From 890066b88f0f6c2bd9c2369f4718fa9ad09aa1c8 Mon Sep 17 00:00:00 2001 From: othocaes Date: Wed, 23 Mar 2016 01:17:40 -0400 Subject: [PATCH] added angular momentum operators and hydrogen atom basics --- HW | 10 + lecture_notes/3-14/3d eigenstates | 5 +- .../3-16/Angular Momentum Commutator | 4 - lecture_notes/3-16/hydrogen atom | 20 + lecture_notes/3-18/overview | 43 + solutions/chap7/prob7 | 88 + solutions/chap7/prob7.ps | 3322 +++++++++++++++++ 7 files changed, 3484 insertions(+), 8 deletions(-) create mode 100644 HW delete mode 100644 lecture_notes/3-16/Angular Momentum Commutator create mode 100644 lecture_notes/3-16/hydrogen atom create mode 100644 lecture_notes/3-18/overview create mode 100644 solutions/chap7/prob7 create mode 100644 solutions/chap7/prob7.ps diff --git a/HW b/HW new file mode 100644 index 0000000..b71b1e9 --- /dev/null +++ b/HW @@ -0,0 +1,10 @@ +Quantum Mechanics Assignments + +Chap 2: 2, 3, 14, 15, 17, 22, 23 + +Chap 5: 2, 6, 8, 12, in-class assignment + +Chap 9: 7, 11, 12, 13, 14 + +Chap 7: 7, 8, one from last class, 11 + due friday 3-18 diff --git a/lecture_notes/3-14/3d eigenstates b/lecture_notes/3-14/3d eigenstates index 9f1a0b5..5e9fb7f 100644 --- a/lecture_notes/3-14/3d eigenstates +++ b/lecture_notes/3-14/3d eigenstates @@ -55,7 +55,4 @@ Understanding a System: With the eigenvalue equation - ❬r❙E,l,mₗ❭ = E❬r❙E,l,mₗ❭ - - - + ❬r❙E,l,mₗ❭ = E❬r❙E,l,mₗ❭ \ No newline at end of file diff --git a/lecture_notes/3-16/Angular Momentum Commutator b/lecture_notes/3-16/Angular Momentum Commutator deleted file mode 100644 index e6371b2..0000000 --- a/lecture_notes/3-16/Angular Momentum Commutator +++ /dev/null @@ -1,4 +0,0 @@ -[L̂𝓏,p̂𝓏] = ιħp̂𝓎 -[L̂𝓏,p̂𝓍] = -ιħp̂𝓎 -[L̂𝓏,p̂𝓏] = 0 -[L̂𝓏,p̂²] = \ No newline at end of file diff --git a/lecture_notes/3-16/hydrogen atom b/lecture_notes/3-16/hydrogen atom new file mode 100644 index 0000000..66a68f2 --- /dev/null +++ b/lecture_notes/3-16/hydrogen atom @@ -0,0 +1,20 @@ +V = E͟_͟Z͟ (?) + r +❬r❙E,l,mₗ❭ ≝ R(𝐫) ≝ U͟(𝐫͟) + r + +∂²/∂r² (u/r) + 2/r ∂/∂r U/r + = ∂/∂r (1/r (∂/∂r u) - u/r²) + 2/r ((1/r ∂/∂r u) - u/r²) + + = 1/r (∂²/∂r² u) - 2/r^2 ∂/∂r u + 2 u/r³ + 2/r² ∂/∂r u - 2u/r³ + + = 1/r (∂²/∂r² u) + + +So, + + (-ħ/2m ∂²/∂r² + Veff(│r│)) = V(│r│) + l͟(l͟+͟1͟)ħ͟²͟ + 2mr² + | ↓ | + Centrifugal + barrier diff --git a/lecture_notes/3-18/overview b/lecture_notes/3-18/overview new file mode 100644 index 0000000..16faa6f --- /dev/null +++ b/lecture_notes/3-18/overview @@ -0,0 +1,43 @@ +Vibrations and rotations of a diatomic molecule + + atoms vibrate about an equilibrium position r₀ + + ⊙~~~~~~~~~⊙ + |← r₀ →| + + (pic MISSEd) potential, with taylor approximation + + (pic MISSEd) A taylor series solution is appropriate to solve this diffEQ. + + Experiments indicate vibration (E~visible light) has more energy than rotation (E~infrared), so vibrations happen much faster. + +The Rotational eigen-value spectrum + + L²,L̂𝓍,L̂𝓎,L̂𝓏 (Hermition operators) + + Say + L̂𝓏❙Ψ❭ = mₗħ❙Ψ❭ + + (pic) mₗ is bound by λ: m has value from -λ to λ + + +Introduced raising and lower operators + + L̂± = L̂𝓍 ± ιL̂𝓎 = (pic) proof = ±ħL± + + L̂𝓏(L̂ ± ❙λ,mₗ❭) => (pic) proof => L̂± ❙λ,mₗ❭ = ❙λ,mₗ±1❭ + + Very important point: + + Let m=l be the maximum value of m + L̂₊❙λ,l❭ = 0 (required because wave function goes to 0 in a forbiddin region) + This is also true for L̂₋❙λ,l′❭ with l′ the minimum value of m + + (pic) L̂₋L̂₊❙λ,l❭ = ... + gives λ = l(l+1) + + (pic) L̂₊L̂₋❙λ,l′❭ = ... + gives λ = l′(l′+1) + + These show that m = -l, -l+1, 0, l + diff --git a/solutions/chap7/prob7 b/solutions/chap7/prob7 new file mode 100644 index 0000000..2afa48b --- /dev/null +++ b/solutions/chap7/prob7 @@ -0,0 +1,88 @@ +Angular momentum system is prepared in the state + + ❙Ψ❭ = 1͟ ❙11❭ - 2͟ ❙10❭ + ι͟2͟ ❙22❭ + ι1͟ ❙20❭ + √10 √10 √10 √10 + + +Possible results of 𝐋² measurement? + +The 𝐋̂² operator was developed in lecture and in the text. The related eigenvalue equation is + + 𝐋̂²❙lm❭ = l(l+1) ħ²❙lm❭ + +The possible measurements of 𝐋̂² for this system are those associated with the initial state vector, i.e. the values lm = 11, 10, 22, 20. + +For lm = 11 & 10, 𝐋̂² = 2ħ² + +For lm = 22 & 20, 𝐋̂² = 6ħ². + +Since the same 𝐋̂² is measures for two states, the sum of the probabilities of measuring those states is the probability of measuring that squared angular momentum. + +𝓟(𝐋̂²=2ħ²) = │❬11❙Ψ❭│² + │❬10❙Ψ❭│². + + ❬11❙Ψ❭ = ❬11❙⎛ 1͟ ❙11❭ - 2͟ ❙10❭ + ι͟2͟ ❙22❭ + ι1͟ ❙20❭ ⎞ + ⎝ √10 √10 √10 √10 ⎠. + +The eigenstates for this system are LaPlace's spherical harmonic functions, which comprise an orthogonal set, I.E.: + + ❬lm❙l′m′❭ = δₗₗ′ δₘₘ′. + + │❬11❙Ψ❭│² = │❬11❙ 1͟ ❙11❭│² = ¹/₁₀. + │ √10 │ + + │❬10❙Ψ❭│² = │❬10❙ 2͟ ❙10❭│² = ⁴/₁₀. + │ √10 │ + +(𝐚) + 𝓟(𝐋̂²=2ħ²) = ½. + +No need to calculate the other set: since the vector is normalized, the probability of measuring 𝐋̂²=6ħ² is also + + 𝓟(𝐋̂²=6ħ²) = ½. + + +For 𝐋̂𝓏 the eigenvalue equation is + + 𝐋̂𝓏❙lm❭ = mħ❙lm❭, + +so the expected measurements are, + +for lm = 11 & 21, 𝐋̂𝓏 = ħ. + +for lm = 10 & 20, 𝐋̂𝓏 = 0. + +In this case, the probabilities will be + + 𝓟(𝐋̂𝓏=ħ) = │❬11❙Ψ❭│² + │❬21❙Ψ❭│² and + + 𝓟(𝐋̂𝓏=0) = │❬10❙Ψ❭│² + │❬20❙Ψ❭│². + +The method has already been demonstrated, so taking the probability components and summing, + +(𝐛) + + 𝓟(𝐋̂𝓏=ħ) = ²/₁₀. + + 𝓟(𝐋̂𝓏=0) = ⁸/₁₀. + +Histogram of probabilities: + + +𝓟 𝓟 + ╭─────────────────┬────────────────╮ +1 │ │ │ 1 + │ │ │ +.8 │ │ ▧ │ .8 + │ │ ▧ │ + │ │ ▧ │ +.5 │ ▧        ▧    │           ▧    │ .5 +   │   ▧        ▧    │           ▧    │     +   │   ▧        ▧    │           ▧    │     +   │   ▧        ▧    │   ▧       ▧    │     +.1 │   ▧        ▧    │   ▧       ▧    │ .1 +   ╰─────────────────┼────────────────╯ + 2ħ² 6ħ² │ 0 ħ + 𝐋̂² │ 𝐋̂𝓏 + + h + diff --git a/solutions/chap7/prob7.ps b/solutions/chap7/prob7.ps new file mode 100644 index 0000000..f126baa --- /dev/null +++ b/solutions/chap7/prob7.ps @@ -0,0 +1,3322 @@ +%!PS-Adobe-3.0 +%%Title: Otho Ulrich: Chap 7 #7 +%%Creator: paps version 0.6.7 by Dov Grobgeld +%%Pages: (atend) +%%BoundingBox: 0 0 595 841 +%%BeginProlog +%%Orientation: Portrait +/papsdict 1 dict def +papsdict begin + +/inch {72 mul} bind def +/mm {1 inch 25.4 div mul} bind def + +% override setpagedevice if it is not defined +/setpagedevice where { + pop % get rid of its dictionary + /setpagesize { + 3 dict begin + /pageheight exch def + /pagewidth exch def + /orientation 0 def + % Exchange pagewidth and pageheight so that pagewidth is bigger + pagewidth pageheight gt { + pagewidth + /pagewidth pageheight def + /pageheight exch def + /orientation 3 def + } if + 2 dict + dup /PageSize [pagewidth pageheight] put + dup /Orientation orientation put + setpagedevice + end + } def +} +{ + /setpagesize { pop pop } def +} ifelse +/duplex { + statusdict /setduplexmode known + { statusdict begin setduplexmode end } {pop} ifelse +} def +/tumble { + statusdict /settumble known + { statusdict begin settumble end } {pop} ifelse +} def +% Turn the page around +/turnpage { + 90 rotate + 0 pageheight neg translate +} def +% User settings +/pagewidth 595 def +/pageheight 841 def +pagewidth pageheight setpagesize +/column_width 523 def +/bodyheight 755 def +/lmarg 36 def +/ytop 791 def +/do_separation_line true def +/do_landscape false def +/do_tumble true def +/do_duplex true def +% Procedures to translate position to first and second column +/lw 20 def % whatever +/setnumcolumns { + /numcolumns exch def + /firstcolumn { /xpos lmarg def /ypos ytop def} def + /nextcolumn { + do_separation_line { + xpos column_width add gutter_width 2 div add % x start + ytop lw add moveto % y start + 0 bodyheight lw add neg rlineto 0 setlinewidth stroke + } if + /xpos xpos column_width add gutter_width add def + /ypos ytop def + } def +} def + +1 setnumcolumns +/showline { + /y exch def + /s exch def + xpos y moveto + column_width 0 rlineto stroke + xpos y moveto /Helvetica findfont 20 scalefont setfont s show +} def +/paps_bop { % Beginning of page definitions + papsdict begin + gsave + do_landscape {turnpage} if + % ps2pdf gets wrong orientation without this! 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