - - Developed normalization condition - -If V(r) is not more singular at the origin than 1/r^2 then the SE has power series solutions. - - Thus for small r we take U(r) → rˢ - -(pic) substitute U(r) = rˢ into S.E. - - -ħ²/2m [(s(s-1) + l(l+1)] + V r² = E r² - - For r → 0 - r² → 0 - V(r) r² → 0 - - ⇒ s(s-1) + l(l+1) = 0 - ⇒ s = l+1 or s = -l - - If s = -l, the normalization conditions - - ∞ │∞ - ∫ r⁻²ˡ dr = 1/(2l-1) 1/(r²ˡ⁻¹) │ → diverges - 0 │0 - - - Uₑ﹐ₗ(r) → (r→0) → rˡ⁺¹ - - Rₑ﹐ₗ(r) → (r→0) → rˡ - - -The Hydrogen Atom -━━━━━━━━━━━━━━━━━ - - V(r) = -e²/r - - For a hydrogenic ion with nuclear charge Z - - V(r) = -Ze²/r - - Eigenfunctions: - - Ψₑ﹐ₗ﹐ₘ(r,θ,φ) = Rₑ﹐ₗ(r) Yₗ﹐ₘ(θ,φ) = Uₑ﹐ₗ/r Yₗ﹐ₘ(θ,φ) + (pic) Developed radial schrodinger equation using U(r) replacement + If V(r) is not more singular at the origin than 1/r^2 then the SE has power series solutions. +Began the Hydrogen Atom diff --git a/lecture_notes/3-28/.ps b/lecture_notes/3-28/.ps new file mode 100644 index 0000000..e69de29 diff --git a/lecture_notes/3-28/3-28-overview.txt b/lecture_notes/3-28/3-28-overview.txt new file mode 100644 index 0000000..fdbe063 --- /dev/null +++ b/lecture_notes/3-28/3-28-overview.txt @@ -0,0 +1 @@ +Developed Theory for The Hydrogen Atom diff --git a/lecture_notes/3-28/hydrogen atom b/lecture_notes/3-28/hydrogen atom index 3e8b813..8ca5ece 100644 --- a/lecture_notes/3-28/hydrogen atom +++ b/lecture_notes/3-28/hydrogen atom @@ -1,9 +1,18 @@ Theory of the Hydrogen Atom ━━━━━━━━━━━━━━━━━━━━━━━━━━━━ + V(𝐫) = -e²/r - Derived from first principles the wave equation of an electron in the hydrogen atom. + For any hydrogenic ion with nuclear charge Z + V(𝐫) = -Ze²/r - -Start with Dirac Notation + Eigenfunctions in spherical coordinates: + + Ψₑ﹐ₗ﹐ₘ(r,θ,φ) = Rₑ﹐ₗ(r) Yₗ﹐ₘ(θ,φ) = Uₑ﹐ₗ/r Yₗ﹐ₘ(θ,φ) + + Derived from first principles the wave equation of an electron in the + hydrogen atom. + + - Start with Dirac Notation - Replace with known general functions - transform to eigenvalue equation in position space - replace pieces with function F and find derivatives @@ -15,4 +24,68 @@ Theory of the Hydrogen Atom These are infinite, though, so - Force Laguerre Polynomial solutions to truncate - - \ No newline at end of file +Derivation +━━━━━━━━━━ + The Hydrogen atom follows the central potential development from the + previous lecture. + + ❬r,θ,φ❙Ψ❭ = R(r)ₑ,ₗ yₗ﹐ₘ(θ,φ) + + Rₑ﹐ₗ(r) = Uₑ﹐ₗ(r) /r + + ρ ≡ √⎛8͟m͟ │E│⎞r + ⎝ ħ² ⎠ + + λ = Z͟e͟² √⎛_͟m͟_͟ ⎞ + ħ ⎝2│E│⎠ + + d͟²͟ U - 1͟ l(l+1) + ⎛λ͟ - 1͟⎞U = 0 + dρ² ρ² ⎝ρ 4⎠ + + + as ρ→0, U→ρˡ⁺¹ + + for ρ→∞, + + d͟²͟ U - 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x +407 2991 623 2991 x +4171 2991 l +4387 2991 4387 2831 x +4387 2664 4171 2664 x +4171 1512 m +623 1512 l +407 1512 407 1679 x +407 1839 623 1839 x +4171 1839 l +4387 1839 4387 1679 x +4387 1512 4171 1512 x +4824 fwd_x +end_ol + } def +/DBA { start_ol +3664 4680 m +1330 4680 l +2721 2641 l +2769 2586 2769 2538 x +2769 2506 2737 2442 x +1282 -176 l +3744 -176 l +3782 675 l +3796 831 3903 831 x +3949 831 l +4071 831 4071 663 x +4071 -336 l +4071 -504 3900 -504 x +1179 -504 l +1008 -504 1008 -352 x +1008 -225 l +1008 -160 1039 -96 x +2486 2562 l +1055 4608 l +1008 4672 1008 4752 x +1008 4855 l +1008 5007 1182 5007 x +3897 5007 l +4071 5007 4071 4838 x +4071 3840 l +4071 3672 3926 3672 x +3881 3672 l +3754 3672 3736 3829 x +3664 4680 l +4824 fwd_x +end_ol + } def +/EBA { start_ol +504 2614 m +504 2767 547 2990 x +591 3213 726 3512 x +862 3811 1062 4053 x +1262 4296 1621 4469 x +1981 4644 2435 4644 x +3241 4644 3816 4089 x +3816 4321 l +3816 4536 3983 4536 x +4143 4536 4143 4318 x +4143 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1725 3201 x +2076 3384 2507 3384 x +2507 3056 m +1885 3056 1501 2651 x +1119 2246 1119 1674 x +1119 1093 1522 691 x +1924 291 2507 291 x +3082 291 3485 696 x +3888 1101 3888 1674 x +3888 2262 3512 2659 x +3137 3056 2507 3056 x +4824 fwd_x +end_ol + } def +/HBA { start_ol +1217 6005 m +1686 6005 l +1686 4038 l +2872 5053 l +3422 5053 l +2337 4132 l +3594 2606 l +3039 2606 l +2021 3871 l +1686 3586 l +1686 2606 l +1217 2606 l +1217 6005 l +4824 fwd_x +end_ol + } def +/IBA { start_ol +970 4285 m +970 4368 1126 4521 x +1282 4674 1625 4822 x +1969 4971 2377 4971 x +2976 4971 3379 4614 x +3783 4258 3783 3737 x +3783 3272 3523 3000 x +3263 2728 2959 2648 x +3431 2454 3714 2112 x +3999 1771 3999 1361 x +3999 751 3507 321 x +3016 -108 2328 -108 x +1857 -108 1317 128 x +778 364 778 564 x +778 621 825 669 x +874 717 930 717 x +986 717 1154 592 x +1321 468 1633 343 x +1945 219 2336 219 x +2880 219 3276 556 x +3672 895 3672 1361 x +3672 1836 3245 2178 x +2820 2520 2233 2520 x +2016 2520 2016 2687 x 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4644 m +1868 4644 1566 4097 x +1263 3550 1263 2800 x +1263 2592 l +3528 2592 l +3528 2800 l +3528 3582 3214 4113 x +2900 4644 2395 4644 x +4824 fwd_x +end_ol + } def +/RBA { start_ol +4359 1040 m +4359 849 4309 686 x +4261 524 4138 356 x +4016 190 3755 94 x +3494 0 3126 0 x +2775 0 l +2775 -1584 l +2448 -1584 l +2448 0 l +2084 0 l +1575 0 1255 277 x +936 556 936 1027 x +936 2984 l +541 2984 l +288 2984 288 3155 x +288 3312 573 3312 x +1303 3312 l +1303 1047 l +1303 723 1510 525 x +1717 327 2041 327 x +2448 327 l +2448 3312 l +3232 3312 l +3551 3312 3786 3208 x +4023 3105 4138 2938 x +4255 2772 4307 2601 x +4359 2430 4359 2255 x +4359 1040 l +3231 335 m +3660 335 3846 561 x +4032 787 4032 1049 x +4032 2262 l +4032 2968 3188 2968 x +2775 2968 l +2775 335 l +3231 335 l +4824 fwd_x +end_ol + } def +/SBA { start_ol +3008 5592 m +3008 -956 l +1803 -956 l +1803 5592 l +3008 5592 l +4824 fwd_x +end_ol + } def +/TBA { start_ol +2823 1440 m +3186 1440 3537 1845 x +3888 2250 3888 2810 x +3888 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+ } def +/VBA { start_ol +2376 1941 m +2376 2533 2613 3240 x +2851 3947 3116 4387 x +3381 4827 3498 4827 x +3561 4827 3607 4779 x +3655 4731 3655 4667 x +3655 4579 3531 4343 x +3408 4107 3259 3823 x +3109 3540 2985 3020 x +2863 2502 2863 1918 x +2863 695 3631 -743 x +3655 -798 3655 -831 x +3655 -895 3604 -942 x +3553 -991 3490 -991 x +3366 -991 3105 -555 x +2844 -120 2610 598 x +2376 1318 2376 1941 x +4824 fwd_x +end_ol + } def +/WBA { start_ol +1224 -831 m +1224 -743 1346 -507 x +1470 -272 1620 11 x +1769 295 1892 814 x +2016 1334 2016 1918 x +2016 3148 1247 4579 x +1224 4635 1224 4667 x +1224 4731 1274 4779 x +1325 4827 1388 4827 x +1513 4827 1774 4390 x +2035 3955 2269 3236 x +2503 2517 2503 1894 x +2503 1302 2264 595 x +2027 -111 1761 -551 x +1497 -991 1380 -991 x +1317 -991 1270 -942 x +1224 -895 1224 -831 x +4824 fwd_x +end_ol + } def +/XBA { start_ol +1447 -288 m +1519 -666 1779 -888 x +2039 -1112 2414 -1112 x +2686 -1112 2921 -1032 x +3157 -954 3285 -867 x +3349 -824 3405 -824 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