From 27552a0f6600f535b3261dde213559f704269d1e Mon Sep 17 00:00:00 2001 From: othocaes Date: Sun, 13 Mar 2016 20:07:19 -0400 Subject: [PATCH] renamed text.pdf, starting chap 9 --- notes/harmonic oscillator | 4 + notes/particle operators | 13 + notes/random | 35 + solutions/chap5/class_prob | 19 + .../chap5/otho_ulrich_chap5_class_prob.ps | 2602 ++++++++++++ solutions/chap5/otho_ulrich_chap5_prob2.ps | 3587 +++++++++++++++++ solutions/chap5/otho_ulrich_chap5_prob6.ps | 3224 +++++++++++++++ solutions/chap5/prob2 | 32 +- solutions/chap5/prob2.ps | 3420 ---------------- solutions/chap5/prob6 | 68 + solutions/chap5/transmission_plot.pdf | Bin 0 -> 20558 bytes solutions/exam1/prob4 | 16 +- ..., Janet] Quantum Mechanics.pdf => text.pdf | Bin 13 files changed, 9584 insertions(+), 3436 deletions(-) create mode 100644 notes/harmonic oscillator create mode 100644 notes/particle operators create mode 100644 notes/random create mode 100644 solutions/chap5/class_prob create mode 100644 solutions/chap5/otho_ulrich_chap5_class_prob.ps create mode 100644 solutions/chap5/otho_ulrich_chap5_prob2.ps create mode 100644 solutions/chap5/otho_ulrich_chap5_prob6.ps delete mode 100644 solutions/chap5/prob2.ps create mode 100644 solutions/chap5/prob6 create mode 100644 solutions/chap5/transmission_plot.pdf rename [McIntyre, David; Manogue, Corinne; Tate, Janet] Quantum Mechanics.pdf => text.pdf (100%) diff --git a/notes/harmonic oscillator b/notes/harmonic oscillator new file mode 100644 index 0000000..f1a7ce5 --- /dev/null +++ b/notes/harmonic oscillator @@ -0,0 +1,4 @@ +Ĥ = T̂ + V̂ = -_͟ħ͟²͟ ∇² + V(𝐫,t). + 2m + +V(𝐫,t) = ½kx² \ No newline at end of file diff --git a/notes/particle operators b/notes/particle operators new file mode 100644 index 0000000..d0ff01e --- /dev/null +++ b/notes/particle operators @@ -0,0 +1,13 @@ + Ĥ = T̂ + V̂ +with + V̂ = V = V(𝐫,t) (potential energy operator) +and + T̂ = _͟𝐩̂͟⋅͟𝐩̂͟ = _͟p̂͟²͟ = -_͟ħ͟²͟ ∇² + 2m 2m 2m +since + p̂ = -ιħ∇. + + +Ĥ = T̂ + V̂ = -_͟ħ͟²͟ ∇² + V(𝐫,t). + 2m + diff --git a/notes/random b/notes/random new file mode 100644 index 0000000..031895b --- /dev/null +++ b/notes/random @@ -0,0 +1,35 @@ + +In position space, + ⌠ ∞ + ❬φₙ❙Ψ❭ ≐ ⎮ dx φₙ⃰(x) Ψ(x). + ⌡-∞ + +But for just 3L/4 to L, + ⌠L + ⎮ dx φₙ⃰(x) Ψ(x). + ⌡3L/4 + +Since this is a particle in an infinite square well, + + φₙ(x) = √⎛2͟⎞ sin⎛n͟π͟x͟⎞ and Ψ(x) = ∑ cₙ φₙ(x). + ⎝L⎠ ⎝ L ⎠ ⁿ + +So the probability to calculate is + + ⌠L + ⎮ dx √⎛2͟⎞ sin⎛n͟π͟x͟⎞ ∑ cₙ φₙ(x). + ⌡3L/4 ⎝L⎠ ⎝ L ⎠ ⁿ + + + + + + + +The evaluation of the last piece of this expression is elucidated by the identity + + sin(u)sin(v) = 1/2 (cos(u-v) - cos(u+v)), i.e. + + sin(2nπ) sin(³/₂nπ) = ¹/₂(cos(1/2nπ) - cos(7/2nπ)). + +So when n is odd, this product disappears, and when n is even one of the sin² terms disappears. The other sin² will always return 0, since regardless of the integer n, its argument is an integer multiple of π. \ No newline at end of file diff --git a/solutions/chap5/class_prob b/solutions/chap5/class_prob new file mode 100644 index 0000000..87bac4c --- /dev/null +++ b/solutions/chap5/class_prob @@ -0,0 +1,19 @@ +The task is to graph the following expression, developed in class. We had a lot of trouble with this, because while it seems straight forward, the parameter values of sa were difficult to pin down for some reason. The produced graph of transmission vs. displacement drops off too quickly according to our standard, but it is within the right ballpark, so I thought it worthwhile to at least include the incorrect result. The following function is plotted: + +The transmission + + T = _____͟1͟_____ , + 1 + sinh(sa) + +with s = √(2͟m͟ₑ͟E͟) and a the displacement from the origin. + ħ + + E = 5 eV + mₑ = 5.11*10⁶ eV/c² + c = 299792458 m/s + ħ = 6.582*10⁽⁻¹⁶⁾ eV s + +So, s = 3.62*10¹⁰ m⁻¹ = 3.62 A⁻¹. + +The parameter space is defined from a=0 to a=4 angstroms. + diff --git a/solutions/chap5/otho_ulrich_chap5_class_prob.ps b/solutions/chap5/otho_ulrich_chap5_class_prob.ps new file mode 100644 index 0000000..07e5c4e --- /dev/null +++ b/solutions/chap5/otho_ulrich_chap5_class_prob.ps @@ -0,0 +1,2602 @@ +%!PS-Adobe-3.0 +%%Title: Lect. 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/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 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7066 x +4854 7185 4963 7185 x +5033 7185 5097 7125 x +5162 7066 5162 6985 x +5162 6905 5133 6865 x +5976 fwd_x +end_ol + } def +/mBA { start_ol +1008 4692 m +1008 5142 1562 5656 x +2117 6170 2900 6170 x +3639 6170 4200 5650 x +4763 5132 4763 4452 x +4763 4003 4494 3609 x +4225 3215 3303 2367 x +1202 441 l +1202 410 l +4392 410 l +4392 770 l +4392 1040 4602 1040 x +4802 1040 4802 770 x +4802 0 l +864 0 l +864 600 l +3222 2815 l +3921 3504 4136 3813 x +4351 4123 4351 4462 x +4351 4981 3915 5370 x +3478 5760 2900 5760 x +2382 5760 1960 5460 x +1540 5161 1418 4711 x +1356 4512 1202 4512 x +1131 4512 1069 4567 x +1008 4622 1008 4692 x +5976 fwd_x +end_ol + } def +/nBA { start_ol +2981 -651 m +2981 -1152 l +-3024 -1152 l +-3024 -651 l +2981 -651 l +0 fwd_x +end_ol + } def +/oBA { start_ol +842 6836 m +813 6877 813 6957 x +813 7036 877 7096 x +942 7157 1012 7157 x +1062 7157 1111 7126 x +1161 7096 1191 7057 x +1220 7026 l +2592 5105 2592 2472 x +2592 -2030 l +2181 -2030 l +2181 2643 l +2181 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a/solutions/chap5/prob2 b/solutions/chap5/prob2 index 801a25c..ab118aa 100644 --- a/solutions/chap5/prob2 +++ b/solutions/chap5/prob2 @@ -45,8 +45,8 @@ So, π͟²͟ħ͟². 𝓟₁=¹/₃ 2mL² - 4͟π͟²͟ħ͟². 𝓟₂=¹/₃ - 2mL² + 2͟π͟²͟ħ͟². 𝓟₂=¹/₃ + mL² 9͟π͟²͟ħ͟². 𝓟₃=¹/₃ 2mL² @@ -59,7 +59,7 @@ The average value of the energy, or the expectation value, is given by │√3 √3 │. ❬Ψ′❙Ĥ❙Ψ′❭ = _͟1͟ (❬φ₁❙ - ❬φ₂❙ - ι❬φ₃❙)_͟1͟ (E₁❙φ₁❭ - E₂❙φ₂❭ + ιE₃❙φ₃❭) - √3 √3 + √3 √3 = ¹/₃ (❬φ₁❙ - ❬φ₂❙ - ι❬φ₃❙)(E₁❙φ₁❭ - E₂❙φ₂❭ + ιE₃❙φ₃❭) @@ -87,16 +87,32 @@ Because the hamiltonian is time-independent, the state vector progresses timewis ❙Ψ′(t)❭ = exp(-ι͟Ĥ͟t͟ )❙Ψ′(t=0)❭ = _͟1͟ exp(-ι͟Ĥ͟t͟ ) (❙φ₁❭ - ❙φ₂❭ + ι ❙φ₃❭). ħ √3 ħ - _͟1͟ ⎛exp(-ι͟E͟₁͟t͟ )❙φ₁❭ - exp(-ι͟E͟₂͟t͟ )❙φ₂❭ + ι exp(-ι͟E͟₃͟t͟ )❙φ₃❭⎞. - √3 ⎝ ħ ħ ħ ⎠ + _͟1͟ ⎛exp(-ι͟E͟₁͟t͟ )❙φ₁❭ - exp(-ι͟E͟₂͟t͟ )❙φ₂❭ + ι exp(-ι͟E͟₃͟t͟ )❙φ₃❭⎞ + √3 ⎝ ħ ħ ħ ⎠ + +(𝐝) + = _͟1͟ ⎛exp(-ι π͟²͟ħ͟t͟ )❙φ₁❭ - exp(-ι 2͟π͟²͟ħ͟t͟ )❙φ₂❭ + ι exp(-ι 9͟π͟²͟ħ͟t͟ )❙φ₃❭⎞ + √3 ⎝ 2mL² mL² 2L² ⎠. +What are the possible measurements at time t = ħ/E₁? -The coefficients for the states have the same magnitudes, so the probability of measuring a particular state at time t=0 is 1/3. +(𝐞) The same value of energy will be measured for each state. Since there has been no change to the coefficients besides a change in phase, and the phase term goes to 1 under the modulus, the probabilities remain the same. +❙Ψ′(t=ħ/E₁)❭ = _͟1͟ ⎛exp(-ι)❙φ₁❭ - exp(-ι E͟₂͟ )❙φ₂❭ + ι exp(-ι E͟₃͟ )❙φ₃❭⎞ + √3 ⎝ E₁ E₁ ⎠ + + Energy Probability + + π͟²͟ħ͟². 𝓟₁=¹/₃ + 2mL² + + 2͟π͟²͟ħ͟². 𝓟₂=¹/₃ + mL² + + 9͟π͟²͟ħ͟². 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energy eigenstate n is + + φₙ(x) = √⎛2͟⎞ sin⎛n͟π͟x͟⎞, so │φₙ(x)│² = ⎛2͟⎞ sin²⎛n͟π͟x͟⎞ + ⎝L⎠ ⎝ L ⎠ ⎝L⎠ ⎝ L ⎠. + +Therefore, with the power reduction identity, + + ⌠L ⌠L + ❬φₙ❙φₙ❭ = ⎛2͟⎞ ⎮ dx sin²⎛n͟π͟x͟⎞ = ⎛2͟⎞ ⎮ dx ⎛1͟ - 1͟ cos(2͟n͟π͟x͟)⎞ + ⎝L⎠ ⎮ ⎝ L ⎠ ⎝L⎠ ⎮ ⎝2 2 L ⎠. + ⌡³/₄L ⌡³/₄L + + ❬φₙ❙φₙ❭ = ⎛2͟⎞ ⎛_͟1͟L͟ - ⎛ _͟1͟L͟ sin(2nπ) - _͟1͟L͟ sin(3͟n͟π͟) ⎞ ⎞ + ⎝L⎠ ⎝ 8 ⎝ 4nπ 4nπ 2 ⎠ ⎠ + + = _͟1͟ L͟ ⎛sin(2nπ) - sin(3͟n͟π͟)⎞ + 16 nπ ⎝ 2 ⎠. + +In general, then, the probability of finding the particle in a particular energy state n is given by + + │❬φₙ❙Ψ❭│² = │_͟1͟ L͟ ⎛sin(2nπ) - sin(3͟n͟π͟)⎞│² + │16 nπ ⎝ 2 ⎠│ + + = _͟1͟ ⎛_͟L͟²͟ ⎞ │⎛sin(2nπ) - sin(3͟n͟π͟)⎞│² + 256 ⎝n²π²⎠ │⎝ 2 ⎠│ + + = _͟1͟ ⎛_͟L͟²͟ ⎞ ⎛sin(2nπ) - sin(3͟n͟π͟)⎞ ⎛sin(2nπ) - sin(3͟n͟π͟)⎞ + 256 ⎝n²π²⎠ ⎝ 2 ⎠ ⎝ 2 ⎠ + + = _͟1͟ ⎛_͟L͟²͟ ⎞ ⎛sin²(2nπ) + sin²(3͟n͟π͟) + 2 sin(2nπ) sin(3͟n͟π͟)⎞ + 256 ⎝n²π²⎠ ⎝ 2 2 ⎠ . + +The third term in the parentheses will always return 0, since one function or the other returns 0 depending on whether n is even or odd. The first term will also always be zero, since regardless of the value of n, the argument is an integer multiple of π. The final term will disappear when n is even, so the probability of finding a particle in the specified region for an energy state n is therefore + + ⎧ + ⎪ 0, n even +P(n) ⎨ + ⎪ _͟1͟ ⎛_͟L͟²͟ ⎞ ⎛sin²(3͟n͟π͟)⎞ + ⎩ 256 ⎝n²π²⎠ ⎝ 2 ⎠ , n odd + +This is wrong, because for n=2, there should be some probability of finding the particle within this region, since the only node is at the center, and the wave function is non-zero across the region and does not cross the axis. This function will return a probability of 0 for n=2, which does not make sense with my statement. For n=1,3, at least a probability can be calculated: + + + +P(n=1) = _͟1͟ ⎛_͟L͟²͟⎞ sin²(3͟π) = _͟1͟ ⎛_͟L͟²͟⎞ = 3.96*10⁻⁴ L² + 256 ⎝ π²⎠ 2 256 ⎝ π²⎠ + +P(n=3) = _͟1͟ ⎛_͟L͟²͟ ⎞ ⎛sin²(9͟π͟)⎞ = _͟1͟ ⎛_͟L͟²͟⎞ = 4.40*10⁻⁵ L² + 256 ⎝9 π²⎠ ⎝ 2 ⎠ 2304 ⎝ π²⎠ + diff --git a/solutions/chap5/transmission_plot.pdf b/solutions/chap5/transmission_plot.pdf new file mode 100644 index 0000000000000000000000000000000000000000..05c3a7fce20cc6fb4ae3ae51b5052c1b0223d5cb GIT binary patch literal 20558 zcmZ^}bx>SQum_ri1eXB8-QC>@zPNiJ=;H1g2ri4evk)ADJA~l2xLeT0-SzEv@BQOd zy?Rx@Iy=+d)6+fEJ#%XN(5OnuuyC+)Bh&O2^>*|&_2waSlCzULn%W=>3$rO&H~`(O z$hlw?bv7AmJ2wj#HW@n;Hw!5XGe>g^QBh=9Hx~;Ndt|TdD*qmkg@D8EYe2sN(SABp zWDzxag7{6#$PZ5u);}m_9$ZZ(QbhQU&ri4vhn=qFlw2se;^2BWi4LfGA-$TPALiYP zecW5e>xSe>m6+y;fxj-9<0Jj0HtE|%U`^n2)?jzdtI+Em*_+GD!@?2NxTgDgPu70* z2!P{F6leT4u^srVPP!ZL{5bJ;y~}Dmn%2I4;$VC;%3^$!X3+h5X8iU_m3cp9Oe#zA z2JNniU30uNuoz1@xLX_#%saBAumAg)%|+&2<>idVthL~lc;zze0zJkDGxO0 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zy4$d0aj^9jz0TsB>K~3CZ?LMOX=w(Zsxro}ZIAulm{S^Ol04;QWR)8pH=R87moZ`D zV(OFW??+R6t`E3An*M&|?u6#iXYTfzqm3$riEicNQ%l~0{;+iK@Xf~0+_N)1XC!CR zJj0`=R3^Bq-CCxi`>}h>%M?Md&qy?dfo(FND?n@ai3p|8+ZGJy~PCsxI2)$Vr*hVPu zvge3hU{Ki$cHjhw-R4HloKdrWwqbKQunk*?#v=(80NZgyf&ifxM+D(fW;=K@$D7AS z!yHvWr4b+sp5O-C3B;fd80!7?LP3B(C>CZ9Ru60t{-`joV3=tS0(~#fi(|K*0b4?1 zu(Bp5FcX*xbcrCe%9!1pkVciGTNM!W#W1khd}|H%t2to67keP!Vk@4v$OW3U8NFo+ z0#L8aQGuUR*=$w^lMOa00T;}l3#L0eeJt$6;{+kC5k?l~78VFX#v=%cg&-hJ93dHidn%B@ z1Qpc4zWa%V#6%DZ4V-|73Qh_D6bTHaP(Gb#AMqm{6+;C51aoi&?b$16%w7pVG{Ay) zfEVZgB74?s@K9g@80fLq5fD5DArNSI8kvX?a0q!7f~;CS$NNFRqgVvOld!ctyS@Bk6>`M~4rzsAyt^y?V+9La@_hsl9)%XzIxJa`3uJ^C(1GL0VDe-NjlyCO4CoAVb0(g|WRgfk znz;$~pIOjg0mx?ha=dnl0{kGDPN(7t6ohF2{tjVIXEE?(CIfFyVNwYU8j(RqP^62* UY@v8|Nl~1F@`1YM))v_R09PCzGXMYp literal 0 HcmV?d00001 diff --git a/solutions/exam1/prob4 b/solutions/exam1/prob4 index f02c19c..2909056 100644 --- a/solutions/exam1/prob4 +++ b/solutions/exam1/prob4 @@ -1,4 +1,4 @@ -Two operators' matrix representations are known in the |1〉, |2〉, |3〉 basis, where a and b are real numbers, and ι is the imaginary unit: +Two operators' matrix representations are known in the |1❭, |2❭, |3❭ basis, where a and b are real numbers, and ι is the imaginary unit:  ≐ B̂ ≐ ⎛ a 0 0 ⎞ ⎛ b 0 0 ⎞ @@ -36,7 +36,7 @@ which reduces to Since the operators commute, they share a set of common eigenstates. The eigenstates of  are apparent from inspection: -|a₁〉 ≐ ⎛1⎞ |a₂〉 ≐ ⎛0⎞ and |a₃〉 ≐ ⎛0⎞ +|a₁❭ ≐ ⎛1⎞ |a₂❭ ≐ ⎛0⎞ and |a₃❭ ≐ ⎛0⎞ ⎜0⎟ ⎜1⎟ ⎜0⎟ ⎝0⎠; ⎝0⎠; ⎝1⎠. @@ -50,25 +50,25 @@ For B̂, the eigenvalues are already known (λ = b,b,-b.), and using the eigenva dicates two possible eigenstates (for the eigenvalue b). One eigenstate is obvious from inspection: - |b₁〉 ≐ ⎛1⎞ - ⎜0⎟ - ⎝0⎠. + |b₁❭ ≐ ⎛1⎞ + ⎜0⎟ + ⎝0⎠. The eigenvalue equation also reveals -ι γ = β. If β = 1 is chosen, then γ = -ι, revealing a second eigenstate, after normalizing: - |b₂〉 ≐ 1 ⎛ 0 ⎞ + |b₂❭ ≐ 1 ⎛ 0 ⎞ √2 ⎜ 1 ⎟ ⎝-ι ⎠. Similarly, when the eigenvalue -b is used, the eigenvalue equation reveals ι γ = β. So, if β = ι, γ = 1. The third eigenstate is therefore, after normalizing, - |b₃〉 ≐ 1 ⎛0⎞ + |b₃❭ ≐ 1 ⎛0⎞ √2 ⎜ι⎟ ⎝1⎠ The complete set of eigenstates of the operator B̂ is - |b₁〉 ≐ ⎛1⎞ |b₂〉 ≐ 1 ⎛ 0 ⎞ |b₃〉 ≐ 1 ⎛0⎞ + |b₁❭ ≐ ⎛1⎞ |b₂❭ ≐ 1 ⎛ 0 ⎞ |b₃❭ ≐ 1 ⎛0⎞ ⎜0⎟ √2 ⎜ 1 ⎟ √2 ⎜ι⎟ ⎝0⎠, ⎝-ι ⎠, and ⎝1⎠. diff --git a/[McIntyre, David; Manogue, Corinne; Tate, Janet] Quantum Mechanics.pdf b/text.pdf similarity index 100% rename from [McIntyre, David; Manogue, Corinne; Tate, Janet] Quantum Mechanics.pdf rename to text.pdf