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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x +3603 1399 3465 1399 x +3218 1379 l +3218 410 l +3539 410 l +3794 410 3794 210 x +3794 0 3529 0 x +2507 0 l +2232 0 2232 210 x +2232 410 2496 410 x +2808 410 l +2808 1379 l +2589 1389 l +2400 1389 2131 1489 x +1864 1590 1545 1805 x +1228 2021 1010 2458 x +792 2895 792 3468 x +792 5205 l +475 5205 l +383 5205 299 5259 x +216 5313 216 5401 x +216 5616 487 5616 x +1202 5616 l +1202 3468 l +1202 2768 1650 2283 x +2099 1800 2581 1800 x +2808 1789 l +2808 5205 l +2580 5205 l +2481 5205 2392 5263 x +2304 5322 2304 5401 x +2304 5616 2589 5616 x +3447 5616 l +3600 5596 3660 5562 x +3722 5528 3722 5401 x +3722 5205 3456 5205 x +3218 5205 l +3218 1789 l +3465 1800 l +5976 fwd_x +end_ol + } def +/JBA { start_ol +2952 2432 m +2952 3172 3252 4057 x +3553 4944 3889 5494 x +4225 6044 4374 6044 x +4452 6044 4511 5983 x +4570 5924 4570 5844 x +4570 5734 4414 5439 x +4258 5144 4066 4789 x +3875 4433 3719 3782 x +3562 3132 3562 2401 x +3562 870 4541 -930 x +4570 -1001 4570 -1040 x +4570 -1120 4506 -1180 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x +2520 1671 2520 2401 x +2520 3943 1541 5734 x +1512 5805 1512 5844 x +1512 5924 1576 5983 x +1640 6044 1719 6044 x +1877 6044 2208 5499 x +2539 4954 2835 4053 x +3130 3152 3130 2371 x +3130 1631 2829 744 x +2529 -140 2192 -690 x +1857 -1240 1708 -1240 x +1630 -1240 1570 -1180 x +1512 -1120 1512 -1040 x +5976 fwd_x +end_ol + } def +/NBA { start_ol +1582 -635 m +3406 3015 l +1588 6670 l +2624 6670 l +4441 3015 l +2624 -635 l +1582 -635 l +6048 fwd_x +end_ol + } def +/OBA { start_ol +5450 1311 m +5450 1071 5389 866 x +5330 661 5179 451 x +5029 240 4709 120 x +4388 0 3937 0 x +3506 0 l +3506 -1944 l +3096 -1944 l +3096 0 l +2646 0 l +2014 0 1618 351 x +1224 703 1224 1294 x +1224 3765 l +741 3765 l +432 3765 432 3980 x +432 4176 784 4176 x +1684 4176 l +1684 1319 l +1684 910 1939 660 x +2194 410 2595 410 x +3096 410 l +3096 4176 l +4061 4176 l +4456 4176 4746 4045 x +5036 3916 5179 3705 x +5322 3495 5386 3279 x +5450 3064 5450 2844 x +5450 1311 l +4062 420 m +4587 420 4813 705 x +5040 991 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x +3798 5760 4392 5170 x +4392 5355 l +4392 5616 4626 5616 x +4802 5616 4802 5349 x +4802 4330 l +4802 4063 4626 4063 x +4403 4063 4392 4303 x +4362 4750 3956 5050 x +3551 5349 2976 5349 x +2363 5349 1963 5019 x +1562 4691 1562 4203 x +1562 3814 1819 3575 x +2076 3335 2457 3251 x +2838 3166 3289 3052 x +3740 2937 4122 2817 x +4503 2697 4761 2363 x +5018 2030 5018 1522 x +5018 794 4443 325 x +3870 -144 2977 -144 x +1955 -144 1346 576 x +1346 277 l +1346 0 1123 0 x +936 0 936 272 x +936 1399 l +936 1671 1132 1671 x +1336 1671 1346 1432 x +1365 943 1838 605 x +2311 266 2967 266 x +3672 266 4140 630 x +4608 994 4608 1542 x +5976 fwd_x +end_ol + } def +/iBA { start_ol +1462 4025 m +1462 3975 1606 3763 x +1750 3550 1750 3381 x +1750 3125 1561 2966 x +1373 2808 1076 2808 x +769 2808 600 2979 x +432 3151 432 3456 x +432 4005 952 4348 x +1473 4692 2325 4692 x +3247 4692 3790 4330 x +4333 3967 4333 3351 x +4333 765 l +4333 456 4543 456 x +4613 456 4654 514 x +4754 655 l +4903 391 l +4374 -144 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3765 l +3091 3765 3091 3975 x +3091 4176 3361 4176 x +4663 4176 l +4933 4176 4933 3975 x +4933 3765 4663 3765 x +4191 3765 l +2621 2453 l +4692 410 l +5143 410 l +5413 410 5413 210 x +5413 0 5143 0 x +3831 0 l +3561 0 3561 210 x +3561 410 3831 410 x +4111 410 l +2310 2192 l +1850 1802 l +5976 fwd_x +end_ol + } def +/lBA { start_ol +2737 6068 m +3988 6068 l +2616 3676 l +2496 3456 2306 3456 x +2185 3456 2100 3530 x +2016 3605 2016 3715 x +2016 3756 2036 3816 x +2737 6068 l +5976 fwd_x +end_ol + } def +/mBA { start_ol +6005 -723 m +6005 -1224 l +0 -1224 l +0 -723 l +6005 -723 l +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 +1271 3146 m +1821 995 l +3822 6984 l +5784 6984 l +6005 6984 6005 6804 x +6005 6624 5784 6624 x +4113 6624 l +1791 0 l +950 2736 l +320 2736 l +100 2736 100 2946 x +100 3146 320 3146 x +1271 3146 l +5976 fwd_x +end_ol + } def +/pBA { start_ol +1279 5296 m 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} def +/KCA { start_ol +2291 1872 m +771 1872 l +504 1872 504 2076 x +504 2282 774 2282 x +2505 2282 l +3025 3312 l +774 3312 l +504 3312 504 3516 x +504 3722 771 3722 x +3202 3722 l +3600 4480 l +3661 4586 3774 4586 x +3866 4586 3938 4529 x +4010 4471 4010 4394 x +4010 4366 3990 4327 x +3661 3722 l +5193 3722 l +5463 3722 5463 3516 x +5463 3312 5193 3312 x +3462 3312 l +2946 2282 l +5193 2282 l +5463 2282 5463 2076 x +5463 1872 5193 1872 x +2738 1872 l +2311 1042 l +2252 936 2142 936 x +2053 936 1987 988 x +1923 1042 1923 1121 x +1923 1140 1944 1199 x +2291 1872 l +5976 fwd_x +end_ol + } def +/LCA { start_ol +3181 2775 m +3283 2975 3444 3067 x +3606 3161 3836 3161 x +4251 3161 4422 2875 x +4593 2589 4593 1797 x +4593 24 l +4075 24 l +4075 1778 l +4075 2428 3992 2585 x +3909 2741 3694 2741 x +3450 2741 3359 2572 x +3269 2404 3269 1778 x +3269 24 l +2751 24 l +2751 1778 l +2751 2433 2662 2587 x +2575 2741 2345 2741 x +2120 2741 2035 2572 x +1949 2404 1949 1778 x +1949 24 l +1432 24 l 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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 0000000..05c3a7f Binary files /dev/null and b/solutions/chap5/transmission_plot.pdf differ 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