From 6a75ede3c79b21d6ca57ccafbdb2b94eaf278852 Mon Sep 17 00:00:00 2001 From: caes Date: Sun, 14 Feb 2016 12:55:35 -0500 Subject: [PATCH] Updated lecture notes up to 2-8 --- lecture_notes/2-8/free particle | 3 + lecture_notes/2-8/observable | 19 + lecture_notes/2-8/translation operator | 54 + lecture_notes/2-8/translation operator.ps | 1216 +++++++++ lecture_notes/2-8/wave mechanics | 35 + lecture_notes/2-8/wave mechanics.ps | 1086 ++++++++ solutions/chap3/prob4/#draft# | 65 + solutions/chap3/prob4/.#draft | 1 + solutions/chap3/prob4/draft | 46 +- solutions/chap3/prob4/draft.ps | 2899 +++++++++++---------- solutions/chap4/draft | 65 + solutions/chap4/draft.ps | 1724 ++++++++++++ 12 files changed, 5864 insertions(+), 1349 deletions(-) create mode 100644 lecture_notes/2-8/free particle create mode 100644 lecture_notes/2-8/observable create mode 100644 lecture_notes/2-8/translation operator create mode 100644 lecture_notes/2-8/translation operator.ps create mode 100644 lecture_notes/2-8/wave mechanics create mode 100644 lecture_notes/2-8/wave mechanics.ps create mode 100644 solutions/chap3/prob4/#draft# create mode 120000 solutions/chap3/prob4/.#draft create mode 100644 solutions/chap4/draft create mode 100644 solutions/chap4/draft.ps diff --git a/lecture_notes/2-8/free particle b/lecture_notes/2-8/free particle new file mode 100644 index 0000000..9400103 --- /dev/null +++ b/lecture_notes/2-8/free particle @@ -0,0 +1,3 @@ +Ĥ = p̂ x²/2m + +In a field Ĥ = p̂²/2m + V̂(x) \ No newline at end of file diff --git a/lecture_notes/2-8/observable b/lecture_notes/2-8/observable new file mode 100644 index 0000000..fd62c6b --- /dev/null +++ b/lecture_notes/2-8/observable @@ -0,0 +1,19 @@ +〈x|Ψ〉 = Ψ(x) + +For an observable A the expectation value is + + 〈A〉 = 〈Ψ|A|Ψ〉 = 〈Ψ|𝟙 A 𝟙|Ψ〉 + + ∞ ∞ + = ∫ ∫ dx dx′ 〈Ψ|x〉 〈x|A|x′〉 〈x′|Ψ〉 + -∞ -∞ + + ∞ + = ∫ dx 〈Ψ|x〉 〈x|A|x〉 〈x|Ψ〉 + -∞ + + ∞ + = ∫ dx a(x) |Ψ(x)|² + -∞ + + \ No newline at end of file diff --git a/lecture_notes/2-8/translation operator b/lecture_notes/2-8/translation operator new file mode 100644 index 0000000..994ae45 --- /dev/null +++ b/lecture_notes/2-8/translation operator @@ -0,0 +1,54 @@ +T̂(a)|x〉 =|x+a〉 + +|Ψ′〉 = T̂(a)|Ψ〉 = T̂(a) ∫ dx′|x′〉〈x′|Ψ〉 + + ∞ + = ∫ dx′|x′+a〉〈x′|Ψ〉 + -∞ + + ∞ + = ∫ δ(x - (x - a′)) Ψ(x′) + -∞ + + = Ψ(x-a) = 〈x|Ψ〉 + +T̂(dx) = 1 - ι/ħ p̂𝓍 dx + +T̂(a) = lim [1 - ι/ħ p𝓍 a/n]ⁿ + n→∞ + + = exp(-ι p𝓍 a/ħ) + +[p] = [m L 1/T] (momentum) + +Commutation Properties +====================== + (x̂ T̂(δx) - T̂ x̂)|Ψ〉 + + ∞ + = (x̂ T̂(δx) - T̂(δx) x̂) ∫ dx 〈x|Ψ〉|x〉 + -∞ + + ∞ ∞ + = ∫ dx x̂ T̂(δx)|x〉〈x|Ψ〉 - ∫ dx T̂(δx) x̂|x〉〈x|Ψ〉 + -∞ -∞ + + ∞ ∞ + = ∫ dx x̂|x+δx〉〈x|Ψ〉 - ∫ dx T̂(δx) x̂|x〉〈x|Ψ〉 + -∞ -∞ + + ∞ ∞ + = ∫ dx (x + δx) 〈x|Ψ〉|x+δx〉 - ∫ dx x 〈x|Ψ〉|x+δx〉 + -∞ -∞ + + = δx|Ψ〉 + + On the other hand... + + x̂ T̂(δx) - T̂\(δx) x̂ = x̂ - ι/ħ δx x̂ p̂𝓍 - x̂ + ι/ħ δx p̂𝓍 x̂ + + = ι/ħ δx [p̂𝓍,x̂] = δx + + [p̂𝓍,x̂] = -ιħ + + [x̂,p̂𝓍] = ιħ (The uncertainty principle) \ No newline at end of file diff --git a/lecture_notes/2-8/translation operator.ps b/lecture_notes/2-8/translation operator.ps new file mode 100644 index 0000000..18227a9 --- /dev/null +++ b/lecture_notes/2-8/translation operator.ps @@ -0,0 +1,1216 @@ +%!PS-Adobe-3.0 +%%Title: translation operator +%%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 725 def +/lmarg 36 def +/ytop 761 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! + /Helvetica findfont setfont 100 100 moveto ( ) show + firstcolumn + end +} def + +/paps_eop { % End of page cleanups + grestore +} def +%%BeginProlog +/papsdict 1 dict def +papsdict begin + +/conicto { + /to_y exch def + /to_x exch def + /conic_cntrl_y exch def + /conic_cntrl_x exch def + currentpoint + /p0_y exch def + /p0_x exch def + /p1_x p0_x conic_cntrl_x p0_x sub 2 3 div mul add def + /p1_y p0_y conic_cntrl_y p0_y sub 2 3 div mul add def + /p2_x p1_x to_x p0_x sub 1 3 div mul add def + /p2_y p1_y to_y p0_y sub 1 3 div mul add def + p1_x p1_y p2_x p2_y to_x to_y curveto +} bind def +/start_ol { gsave } bind def +/end_ol { closepath fill grestore } bind def +/draw_char { fontdict begin gsave 0.001000 dup scale last_x last_y translate load exec end grestore} def +/goto_xy { fontdict begin /last_y exch def /last_x exch def end } def +/goto_x { fontdict begin /last_x exch def end } def +/fwd_x { fontdict begin /last_x exch last_x add def end } def +/c /curveto load def +/x /conicto load def +/l /lineto load def +/m /moveto load def +end +/paps_exec { + 1 dict begin + /ps exch def + /len ps length def + /pos 0 def + + % Loop over all the characters of the string + { + pos len eq {exit} if + + % Get character at pos + /ch ps pos 1 getinterval def + + % check for + + (+) ch eq { + /pos 1 pos add def + /xp ps pos 8 getinterval cvi def + /yp ps pos 8 add 8 getinterval cvi def + /pos 16 pos add def + papsdict begin xp yp goto_xy end + } { + (*) ch eq { + /pos 1 pos add def + /xp ps pos 8 getinterval cvi def + /pos 8 pos add def + papsdict begin xp goto_x end + } { (>) ch eq { + /pos 1 pos add def + /xp ps pos 4 getinterval cvi def + /pos 4 pos add def + papsdict begin xp 2 mul fwd_x end + } { (-) ch eq { + /pos 1 pos add def + /xp ps pos 4 getinterval cvi def + /pos 4 pos add def + papsdict begin xp neg 2 mul fwd_x end + } { + % Must be a 3 char sym. 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} def +/WAA { start_ol +6851 3053 m +5143 1344 l +4614 1873 l +5554 2812 l +387 2812 l +387 3775 l +5554 3775 l +4614 4714 l +5143 5242 l +6851 3534 l +6851 3053 l +7200 fwd_x +end_ol + } def +/XAA { start_ol +3096 -406 m +3096 8849 l +3096 9474 3647 10064 x +4029 10470 4880 10512 x +5666 10548 6501 9869 x +5878 9200 l +5338 9594 4880 9576 x +4176 9546 4176 8686 x +4176 -569 l +4176 -1194 3630 -1784 x +3249 -2190 2397 -2232 x +1609 -2266 776 -1589 x +1398 -920 l +1939 -1314 2397 -1296 x +3096 -1266 3096 -406 x +7200 fwd_x +end_ol + } def +/YAA { start_ol +5040 5679 m +5040 9144 l +6120 9144 l +6120 0 l +5040 0 l +5040 873 l +4770 376 4320 115 x +3870 -144 3282 -144 x +2089 -144 1404 771 x +720 1688 720 3299 x +720 4887 1407 5791 x +2095 6696 3282 6696 x +3876 6696 4329 6435 x +4781 6175 5040 5679 x +1872 3276 m +1872 2045 2270 1418 x +2668 792 3447 792 x +4225 792 4632 1424 x +5040 2056 5040 3276 x +5040 4501 4632 5130 x +4225 5760 3447 5760 x +2668 5760 2270 5133 x +1872 4506 1872 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339800ZAATAACAAAAADAA+ 158400 339800ZAAAAACAATAADAA)paps_exec +(+ 79200 325760SAA)paps_exec +()paps_exec +(+ 79200 297680SAA)paps_exec +(+ 64800 283640FAA+ 79200 283640XAA+ 93600 283640YAAAAA+ 115200 283640CAAZAAAAACAATAADAACAArAA+ 180000 283640FAA+ 194400 283640fAA)paps_exec +(+ 72000 269600bAASAA)paps_exec +paps_eop +showpage +%%Pages: 1 +%%Trailer +%%EOF diff --git a/solutions/chap3/prob4/#draft# b/solutions/chap3/prob4/#draft# new file mode 100644 index 0000000..a624b95 --- /dev/null +++ b/solutions/chap3/prob4/#draft# @@ -0,0 +1,65 @@ +A spin-1/2 particle has a magnetic moment 𝛍 and is placed in a uniform magnetic field 𝐁, which is aligned with the z axis, so 𝐁 = B𝓏 ẑ. It is known that the Hamiltonian operator for this sytem commutes with the spin component operator in the z direction but not with spin component operators in the x and y directions. The student is tasked with demonstrating this prediction. + +The commutator is defined as [û,v̂] = ûv̂-v̂û, where û and v̂ are operators. + +If the product of û and v̂ is independent of order, the commutator will be zero, and the operators can be said to commute. To test whether the Hamiltonian and the spin operators commute, one need express them in a common basis and compute the commutator. In this case, the question is whether the Hamiltonian and spin operators commute in the z direction, so they shall be expressed in the z basis. + +The spin operator in the z basis Ŝ𝓏 is well-known and diagonalized in the z basis because the magnetic field is oriented in the z direction. + + Ŝ𝓏 ≐ ħ/2 ( 1 0 ) + ( 0 -1 ) . + +The Hamiltonian H is the total energy of the system. The only contributing term for this system is the potential energy V = -𝛍⋅𝐁, where 𝛍 and 𝐁 are the physical terms described in the introduction. The magnetic moment of a magnetic dipole, such as that seen with a spin-1/2 particle, is + + 𝛍 = g q/2mₑ 𝐒, + +with 𝐒 the intrinsic spin vector for the case of no orbital angular momentum, and the remaining factors constant, intrinsic values of the particle. Then, + + 𝛍 = k 𝐒, + +The Hamiltonian H is therefore, because 𝐁 = B𝓏 ẑ, + + H = V = -𝛍⋅𝐁 = -k S𝓏 B𝓏. + +The Hamiltonian is measureable, and therefore is an operator. The magnetic field strength is constant in this system, so the Hamiltonian operator is related to the spin operator in the z basis by + + Ĥ = -k B𝓏 Ŝ𝓏. + +Therefore, the Hamiltonian + + Ĥ ≐ -k B𝓏 ħ/2 ( 1 0 ) + ( 0 -1 ) . + +The commutator can now be computed. This computation is included on an attached sheet. The computation indicates that Ŝ𝓏Ĥ = ĤŜ𝓏, and therefore the commutator is zero. The Hamiltonian and the spin operator in the z direction commute. A similar computation for the x and y directions should indicate a lack of commutability. + +The spin operators in the x direction and z direction in the z basis, with ι the imaginary unit, + + Ŝ𝓍 ≐ ħ/2 ( 0 1 ) + ( 1 0 ) + + Ŝ𝓎 ≐ ħ/2 ( 0 -ι ) + ( ι 0 ) + +As with the spin component in the z direction, when multiplying the Hamiltonian and spin in x or spin in y operators, the multiplicative factors associate outside the matrices, and only the matrix multiplication determines the commutability of the operators, so, the operators are commutable if their matrix components are commutable. + +For Ĥ and Ŝ𝓍: + + ( 1 0 ) ( 0 1 ) = ( 0 1 ) + ( 0 -1 ) ( 1 0 ) ( -1 0 ) + + ( 0 1 ) ( 1 0 ) = ( 0 -1 ) + ( 1 0 ) ( 0 -1 ) ( 1 0 ) + +Since these matrix products are not equal, their difference is not zero. Therefore, these operators do not commute. + +For Ĥ and Ŝ𝓎: + + ( 1 0 ) ( 0 -ι ) = ( 0 -ι ) + ( 0 -1 ) ( ι 0 ) ( -ι 0 ) + + ( 0 -ι ) ( 1 0 ) = ( 0 ι ) + ( ι 0 ) ( 0 -1 ) ( ι 0 ) + +Again, these matrix products are not equal, so these operators do not commute. + +As proved earlier, the S𝓏 \ No newline at end of file diff --git a/solutions/chap3/prob4/.#draft b/solutions/chap3/prob4/.#draft new file mode 120000 index 0000000..32d1bbb --- /dev/null +++ b/solutions/chap3/prob4/.#draft @@ -0,0 +1 @@ +caes@calculo-agilis.4326:1455297843 \ No newline at end of file diff --git a/solutions/chap3/prob4/draft b/solutions/chap3/prob4/draft index 0ba2890..5c6b7a5 100644 --- a/solutions/chap3/prob4/draft +++ b/solutions/chap3/prob4/draft @@ -1,21 +1,17 @@ -A spin-1/2 particle has a magnetic moment 𝛍 and is placed in a uniform magnetic field 𝐁, which is aligned with the z axis, so 𝐁 = B𝓏 ẑ. It is known that the Hamiltonian operator for this sytem commutes with the spin component operator in the z direction but not with spin component operators in the x and y directions. The student is tasked with demonstrating this prediction. +A spin-1/2 particle has a magnetic moment 𝛍 and is placed in a uniform magnetic field 𝐁, which is aligned with the z axis, so 𝐁 = B𝓏 ẑ. It is known that the Hamiltonian operator for this system commutes with the spin component operator in the z direction but not with spin component operators in the x and y directions. How and why is this important? -The commutator is defined as [û,v̂] = ûv̂-v̂û, where û and v̂ are operators. +The commutator is defined as [û,v̂] = ûv̂-v̂û, where û and v̂ are operators. -If the product of û and v̂ is independent of order, the commutator will be zero, and the operators can be said to commute. To test whether the Hamiltonian and the spin operators commute, one need express them in a common basis and compute the commutator. In this case, the question is whether the Hamiltonian and spin operators commute in the z direction, so they shall be expressed in the z basis. +If the product of û and v̂ is independent of order, the commutator will be zero, and the operators can be said to commute. To test whether the Hamiltonian and the spin operators commute, one need express them in a common basis and compute the commutator. The z basis should serve as a natural basis for the comparison. -The spin operator in the z basis Ŝ𝓏 is well-known and diagonalized in the z basis because the magnetic field is oriented in the z direction. +The spin operator in the z basis Ŝ𝓏 is diagonalized in the z basis because the magnetic field is oriented in the z direction. This result is known from the Stern-Gerlach experiment. Ŝ𝓏 ≐ ħ/2 ( 1 0 ) ( 0 -1 ) . -The Hamiltonian H is the total energy of the system. The only contributing term for this system is the potential energy V = -𝛍⋅𝐁, where 𝛍 and 𝐁 are the physical terms described in the introduction. The magnetic moment of a magnetic dipole, such as that seen with a spin-1/2 particle, is - - 𝛍 = g q/2mₑ 𝐒, +The Hamiltonian H is the total energy of the system. The only contributing term for this system is the potential energy V = -𝛍⋅𝐁, where 𝛍 and 𝐁 are the physical terms described in the introduction. The magnetic moment of a magnetic dipole, such as that seen with a spin-1/2 particle, is 𝛍 = g q/2mₑ 𝐒, with 𝐒 the intrinsic spin vector for the case of no orbital angular momentum, and the remaining factors constant, intrinsic values of the particle. Then, with -with 𝐒 the intrinsic spin vector for the case of no orbital angular momentum, and the remaining factors constant, intrinsic values of the particle. Then, - - 𝛍 = k 𝐒, + k ≝ g q/2mₑ, 𝛍 = k 𝐒. The Hamiltonian H is therefore, because 𝐁 = B𝓏 ẑ, @@ -28,38 +24,50 @@ The Hamiltonian is measureable, and therefore is an operator. The magnetic field Therefore, the Hamiltonian Ĥ ≐ -k B𝓏 ħ/2 ( 1 0 ) - ( 0 -1 ) . + ( 0 -1 ) . -The commutator can now be computed. This computation is included on an attached sheet. The computation indicates that Ŝ𝓏Ĥ = ĤŜ𝓏, and therefore the commutator is zero. The Hamiltonian and the spin operator in the z direction commute. A similar computation for the x and y directions should indicate a lack of commutability. +The commutator can now be computed. A constant factor of -k B𝓏 ħ²/4 can be collected, leaving only the matrix multiplication to determine commutability. + + [Ĥ,Ŝ𝓏] = Ĥ Ŝ𝓏 - Ŝ𝓏 Ĥ. + + K ≝ -k B𝓏 ħ²/4. + + Ĥ Ŝ𝓏 ≐ K ( 1 0 ) ( 1 0 ) = K ( 1 0 ) + ( 0 -1 ) ( 0 -1 ) ( 0 1 ) . + + Ŝ𝓏 Ĥ ≐ K ( 1 0 ) ( 1 0 ) = K ( 1 0 ) + ( 0 -1 ) ( 0 -1 ) ( 0 1 ) . + + The computation indicates that Ŝ𝓏 Ĥ = Ĥ Ŝ𝓏, and therefore the commutator is zero. The Hamiltonian and the spin operator in the z direction commute. A similar computation for the x and y directions should indicate a lack of commutability. The spin operators in the x direction and z direction in the z basis, with ι the imaginary unit, Ŝ𝓍 ≐ ħ/2 ( 0 1 ) - ( 1 0 ) + ( 1 0 ) . Ŝ𝓎 ≐ ħ/2 ( 0 -ι ) - ( ι 0 ) + ( ι 0 ) . As with the spin component in the z direction, when multiplying the Hamiltonian and spin in x or spin in y operators, the multiplicative factors associate outside the matrices, and only the matrix multiplication determines the commutability of the operators, so, the operators are commutable if their matrix components are commutable. For Ĥ and Ŝ𝓍: ( 1 0 ) ( 0 1 ) = ( 0 1 ) - ( 0 -1 ) ( 1 0 ) ( -1 0 ) + ( 0 -1 ) ( 1 0 ) ( -1 0 ) ; and ( 0 1 ) ( 1 0 ) = ( 0 -1 ) - ( 1 0 ) ( 0 -1 ) ( 1 0 ) + ( 1 0 ) ( 0 -1 ) ( 1 0 ). Since these matrix products are not equal, their difference is not zero. Therefore, these operators do not commute. For Ĥ and Ŝ𝓎: ( 1 0 ) ( 0 -ι ) = ( 0 -ι ) - ( 0 -1 ) ( ι 0 ) ( -ι 0 ) + ( 0 -1 ) ( ι 0 ) ( -ι 0 ) ; and ( 0 -ι ) ( 1 0 ) = ( 0 ι ) - ( ι 0 ) ( 0 -1 ) ( ι 0 ) + ( ι 0 ) ( 0 -1 ) ( ι 0 ). Again, these matrix products are not equal, so these operators do not commute. - +When an observable, such as the spin component operator in the z direction Ŝ𝓏, commutes with the Hamiltonian, it is referred to as a constant of motion. Such a quantity has specific values its measurement will consistently result in. 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+5184 0 l +5184 4176 l +2016 4176 l +2016 0 l +792 0 l +792 8784 l +7200 fwd_x end_ol } def /LBA { start_ol -1584 10152 m -7560 10152 l -7560 9000 l -2952 9000 l -2952 6048 l -7128 6048 l -7128 4896 l -2952 4896 l -2952 0 l -1584 0 l -1584 10152 l -8424 fwd_x +2776 4608 m +5184 4608 l +5184 3960 l +1872 3960 l +1872 4604 l +3186 5927 l +3775 6487 4011 6840 x +4248 7193 4248 7488 x +4248 7841 3955 8060 x +3664 8280 3186 8280 x +2899 8280 2569 8190 x +2239 8100 1872 7920 x +1872 8568 l +2256 8748 2615 8838 x +2974 8928 3323 8928 x +4164 8928 4674 8541 x +5184 8154 5184 7531 x +5184 7200 4957 6853 x +4732 6508 4008 5827 x +2776 4608 l +7200 fwd_x end_ol } def /MBA { start_ol -3312 7200 m -5040 7200 l -5040 5112 l -3312 5112 l -3312 7200 l -3312 2088 m -5040 2088 l -5040 0 l -3312 0 l -3312 2088 l -8424 fwd_x +4248 7695 m +1515 3024 l +4248 3024 l +4248 7695 l +4098 8784 m +5472 8784 l +5472 3024 l +6624 3024 l +6624 2088 l +5472 2088 l +5472 0 l +4248 0 l +4248 2088 l +576 2088 l +576 3213 l +4098 8784 l +7200 fwd_x +end_ol + } def +/NBA { start_ol +792 8784 m +2016 8784 l +2016 4876 l +5720 8784 l +7112 8784 l +3701 5195 l +7212 0 l +5780 0 l +2920 4401 l +2016 3435 l +2016 0 l +792 0 l +792 8784 l +7200 fwd_x +end_ol + } def +/OBA { start_ol +4104 6552 m +4104 2274 l +4104 1418 4305 1147 x +4519 864 5160 864 x +5688 864 l +5688 0 l +5037 0 l +3984 0 3504 559 x +3029 1129 3024 2363 x +3024 5688 l +1800 5688 l +1800 6552 l +4104 6552 l +7200 fwd_x +end_ol + } def +/PBA { start_ol +4960 4140 m +5224 4140 5391 3974 x +5557 3810 5557 3542 x +5557 3035 5063 3035 x +4800 3035 4685 3197 x +4570 3359 4552 3521 x +4536 3683 4498 3683 x +4362 3683 4008 3174 x +3655 2667 3481 2305 x +3481 1892 3332 1190 x +3182 488 3182 406 x +3182 144 3378 144 x +3570 144 3774 257 x +3978 372 4278 696 x +4578 1021 4692 1159 x +4806 1298 5178 1779 x +5270 1896 5315 1955 x +5465 1861 l +5407 1792 5161 1459 x +4915 1126 4818 1003 x +4722 882 4480 618 x +4240 355 4077 245 x +3915 136 3685 31 x 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m +4464 1800 l +4464 587 l +3312 -1656 l +2376 -1656 l +2952 587 l +2952 1800 l +2880 6192 m +4392 6192 l +4392 4392 l +2880 4392 l +2880 6192 l +7200 fwd_x +end_ol + } def +/UBA { start_ol +0 8784 m +1162 8784 l +2004 1662 l +3008 6408 l +4252 6408 l +5267 1650 l +6109 8784 l +7272 8784 l +5956 0 l +4830 0 l +3632 5248 l +2441 0 l +1315 0 l +0 8784 l +7200 fwd_x end_ol } def end end @@ -1618,116 +1851,122 @@ end end %%EndPrologue %%Page: 1 1 paps_bop -(+ 36000 744656AAA+ 52848 744656CAADAAEAAFAAGAAHAAIAAJAA+ 128664 744656DAAKAALAAMAAEAANAAOAAPAA+ 204480 744656QAAKAACAA+ 238176 744656KAA+ 255024 744656RAAKAASAAFAAPAAMAAEAANAA+ 330840 744656RAATAARAAPAAFAAMAA+ 389808 744656UAA+ 401832 744656KAAFAAWAA+ 435528 744656EAACAA+ 460800 744656DAAOAAKAANAAPAAWAA+ 519768 744656EAAFAA)paps_exec -(+ 36000 728312KAA+ 52848 728312YAAFAAEAAZAATAALAARAA+ 120240 728312RAAKAASAAFAAPAAMAAEAANAA+ 196056 728312ZAAEAAPAAOAAWAA+ 246600 728312aAAbAA+ 272808 728312cAAQAAEAANAAQAA+ 323352 728312EAACAA+ 348624 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417600 676760tAAKAALAAnAAjAA)paps_exec paps_eop showpage %%Pages: 3 diff --git a/solutions/chap4/draft b/solutions/chap4/draft new file mode 100644 index 0000000..0ba2890 --- /dev/null +++ b/solutions/chap4/draft @@ -0,0 +1,65 @@ +A spin-1/2 particle has a magnetic moment 𝛍 and is placed in a uniform magnetic field 𝐁, which is aligned with the z axis, so 𝐁 = B𝓏 ẑ. It is known that the Hamiltonian operator for this sytem commutes with the spin component operator in the z direction but not with spin component operators in the x and y directions. The student is tasked with demonstrating this prediction. + +The commutator is defined as [û,v̂] = ûv̂-v̂û, where û and v̂ are operators. + +If the product of û and v̂ is independent of order, the commutator will be zero, and the operators can be said to commute. To test whether the Hamiltonian and the spin operators commute, one need express them in a common basis and compute the commutator. In this case, the question is whether the Hamiltonian and spin operators commute in the z direction, so they shall be expressed in the z basis. + +The spin operator in the z basis Ŝ𝓏 is well-known and diagonalized in the z basis because the magnetic field is oriented in the z direction. + + Ŝ𝓏 ≐ ħ/2 ( 1 0 ) + ( 0 -1 ) . + +The Hamiltonian H is the total energy of the system. The only contributing term for this system is the potential energy V = -𝛍⋅𝐁, where 𝛍 and 𝐁 are the physical terms described in the introduction. The magnetic moment of a magnetic dipole, such as that seen with a spin-1/2 particle, is + + 𝛍 = g q/2mₑ 𝐒, + +with 𝐒 the intrinsic spin vector for the case of no orbital angular momentum, and the remaining factors constant, intrinsic values of the particle. Then, + + 𝛍 = k 𝐒, + +The Hamiltonian H is therefore, because 𝐁 = B𝓏 ẑ, + + H = V = -𝛍⋅𝐁 = -k S𝓏 B𝓏. + +The Hamiltonian is measureable, and therefore is an operator. The magnetic field strength is constant in this system, so the Hamiltonian operator is related to the spin operator in the z basis by + + Ĥ = -k B𝓏 Ŝ𝓏. + +Therefore, the Hamiltonian + + Ĥ ≐ -k B𝓏 ħ/2 ( 1 0 ) + ( 0 -1 ) . + +The commutator can now be computed. This computation is included on an attached sheet. The computation indicates that Ŝ𝓏Ĥ = ĤŜ𝓏, and therefore the commutator is zero. The Hamiltonian and the spin operator in the z direction commute. A similar computation for the x and y directions should indicate a lack of commutability. + +The spin operators in the x direction and z direction in the z basis, with ι the imaginary unit, + + Ŝ𝓍 ≐ ħ/2 ( 0 1 ) + ( 1 0 ) + + Ŝ𝓎 ≐ ħ/2 ( 0 -ι ) + ( ι 0 ) + +As with the spin component in the z direction, when multiplying the Hamiltonian and spin in x or spin in y operators, the multiplicative factors associate outside the matrices, and only the matrix multiplication determines the commutability of the operators, so, the operators are commutable if their matrix components are commutable. + +For Ĥ and Ŝ𝓍: + + ( 1 0 ) ( 0 1 ) = ( 0 1 ) + ( 0 -1 ) ( 1 0 ) ( -1 0 ) + + ( 0 1 ) ( 1 0 ) = ( 0 -1 ) + ( 1 0 ) ( 0 -1 ) ( 1 0 ) + +Since these matrix products are not equal, their difference is not zero. Therefore, these operators do not commute. + +For Ĥ and Ŝ𝓎: + + ( 1 0 ) ( 0 -ι ) = ( 0 -ι ) + ( 0 -1 ) ( ι 0 ) ( -ι 0 ) + + ( 0 -ι ) ( 1 0 ) = ( 0 ι ) + ( ι 0 ) ( 0 -1 ) ( ι 0 ) + +Again, these matrix products are not equal, so these operators do not commute. + + diff --git a/solutions/chap4/draft.ps b/solutions/chap4/draft.ps new file mode 100644 index 0000000..49337d4 --- /dev/null +++ b/solutions/chap4/draft.ps @@ -0,0 +1,1724 @@ +%!PS-Adobe-3.0 +%%Title: draft +%%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 725 def +/lmarg 36 def +/ytop 761 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! + /Helvetica findfont setfont 100 100 moveto ( ) show + firstcolumn + end +} def + +/paps_eop { % End of page cleanups + grestore +} def +%%BeginProlog +/papsdict 1 dict def +papsdict begin + +/conicto { + /to_y exch def + /to_x exch def + /conic_cntrl_y exch def + /conic_cntrl_x exch def + currentpoint + /p0_y exch def + /p0_x exch def + /p1_x p0_x conic_cntrl_x p0_x sub 2 3 div mul add def + /p1_y p0_y conic_cntrl_y p0_y sub 2 3 div mul add def + /p2_x p1_x to_x p0_x sub 1 3 div mul add def + /p2_y p1_y to_y p0_y sub 1 3 div mul add def + p1_x p1_y p2_x p2_y to_x to_y curveto +} bind def +/start_ol { gsave } bind def +/end_ol { closepath fill grestore } bind def +/draw_char { fontdict begin gsave 0.001000 dup scale last_x last_y translate load exec end grestore} def +/goto_xy { fontdict begin /last_y exch def /last_x exch def end } def +/goto_x { fontdict begin /last_x exch def end } def +/fwd_x { fontdict begin /last_x exch last_x add def end } def +/c /curveto load def +/x /conicto load def +/l /lineto load def +/m /moveto load def +end +/paps_exec { + 1 dict begin + /ps exch def + /len ps length def + /pos 0 def + + % Loop over all the characters of the string + { + pos len eq {exit} if + + % Get character at pos + /ch ps pos 1 getinterval def + + % check for + + (+) ch eq { + /pos 1 pos add def + /xp ps pos 8 getinterval cvi def + /yp ps pos 8 add 8 getinterval cvi def + /pos 16 pos add def + papsdict begin xp yp goto_xy end + } { + (*) ch eq { + /pos 1 pos add def + /xp ps pos 8 getinterval cvi def + /pos 8 pos add def + papsdict begin xp goto_x end + } { (>) ch eq { + /pos 1 pos add def + /xp ps pos 4 getinterval cvi def + /pos 4 pos add def + papsdict begin xp 2 mul fwd_x end + } { (-) ch eq { + /pos 1 pos add def + /xp ps pos 4 getinterval cvi def + /pos 4 pos add def + papsdict begin xp neg 2 mul fwd_x end + } { + % Must be a 3 char sym. 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+/EBA { start_ol +5448 2023 m +5448 1730 l +2500 1730 l +2500 1713 l +2500 1108 2853 779 x +3205 451 3852 451 x +4174 451 4529 541 x +4884 632 5289 820 x +5289 221 l +4902 81 4541 10 x +4180 -59 3845 -59 x +2876 -59 2333 453 x +1791 967 1791 1872 x +1791 2751 2322 3277 x +2853 3803 3746 3803 x +4532 3803 4990 3327 x +5448 2851 5448 2023 x +4767 2206 m +4755 2734 4485 3013 x +4215 3292 3715 3292 x +3223 3292 2902 3004 x +2583 2716 2524 2206 x +4767 2206 l +7200 fwd_x +end_ol + } def +/FBA { start_ol +3162 8281 m +3740 8281 4366 8088 x +4992 7896 5077 7896 x +5227 7896 5308 7984 x +5389 8071 5447 8293 x +5808 8293 l +5808 5760 l +5472 5760 l +5125 6869 4509 7394 x +3892 7920 3162 7920 x +2543 7920 2184 7594 x +1827 7269 1827 6716 x +1827 6283 2088 5999 x +2349 5717 3076 5356 x +4856 4477 l +5413 4188 5773 3574 x +6134 2961 6134 2287 x +6134 1172 5320 478 x +4505 -216 3186 -216 x +2522 -216 1865 -29 x +1208 157 1085 157 x +808 157 708 -216 x +360 -216 l +360 2665 l +708 2665 l +1350 145 3223 145 x +3927 145 4339 517 x +4752 891 4752 1517 x +4752 1986 4473 2287 x +4194 2588 3393 2961 x +2579 3346 l +1487 3864 995 4459 x +504 5055 504 5850 x +504 6987 1213 7634 x +1923 8281 3162 8281 x +6696 fwd_x +end_ol + } def +/GBA { start_ol +5904 8496 m +5904 7272 l +5368 7628 4829 7809 x +4290 7992 3744 7992 x +2910 7992 2426 7599 x +1944 7207 1944 6541 x +1944 5958 2259 5650 x +2574 5344 3435 5139 x +4047 4998 l +5280 4711 5844 4095 x +6408 3479 6408 2418 x +6408 1170 5639 513 x +4871 -144 3404 -144 x +2793 -144 2175 -18 x +1559 108 936 360 x +936 1656 l +1600 1208 2193 1000 x +2786 792 3388 792 x +4272 792 4764 1197 x +5256 1603 5256 2331 x +5256 2994 4917 3343 x +4579 3693 3741 3882 x +3117 4028 l +1895 4303 1343 4858 x +792 5414 792 6349 x +792 7519 1579 8223 x +2367 8928 3673 8928 x +4177 8928 4732 8820 x +5288 8712 5904 8496 x +7200 fwd_x +end_ol + } def +/HBA { start_ol +3043 11160 m +4156 11160 l +5400 9576 l +4575 9576 l +3596 10635 l +2624 9576 l +1800 9576 l +3043 11160 l +792 8784 m +2016 8784 l +2016 5184 l +5184 5184 l +5184 8784 l +6408 8784 l +6408 0 l +5184 0 l +5184 4176 l +2016 4176 l +2016 0 l +792 0 l +792 8784 l +7200 fwd_x +end_ol + } def +/IBA { start_ol +4104 6552 m +4104 2274 l +4104 1418 4305 1147 x +4519 864 5160 864 x +5688 864 l +5688 0 l +5037 0 l +3984 0 3504 559 x +3029 1129 3024 2363 x +3024 5688 l +1800 5688 l +1800 6552 l +4104 6552 l +7200 fwd_x +end_ol + } def +/JBA { start_ol +4960 4140 m +5224 4140 5391 3974 x +5557 3810 5557 3542 x +5557 3035 5063 3035 x +4800 3035 4685 3197 x +4570 3359 4552 3521 x +4536 3683 4498 3683 x +4362 3683 4008 3174 x +3655 2667 3481 2305 x +3481 1892 3332 1190 x +3182 488 3182 406 x +3182 144 3378 144 x +3570 144 3774 257 x +3978 372 4278 696 x +4578 1021 4692 1159 x +4806 1298 5178 1779 x +5270 1896 5315 1955 x +5465 1861 l +5407 1792 5161 1459 x +4915 1126 4818 1003 x +4722 882 4480 618 x +4240 355 4077 245 x +3915 136 3685 31 x +3457 -72 3240 -72 x +2376 -72 2099 1510 x +1245 -72 560 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367880ABA+ 280800 367880BBA)paps_exec +()paps_exec +(+ 36000 339800GBAEAAFAANAAPAA+ 79200 339800MAAQAAPAACAAPAA+ 122400 339800RAAKAAMAALAAEAAeAA+ 172800 339800DAALAATAAWAAXAANAAMAACAA+ 237600 339800KAALAAPAA+ 266400 339800FAATAAMAA+ 295200 339800PAAvAAXAAKAAOAAbAA+ 345600 339800MAAQAAPAAEAALAA+ 388800 339800WAAEAAYAAYAAPAALAAPAAFAANAAPAA+ 468000 339800EAACAA+ 489600 339800FAATAAMAA)paps_exec +(+ 36000 325760dAAPAALAATAAjAA+ 79200 325760pAAQAAPAALAAPAAYAATAALAAPAAbAA+ 158400 325760MAAQAAPAACAAPAA+ 201600 325760TAADAAPAALAAKAAMAATAALAACAA+ 273600 325760WAATAA+ 295200 325760FAATAAMAA+ 324000 325760NAATAARAARAAXAAMAAPAAjAA)paps_exec +()paps_exec +(+ 36000 297680LBATAALAA+ 64800 297680HBA+ 79200 297680KAAFAAWAA+ 108000 297680wAAKBAMBA)paps_exec +()paps_exec +(+ 64800 269600zAA+ 79200 269600HAA+ 100800 269600ABA+ 115200 269600BBA+ 129600 269600zAA+ 144000 269600ABA+ 158400 269600GAAIBA+ 180000 269600BBA+ 201600 269600fAA+ 223200 269600zAA+ 237600 269600ABA+ 259200 269600GAAIBA+ 280800 269600BBA)paps_exec +(+ 64800 255560zAA+ 79200 255560ABA+ 93600 255560GAAHAA+ 115200 255560BBA+ 129600 255560zAA+ 144000 255560IBA+ 165600 255560ABA+ 180000 255560BBA+ 223200 255560zAA+ 237600 255560GAAIBA+ 266400 255560ABA+ 280800 255560BBA)paps_exec +()paps_exec +(+ 64800 227480zAA+ 79200 227480ABA+ 93600 227480GAAIBA+ 115200 227480BBA+ 129600 227480zAA+ 144000 227480HAA+ 165600 227480ABA+ 180000 227480BBA+ 201600 227480fAA+ 223200 227480zAA+ 237600 227480ABA+ 259200 227480IBA+ 273600 227480BBA)paps_exec +(+ 64800 213440zAA+ 79200 213440IBA+ 100800 213440ABA+ 115200 213440BBA+ 129600 213440zAA+ 144000 213440ABA+ 158400 213440GAAHAA+ 180000 213440BBA+ 223200 213440zAA+ 237600 213440IBA+ 259200 213440ABA+ 273600 213440BBA)paps_exec +()paps_exec +(+ 36000 185360AAASAAKAAEAAFAAbAA+ 86400 185360MAAQAAPAACAAPAA+ 129600 185360RAAKAAMAALAAEAAeAA+ 180000 185360DAALAATAAWAAXAANAAMAACAA+ 244800 185360KAALAAPAA+ 273600 185360FAATAAMAA+ 302400 185360PAAvAAXAAKAAOAAbAA+ 352800 185360CAATAA+ 374400 185360MAAQAAPAACAAPAA+ 417600 185360TAADAAPAALAAKAAMAATAALAACAA+ 489600 185360WAATAA+ 511200 185360FAATAAMAA)paps_exec +(+ 36000 171320NAATAARAARAAXAAMAAPAAjAA)paps_exec +()paps_exec +()paps_exec +paps_eop +showpage +%%Pages: 2 +%%Trailer +%%EOF