commit 4eb811263c19fa511950a5d4a02fc3ce37f8fe64 Author: caes Date: Fri Feb 12 12:26:39 2016 -0500 Initialization diff --git a/Postulates b/Postulates new file mode 100644 index 0000000..97e0aab --- /dev/null +++ b/Postulates @@ -0,0 +1,14 @@ +(1) A general state vector is represented by |Ψ> (a ket) + +(2) A physical observable is represented by an operator A that acts on |Ψ> + +(3) The possible measurements of an observable are the eigenvalues aₙ of the operator A + +(4) The probability of measuring eigenvalue aₙ is given by Paₙ=||² + +(5) After a measurement A that yields aₙ, the quantum system is in a new state that is the normalized projection of the original system ket onto the ket (or kets) corresponding to the result of the measurement: + |Ψ'> = Pn|Ψ> / √(<Ψ|Pn|Ψ>) + +(6) The time-evolution of the quantum system is determined by the Hamiltonian operator H(t) through the Schrodinger equation + ιħ(d/dt)|Ψ(t)> = H(t)|Ψ(t)> + diff --git a/book_notes/chap2/S² operator b/book_notes/chap2/S² operator new file mode 100644 index 0000000..00bd8d1 --- /dev/null +++ b/book_notes/chap2/S² operator @@ -0,0 +1,10 @@ +𝐒² = S𝓍² + S𝓎² + S𝓏² + +𝐒²|Ψ〉 = ¾ħ²|Ψ〉 + +〈𝐒〉 = ¾ħ² + +|𝐒| = √〈𝐒²〉 = √(¾)ħ + +Vector Model, pg. 58 + diff --git a/book_notes/chap2/commutators b/book_notes/chap2/commutators new file mode 100644 index 0000000..a7fd457 --- /dev/null +++ b/book_notes/chap2/commutators @@ -0,0 +1,5 @@ +The commutator of two operators A and B is notated as and defined to be + [A,B] = AB - BA . + +A and B commute when [A,B] = 0, i.e. AB = BA. + diff --git a/book_notes/chap2/operators b/book_notes/chap2/operators new file mode 100644 index 0000000..b042708 --- /dev/null +++ b/book_notes/chap2/operators @@ -0,0 +1,35 @@ +Eigenvalue equations for the S𝓏 operator in a spin-1/2 system: + S𝓏|+> = +ħ/2|+> + S𝓏|-> = -ħ/2|-> + +Matrix form of an operator: + S𝓏 ≐ ( a b ) + ( c d ) + +Eigenvalue equations in matrix form: + + ( a b )( 1 ) = +ħ/2 ( 1 ) + ( c d )( 0 ) ( 0 ) + + ( a b )( 0 ) = -ħ/2 ( 0 ) + ( c d )( 1 ) ( 1 ) + +Can show with matrix multiplication operations that + S𝓏 ≐ ħ/2 ( 1 0 ) + ( 0 -1 ) + +Some Properties: + An operator is always diagonal in its own basis. + Eigenvectors are unit vectirs in their own basis. + + S𝓏 ≐ ħ/2 ( 1 0 ) |+> = ( 1 ) |-> = ( 0 ) + ( 0 -1 ) ( 0 ) ( 1 ) + + +Diagonalizing an Operator + Diagonize a matrix --> find the eigenvalues and eigenvector + + S𝓍 ≐ ħ/2 ( 0 1 ) + ( 1 0 ) + + (1) 1⁄2 ⁵⁄ₐ \ No newline at end of file diff --git a/book_notes/chap2/spin-1 system b/book_notes/chap2/spin-1 system new file mode 100644 index 0000000..d4c3cee --- /dev/null +++ b/book_notes/chap2/spin-1 system @@ -0,0 +1,17 @@ +Eigenvalue Equations for Spin-1 System: + + 𝐒𝓏|1〉 = ħ|1〉 + 𝐒𝓏|0〉 = 0ħ|0〉 + 𝐒𝓏|-1〉 = ħ|-1〉 + + ħ ( 1 0 0 ) +S𝓏 ≐ --- ( 0 1 0 ) + √2 ( 0 0 1 ) + + ħ ( 0 1 0 ) +S𝓍 ≐ --- ( 1 0 1 ) + √2 ( 0 1 0 ) + + ħ ( 0 -ι 0 ) +S𝓎 ≐ --- ( ι 0 -ι ) + √2 ( 0 ι 0 ) diff --git a/book_notes/chap3/Bohr Frequency b/book_notes/chap3/Bohr Frequency new file mode 100644 index 0000000..5789ce2 --- /dev/null +++ b/book_notes/chap3/Bohr Frequency @@ -0,0 +1,14 @@ +Ψ(t) is the time-revolving state function, + Ψ(t) = c₁ exp(-ι E₁/ħ t) + c₂ exp(-ι E₂/ħ t) + ... + +The eigenstate of A corresponding to the eigenvalue a₁ + |a₁〉 = α₁|E₁〉 + α₂|E₂〉 + +The probability of measuring the eigenvalue a₁ is + P(a₁) = |〈a₁|Ψ(t)〉|² + ↓ Derivation on page 71 + P(a₁) = |α₁|²|c₁|² + |α₂|²|c₂|² + Re(α₁c₁⃰α₂⃰c₂ exp(-ι (E₂-E₁)/ħ t)) + +The Bohy Frequency (angular frequency) of two states E₁ and E₂ is thus revealed: + ω₂₁ = (E₂ - E₁)/ħ + diff --git a/book_notes/chap3/Time-Independent Hamiltonian Recipe b/book_notes/chap3/Time-Independent Hamiltonian Recipe new file mode 100644 index 0000000..fd83090 --- /dev/null +++ b/book_notes/chap3/Time-Independent Hamiltonian Recipe @@ -0,0 +1,7 @@ +(1) Diagonalize H (find the eigenvalues of Eₙ and eigenvectors |Eₙ〉). + +(2) Write |Ψ(0)〉 in terms of the energy eigenstates |Eₙ〉. + +(3) Multiple each eigenstate coefficient by exp(-ι Eₙ/ħ t) to get |Ψ(t)〉. + +(4) Calculate the probability P(aⱼ) = |〈aⱼ|Ψ(t)〉|² diff --git a/lecture_notes/1-11/Determing Coefficients.jpg b/lecture_notes/1-11/Determing Coefficients.jpg new file mode 100644 index 0000000..b1fea7a Binary files /dev/null and b/lecture_notes/1-11/Determing Coefficients.jpg differ diff --git a/lecture_notes/1-11/Eigenstates.jpg b/lecture_notes/1-11/Eigenstates.jpg new file mode 100644 index 0000000..cd91906 Binary files /dev/null and b/lecture_notes/1-11/Eigenstates.jpg differ diff --git a/lecture_notes/1-11/Heisenberg b/lecture_notes/1-11/Heisenberg new file mode 100644 index 0000000..a04ee24 --- /dev/null +++ b/lecture_notes/1-11/Heisenberg @@ -0,0 +1,29 @@ +Talked about Movie. + +Ket vectors/ Basis using hero/villain (pic) + +Orthogonal Basis using Bra Ket notation + +Orthogonal basis + + < villain | villain > = < hero | hero > = 1 + +Determining coefficients using projections in braket notation (pic) + +Probability from complex numbers + + C_v* C_v = |C_v|^2 + +Playing around with braket notation + +---------------------------------------------------- + +Observables and Operators + + Eigen-kets, or eigen-states (pic) + + Operators change the state when applied to a state + + Operators refer to measurements (pic) + + \ No newline at end of file diff --git a/lecture_notes/1-11/Hero-Villian Basis.jpg b/lecture_notes/1-11/Hero-Villian Basis.jpg new file mode 100644 index 0000000..d2a3ca0 Binary files /dev/null and b/lecture_notes/1-11/Hero-Villian Basis.jpg differ diff --git a/lecture_notes/1-11/Operators.jpg b/lecture_notes/1-11/Operators.jpg new file mode 100644 index 0000000..de0c571 Binary files /dev/null and b/lecture_notes/1-11/Operators.jpg differ diff --git a/lecture_notes/1-13/Basis b/lecture_notes/1-13/Basis new file mode 100644 index 0000000..55a0a8f --- /dev/null +++ b/lecture_notes/1-13/Basis @@ -0,0 +1,14 @@ +Quantum State Basis +------------------- + + |Ψ> = C_+ |+> + C_- |-> with |C_+|^2 + |C_-|^2 = 1 + + Example (pic) + + Spin in x, y, and z (pic) + + Coefficients in general + |+x> = C_+ |+z> + C_- |+z> + C_+ = e^(i δ_+) / √2 , C_- = e^(i δ_-) / √2 + + <+x| S_z |+x> = = expectation value \ No newline at end of file diff --git a/lecture_notes/1-13/Stern-Gerlach b/lecture_notes/1-13/Stern-Gerlach new file mode 100644 index 0000000..52ea949 --- /dev/null +++ b/lecture_notes/1-13/Stern-Gerlach @@ -0,0 +1,10 @@ +Explanation of STern-Gerlach Experiment (pic) +============================================= + + - Electrons have intrinsic magnetic moment + + Showed how magnetic moment is related to angular momentum (pic) + + mu-bar_s = g q /2 /m /c S-bar g is measured for any particle, S-bar is spin angular momentum + + S_z = ± h-bar / 2 diff --git a/lecture_notes/1-15/Applications b/lecture_notes/1-15/Applications new file mode 100644 index 0000000..59a1016 --- /dev/null +++ b/lecture_notes/1-15/Applications @@ -0,0 +1,19 @@ +Revisited Stern-Gerlach setup, now using three machines in series. + +Introduced incompatible operators. + +Reintroduced normalized and phased notation, using δ. (pic) + +Combined y and x to view result of <+y|+x> + + +Talked about projections on other axes (pic x2) + +Practice problem (pic) performed partially in class + + what is the probability that the S_y will be measured to be +ħ∕2 + Should be .93 + To find, calculate <+y|Ψ> to find probability. + + What is ? I.e. the expecation value for S_y? + <Ψ|S_y|Ψ> gives expectation value, or use <+y|Ψ> and <-y|Ψ> to find coefficients. (pic) diff --git a/lecture_notes/1-22/IMG_20160122_130917.jpg b/lecture_notes/1-22/IMG_20160122_130917.jpg new file mode 100644 index 0000000..fe239aa Binary files /dev/null and b/lecture_notes/1-22/IMG_20160122_130917.jpg differ diff --git a/lecture_notes/1-22/IMG_20160122_132503.jpg b/lecture_notes/1-22/IMG_20160122_132503.jpg new file mode 100644 index 0000000..186c1a5 Binary files /dev/null and b/lecture_notes/1-22/IMG_20160122_132503.jpg differ diff --git a/lecture_notes/1-22/IMG_20160122_132757.jpg b/lecture_notes/1-22/IMG_20160122_132757.jpg new file mode 100644 index 0000000..e08ecd9 Binary files /dev/null and b/lecture_notes/1-22/IMG_20160122_132757.jpg differ diff --git a/lecture_notes/1-22/IMG_20160122_133259.jpg b/lecture_notes/1-22/IMG_20160122_133259.jpg new file mode 100644 index 0000000..3a55c24 Binary files /dev/null and b/lecture_notes/1-22/IMG_20160122_133259.jpg differ diff --git a/lecture_notes/1-22/IMG_20160122_133907.jpg b/lecture_notes/1-22/IMG_20160122_133907.jpg new file mode 100644 index 0000000..83756e8 Binary files /dev/null and b/lecture_notes/1-22/IMG_20160122_133907.jpg differ diff --git a/lecture_notes/1-22/IMG_20160122_133909.jpg b/lecture_notes/1-22/IMG_20160122_133909.jpg new file mode 100644 index 0000000..1daedee Binary files /dev/null and b/lecture_notes/1-22/IMG_20160122_133909.jpg differ diff --git a/lecture_notes/1-22/IMG_20160122_134436.jpg b/lecture_notes/1-22/IMG_20160122_134436.jpg new file mode 100644 index 0000000..ffc3242 Binary files /dev/null and b/lecture_notes/1-22/IMG_20160122_134436.jpg differ diff --git a/lecture_notes/1-22/IMG_20160122_135204.jpg b/lecture_notes/1-22/IMG_20160122_135204.jpg new file mode 100644 index 0000000..db50f98 Binary files /dev/null and b/lecture_notes/1-22/IMG_20160122_135204.jpg differ diff --git a/lecture_notes/1-22/Matrix Forms b/lecture_notes/1-22/Matrix Forms new file mode 100644 index 0000000..695859a --- /dev/null +++ b/lecture_notes/1-22/Matrix Forms @@ -0,0 +1,15 @@ +States can be represented as vectors + +|Ψ> = ½|+> + (i√3)/2|-> = ½ ( 1 ) + (i√3)/2 ( 0 ) = ( 1/2 ) + ( 0 ) ( 1 ) ( (i√3)/2 ) + +Operators are matrices + +Showed how Â|Ψ>=|φ> translates to a matrix in the z basis (pic x2) + +Example with S𝓏 operator (pic) +Example with S𝓍 operator (pic) + +Often we know operator but not basis. How to find basis? (pic) + i.e. find the eigen-basis S|α>=λ|α> + Only non-trivial if determinant is zero. Solve for basis. (pic) diff --git a/lecture_notes/1-27/Expectation Value Time Dependence b/lecture_notes/1-27/Expectation Value Time Dependence new file mode 100644 index 0000000..1811127 --- /dev/null +++ b/lecture_notes/1-27/Expectation Value Time Dependence @@ -0,0 +1,13 @@ +Computed Derivative (pic) d/dt (pic) + +Discovered/Introduced Commutator +Cpmpatible Observables occur when commutator is equal to 0. + +[Ĥ,Â] = Ĥ - ÂĤ + +Constant of Motion +------------------ + if d/dt  = 0 and [Ĥ,Â] = 0, + then  is a constant of motion + + Thm: Two operators such that [Â,B̂] always have common Eigenstates diff --git a/lecture_notes/1-27/Spin-Half Precession b/lecture_notes/1-27/Spin-Half Precession new file mode 100644 index 0000000..c423a8c --- /dev/null +++ b/lecture_notes/1-27/Spin-Half Precession @@ -0,0 +1,20 @@ +Precession of a Spin-1/2 Particle in a Constant Magnetic Field +============================================================== + Find Hamiltoninan (pic) + Find eigenstates of the Hamiltonian (pic) + Find time-progression expression + + Û(t) = exp(-ι Ĥ t/ħ) = exp(-ι ωₒt S𝓏/ħ) = exp(-ι θ S𝓏/ħ) + Û(t) = R̂(σk̂) + ⸜ ⸝ + Rotational Operator + + So for arbitrary t (pic) + + + Projection along 𝓏-axis? <+𝓏|Ψ(t)> + +--->Why for (-x)-axis does this turn out to be sin²(ω₀ t/2) + +Hw->Find projection of 𝓎-axis + diff --git a/lecture_notes/1-27/Time-Independent SE b/lecture_notes/1-27/Time-Independent SE new file mode 100644 index 0000000..5954c1e --- /dev/null +++ b/lecture_notes/1-27/Time-Independent SE @@ -0,0 +1,14 @@ +Time-Independent Schrodinger Equation + +Û(t) = lim [1 - ι/ħ H t/n] ^n (pic) + n→∞ + +Û(t) = exp(-ι Ĥ t/n) + +|Ψ(t)> = exp(-ι Ĥ t/n) + +Ĥ|E> = E|E> + +exp(-ι Ĥ t/n) |E> = exp(-ι E t/n) |E> (pic) + +|E> = ? diff --git a/lecture_notes/1-28/Work a Hamiltonian Problem b/lecture_notes/1-28/Work a Hamiltonian Problem new file mode 100644 index 0000000..8cec1f3 --- /dev/null +++ b/lecture_notes/1-28/Work a Hamiltonian Problem @@ -0,0 +1,18 @@ +Magnetic Resonance Problem (pic 1) + + This problem has time dependence (pic 2) + +Time-dependent Ψ? (pic 3) + +Plug in to Schrodinger Equation (pic 3) + +Solve S.E. with unknown functions. (pic 4,5) + Plug back into S.E. from matrices + ↓ + Nice concise equation system (pic 6) + ↓ Time derivative of both sides (pic 7) + Final Solution? Yes, and apply initial conditions. + + Minus sign is switched somewhere in these pictures + + \ No newline at end of file diff --git a/notes/eigenstates b/notes/eigenstates new file mode 100644 index 0000000..89eb12c --- /dev/null +++ b/notes/eigenstates @@ -0,0 +1,22 @@ +A(x) is a real-space vector +Ψ(x) is a wave function +A acts on Ψ + +Some Ψ, when acted upon by A, result in a multiple of the original Ψ. These Ψ are called eigenstates and may be denoted Ψₐ. In algebraic terms, + + A Ψₐ(x) = a Ψₐ(x), where a is complex. + +Ψₐ is called an eigenstate of Α corresponding to the eigenvalue a. + +If A is a Hermitian operator corresponding to some physical dynamical variable: + ∞ ∞ +〈A〉 = ∫ Ψₐ⃰ A Ψₐ dx = a ∫ Ψₐ˟ Ψₐ dx = a + -∞ -∞ +ₐ + ∞ ∞ ∞ +〈A²〉 = ∫ Ψₐ⃰ A² Ψₐ dx = a ∫ Ψ⃰ₐ A Ψₐ dx = a² ∫ Ψₐ˟ Ψₐ dx = a² + -∞ -∞ -∞ + +σ² = 〈A²〉 - 〈A〉² = a² - a² = 0 + ᴬ + diff --git a/solutions/chap3/prob4/draft b/solutions/chap3/prob4/draft new file mode 100644 index 0000000..0ba2890 --- /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. + + diff --git a/solutions/chap3/prob4/draft.ps b/solutions/chap3/prob4/draft.ps new file mode 100644 index 0000000..7934429 --- /dev/null +++ b/solutions/chap3/prob4/draft.ps @@ -0,0 +1,1735 @@ +%!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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+5009 7848 5577 7722 x +6147 7596 6696 7344 x +8424 fwd_x +end_ol + } def +/DAA { start_ol +2520 959 m +2520 -2880 l +1296 -2880 l +1296 7632 l +2520 7632 l +2520 6672 l +2836 7246 3361 7546 x +3888 7848 4575 7848 x +5971 7848 6765 6771 x +7560 5694 7560 3789 x +7560 1917 6762 850 x +5964 -216 4575 -216 x +3874 -216 3348 84 x +2822 385 2520 959 x +6264 3816 m +6264 5278 5796 6023 x +5328 6768 4405 6768 x +3476 6768 2998 6019 x +2520 5271 2520 3816 x +2520 2367 2998 1615 x +3476 864 4405 864 x +5328 864 5796 1608 x +6264 2353 6264 3816 x +8424 fwd_x +end_ol + } def +/EAA { start_ol +1728 7632 m +4896 7632 l +4896 1008 l +7416 1008 l +7416 0 l +1152 0 l +1152 1008 l +3672 1008 l +3672 6624 l +1728 6624 l +1728 7632 l +3672 10584 m +4896 10584 l +4896 9000 l +3672 9000 l +3672 10584 l +8424 fwd_x +end_ol + } def +/FAA { start_ol +7128 4758 m +7128 0 l +5904 0 l +5904 4758 l +5904 5793 5535 6280 x +5167 6768 4381 6768 x +3486 6768 3002 6140 x +2520 5512 2520 4340 x +2520 0 l +1296 0 l 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+5836 4285 5055 3499 x +4275 2715 2677 1152 x +8424 fwd_x +end_ol + } def +/KAA { start_ol +4847 3816 m +4421 3816 l +3297 3816 2728 3434 x +2160 3054 2160 2299 x +2160 1618 2585 1240 x +3011 864 3765 864 x +4826 864 5433 1575 x +6041 2286 6048 3539 x +6048 3816 l +4847 3816 l +7272 4348 m +7272 0 l +6048 0 l +6048 1140 l +5640 445 5023 114 x +4406 -216 3523 -216 x +2343 -216 1639 444 x +936 1105 936 2215 x +936 3495 1796 4159 x +2657 4824 4323 4824 x +6048 4824 l +6048 5023 l +6041 5940 5580 6354 x +5118 6768 4107 6768 x +3459 6768 2799 6584 x +2138 6401 1512 6048 x +1512 7272 l +2208 7560 2846 7704 x +3484 7848 4084 7848 x +5033 7848 5704 7565 x +6377 7283 6793 6718 x +7053 6373 7162 5866 x +7272 5360 7272 4348 x +8424 fwd_x +end_ol + } def +/LAA { start_ol +7848 6048 m +7443 6386 7023 6540 x +6604 6696 6103 6696 x +4921 6696 4296 5947 x +3672 5200 3672 3789 x +3672 0 l +2448 0 l +2448 7632 l +3672 7632 l +3672 6136 l +3988 6964 4644 7405 x +5299 7848 6199 7848 x +6666 7848 7071 7725 x +7476 7602 7848 7344 x +7848 6048 l +8424 fwd_x +end_ol + } def +/MAA { start_ol +4176 9792 m +4176 7632 l +7056 7632 l +7056 6624 l +4176 6624 l +4176 2509 l +4176 1670 4500 1338 x +4824 1008 5629 1008 x +7056 1008 l +7056 0 l +5505 0 l +4104 0 3528 564 x +2952 1129 2952 2509 x +2952 6624 l +936 6624 l +936 7632 l +2952 7632 l +2952 9792 l +4176 9792 l +8424 fwd_x +end_ol + } def +/NAA { start_ol +7200 360 m +6696 72 6160 -72 x +5625 -216 5065 -216 x +3294 -216 2295 853 x +1296 1923 1296 3816 x +1296 5708 2295 6778 x +3294 7848 5065 7848 x +5618 7848 6142 7689 x +6667 7531 7200 7200 x +7200 5904 l +6699 6360 6195 6564 x +5691 6768 5053 6768 x +3867 6768 3229 6001 x +2592 5236 2592 3816 x +2592 2401 3233 1632 x +3874 864 5053 864 x +5711 864 6232 1056 x +6754 1249 7200 1656 x +7200 360 l +8424 fwd_x +end_ol + } def +/OAA { start_ol +4320 2698 m +4320 1860 4626 1434 x +4933 1008 5533 1008 x +6984 1008 l +6984 0 l +5412 0 l +4307 0 3701 704 x +3096 1409 3096 2698 x +3096 9576 l +1080 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7129 x +7776 6411 7776 4423 x +7776 0 l +6624 0 l +6624 4368 l +6624 5983 6441 6375 x +6259 6768 5779 6768 x +5229 6768 5026 6348 x +4824 5929 4824 4368 x +4824 0 l +3672 0 l +3672 4368 l +3672 6004 3475 6385 x +3279 6768 2768 6768 x +2264 6768 2067 6348 x +1872 5929 1872 4368 x +1872 0 l +720 0 l +720 7632 l +1872 7632 l +1872 6978 l +2099 7402 2440 7625 x +2781 7848 3214 7848 x +3737 7848 4084 7601 x +4432 7356 4624 6856 x +8424 fwd_x +end_ol + } def +/SAA { start_ol +5904 3894 m +5904 5304 5437 6035 x +4970 6768 4077 6768 x +3143 6768 2651 6035 x +2160 5304 2160 3894 x +2160 2485 2655 1746 x +3150 1008 4090 1008 x +4970 1008 5437 1749 x +5904 2491 5904 3894 x +7128 492 m +7128 -1167 6318 -2023 x +5509 -2880 3938 -2880 x +3421 -2880 2856 -2769 x +2292 -2660 1728 -2448 x +1728 -1224 l +2398 -1554 2946 -1713 x +3494 -1872 3953 -1872 x +4972 -1872 5438 -1351 x +5904 -830 5904 302 x +5904 358 l +5904 1233 l +5602 573 5080 250 x +4559 -72 3812 -72 x +2468 -72 1666 1004 x +864 2081 864 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5446 5647 x +5904 6026 5904 7114 x +5904 8092 5546 8545 x +5189 9000 4447 9000 x +4032 9000 3853 8848 x +3675 8697 3675 8326 x +9360 fwd_x +end_ol + } def +/bAA { start_ol +3456 2088 m +5184 2088 l +5184 673 l +3888 -1944 l +2808 -1944 l +3456 673 l +3456 2088 l +8424 fwd_x +end_ol + } def +/cAA { start_ol +0 7632 m +1243 7632 l +2576 1463 l +3668 5400 l +4741 5400 l +5847 1463 l +7180 7632 l +8424 7632 l +6634 0 l +5431 0 l +4208 4181 l +2992 0 l +1789 0 l +0 7632 l +8424 fwd_x +end_ol + } def +/dAA { start_ol +1512 7632 m +7056 7632 l +7056 6486 l +2670 1008 l +7056 1008 l +7056 0 l +1368 0 l +1368 1155 l +5733 6624 l +1512 6624 l +1512 7632 l +8424 fwd_x +end_ol + } def +/eAA { start_ol +7560 7632 m +4854 3979 l +7848 0 l +6401 0 l +4172 3060 l +1950 0 l +504 0 l +3497 3979 l +792 7632 l +2169 7632 l +4172 4872 l +6162 7632 l +7560 7632 l +8424 fwd_x +end_ol + } def +/fAA { start_ol +576 3600 m +7776 3600 l +7776 2448 l +576 2448 l +576 3600 l +576 6336 m +7776 6336 l +7776 5184 l +576 5184 l +576 6336 l +8424 fwd_x +end_ol + } def +/gAA { start_ol +2520 4824 m +2520 1152 l +4167 1152 l +5338 1152 5836 1564 x +6336 1977 6336 2923 x +6336 3904 5810 4363 x +5285 4824 4167 4824 x +2520 4824 l +2520 9000 m +2520 5976 l +4138 5976 l +5144 5976 5595 6351 x +6048 6728 6048 7569 x +6048 8328 5603 8663 x +5159 9000 4138 9000 x +2520 9000 l +1152 10152 m +4167 10152 l +5726 10152 6571 9478 x +7416 8804 7416 7572 x +7416 6639 6973 6102 x +6532 5564 5648 5428 x +6644 5278 7210 4575 x +7776 3872 7776 2785 x +7776 1406 6868 703 x +5961 0 4167 0 x +1152 0 l +1152 10152 l +8424 fwd_x +end_ol + } def +/hAA { start_ol +3681 4101 m +3681 3765 3345 3281 x +3011 2797 3011 2783 x +3011 2657 3416 2657 x +3807 2657 4352 2797 x +5581 4829 6125 4829 x +6377 4829 6377 4577 x +6377 4241 5902 3679 x +5428 3119 4561 2670 x +3876 1519 3198 1024 x +3840 784 4036 402 x +4874 667 6061 2210 x +6243 2076 l +4986 495 4119 219 x +4245 -151 4245 -509 x +4245 -1584 3324 -2286 x +2403 -2988 1229 -2988 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+/jAA { start_ol +3312 2088 m +5040 2088 l +5040 0 l +3312 0 l +3312 2088 l +8424 fwd_x +end_ol + } def +/kAA { start_ol +1368 10152 m +7056 10152 l +7056 9000 l +4896 9000 l +4896 1152 l +7056 1152 l +7056 0 l +1368 0 l +1368 1152 l +3528 1152 l +3528 9000 l +1368 9000 l +1368 10152 l +8424 fwd_x +end_ol + } def +/lAA { start_ol +1584 10584 m +2808 10584 l +2808 4462 l +6168 7632 l +7727 7632 l +4658 4756 l +8208 0 l +6642 0 l +3760 3938 l +2808 3060 l +2808 0 l +1584 0 l +1584 10584 l +8424 fwd_x +end_ol + } def +/mAA { start_ol +936 10152 m +2304 10152 l +2304 5976 l +6120 5976 l +6120 10152 l +7488 10152 l +7488 0 l +6120 0 l +6120 4824 l +2304 4824 l +2304 0 l +936 0 l +936 10152 l +8424 fwd_x +end_ol + } def +/nAA { start_ol +5853 2482 m +5542 1684 5062 381 x +4393 -1419 4163 -1815 x +3852 -2347 3385 -2613 x +2919 -2880 2297 -2880 x +1296 -2880 l +1296 -1800 l +2032 -1800 l +2580 -1800 2891 -1481 x +3202 -1163 3683 165 x +720 7632 l +2073 7632 l +4318 1662 l +6529 7632 l +7848 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x +8424 fwd_x +end_ol + } def +/xAA { start_ol +576 3600 m +7776 3600 l +7776 2448 l +576 2448 l +576 3600 l +576 6336 m +7776 6336 l +7776 5184 l +576 5184 l +576 6336 l +3312 8928 m +5040 8928 l +5040 6840 l +3312 6840 l +3312 8928 l +8424 fwd_x +end_ol + } def +/yAA { start_ol +7128 4703 m +7128 0 l +5904 0 l +5904 4703 l +5904 5767 5535 6267 x +5167 6768 4381 6768 x +3486 6768 3002 6127 x +2520 5487 2520 4290 x +2520 0 l +1296 0 l +1296 8712 l +432 8712 l +432 9720 l +1296 9720 l +1296 10584 l +2520 10584 l +2520 9720 l +4896 9720 l +4896 8712 l +2520 8712 l +2520 6453 l +2856 7139 3432 7493 x +4008 7848 4797 7848 x +5969 7848 6548 7067 x +7128 6286 7128 4703 x +8424 fwd_x +end_ol + } def +/zAA { start_ol +5976 10584 m +5069 9027 4622 7480 x +4176 5933 4176 4362 x +4176 2799 4622 1248 x +5069 -301 5976 -1872 x +4884 -1872 l +3872 -246 3376 1292 x +2880 2832 2880 4362 x +2880 5886 3376 7428 x +3872 8971 4884 10584 x +5976 10584 l +8424 fwd_x +end_ol + } def +/ABA { start_ol +3312 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l +8424 fwd_x +end_ol + } def +/DBA { start_ol +3312 5904 m +5040 5904 l +5040 3816 l +3312 3816 l +3312 5904 l +8424 fwd_x +end_ol + } def +/EBA { start_ol +6329 2332 m +6329 1991 l +2905 1991 l +2905 1971 l +2905 1267 3314 885 x +3723 504 4474 504 x +4848 504 5261 609 x +5674 715 6144 933 x +6144 238 l +5694 74 5275 -6 x +4856 -88 4467 -88 x +3342 -88 2711 507 x +2080 1104 2080 2154 x +2080 3178 2696 3787 x +3314 4398 4351 4398 x +5265 4398 5797 3846 x +6329 3294 6329 2332 x +5538 2543 m +5524 3157 5211 3480 x +4897 3804 4317 3804 x +3744 3804 3372 3470 x +3001 3136 2932 2543 x +5538 2543 l +8424 fwd_x +end_ol + } def +/FBA { start_ol +3685 9634 m +4342 9634 5054 9410 x +5766 9187 5862 9187 x +6043 9187 6140 9289 x +6238 9391 6307 9649 x +6727 9649 l +6727 6696 l +6336 6696 l +5937 7989 5230 8602 x +4523 9216 3685 9216 x +2990 9216 2587 8835 x +2184 8456 2184 7809 x +2184 7303 2477 6972 x +2770 6642 3589 6220 x +5596 5193 l +6266 4855 6672 4137 x +7078 3421 7078 2633 x +7078 1332 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7200 l +3312 2088 m +5040 2088 l +5040 0 l +3312 0 l +3312 2088 l +8424 fwd_x +end_ol + } def +end end +%%EndPrologue +%%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 728312KAAOAAEAASAAFAAPAAWAA+ 416016 728312cAAEAAMAAQAA+ 458136 728312MAAQAAPAA+ 491832 728312dAA)paps_exec +(+ 36000 711968KAAeAAEAACAAbAA+ 86544 711968CAATAA+ 111816 711968aAA+ 124704 711968fAA+ 141552 711968gAAhAA+ 159480 711968iAAjAA+ 184752 711968kAAMAA+ 210024 711968EAACAA+ 235296 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246600 434120dAA+ 263448 434120oAAKAACAAEAACAA+ 313992 434120wAAhAA+ 331920 434120EAACAA+ 357192 434120cAAPAAOAAOAAGAAlAAFAATAAcAAFAA+ 449856 434120KAAFAAWAA)paps_exec +(+ 36000 417776WAAEAAKAASAATAAFAAKAAOAAEAAdAAPAAWAA+ 145512 417776EAAFAA+ 170784 417776MAAQAAPAA+ 204480 417776dAA+ 221328 417776oAAKAACAAEAACAA+ 271872 417776oAAPAANAAKAAYAACAAPAA+ 339264 417776MAAQAAPAA+ 372960 417776RAAKAASAAFAAPAAMAAEAANAA+ 448776 417776ZAAEAAPAAOAAWAA+ 499320 417776EAACAA)paps_exec +(+ 36000 401432TAALAAEAAPAAFAAMAAPAAWAA+ 111816 401432EAAFAA+ 137088 401432MAAQAAPAA+ 170784 401432dAA+ 187632 401432WAAEAALAAPAANAAMAAEAATAAFAAjAA)paps_exec +()paps_exec +(+ 69696 368744wAAhAA+ 87624 368744xAA+ 104472 368744yAAIAAJAA+ 146592 368744zAA+ 163440 368744HAA+ 197136 368744ABA+ 213984 368744BBA)paps_exec +(+ 145512 352400zAA+ 162360 352400ABA+ 187632 352400GAAHAA+ 212904 352400BBA+ 229752 352400jAA)paps_exec +()paps_exec +(+ 36000 319712pAAQAAPAA+ 69696 319712mAAKAARAAEAAOAAMAATAAFAAEAAKAAFAA+ 170784 319712mAA+ 187632 319712EAACAA+ 212904 319712MAAQAAPAA+ 246600 319712MAATAAMAAKAAOAA+ 297144 319712PAAFAAPAALAASAAnAA+ 356112 319712TAAZAA+ 381384 319712MAAQAAPAA+ 415080 319712CAAnAACAAMAAPAARAAjAA+ 482472 319712pAAQAAPAA+ 516168 319712TAAFAAOAAnAA)paps_exec +(+ 36000 303368NAATAAFAAMAALAAEAAoAAYAAMAAEAAFAASAA+ 145512 303368MAAPAALAARAA+ 187632 303368ZAATAALAA+ 221328 303368MAAQAAEAACAA+ 263448 303368CAAnAACAAMAAPAARAA+ 322416 303368EAACAA+ 347688 303368MAAQAAPAA+ 381384 303368DAATAAMAAPAAFAAMAAEAAKAAOAA+ 465624 303368PAAFAAPAALAASAAnAA+ 524592 303368CBA+ 541440 303368fAA)paps_exec +(+ 36000 287024GAAUAADBAaAAbAA+ 87552 287024cAAQAAPAALAAPAA+ 138096 287024UAA+ 150120 287024KAAFAAWAA+ 183816 287024aAA+ 196704 287024KAALAAPAA+ 230400 287024MAAQAAPAA+ 264096 287024DAAQAAnAACAAEAANAAKAAOAA+ 339912 287024MAAPAALAARAACAA+ 390456 287024WAAPAACAANAALAAEAAoAAPAAWAA+ 474696 287024EAAFAA+ 499968 287024MAAQAAPAA)paps_exec +(+ 36000 270680EAAFAAMAALAATAAWAAYAANAAMAAEAATAAFAAjAA+ 153936 270680pAAQAAPAA+ 187632 270680RAAKAASAAFAAPAAMAAEAANAA+ 263448 270680RAATAARAAPAAFAAMAA+ 322416 270680TAAZAA+ 347688 270680KAA+ 364536 270680RAAKAASAAFAAPAAMAAEAANAA+ 440352 270680WAAEAADAATAAOAAPAAbAA+ 507744 270680CAAYAANAAQAA)paps_exec +(+ 36000 254336KAACAA+ 61272 254336MAAQAAKAAMAA+ 103392 254336CAAPAAPAAFAA+ 145512 254336cAAEAAMAAQAA+ 187632 254336KAA+ 204480 254336CAADAAEAAFAAGAAHAAIAAJAA+ 280296 254336DAAKAALAAMAAEAANAAOAAPAAbAA+ 364536 254336EAACAA)paps_exec +()paps_exec +(+ 50112 221648UAA+ 62136 221648fAA+ 78984 221648SAA+ 95832 221648vAAIAAJAARAAEBA+ 146376 221648FBAbAA)paps_exec +()paps_exec +(+ 36000 188960cAAEAAMAAQAA+ 78120 188960FBA+ 89424 188960MAAQAAPAA+ 123120 188960EAAFAAMAALAAEAAFAACAAEAANAA+ 207360 188960CAADAAEAAFAA+ 249480 188960sAAPAANAAMAATAALAA+ 308448 188960ZAATAALAA+ 342144 188960MAAQAAPAA+ 375840 188960NAAKAACAAPAA+ 417960 188960TAAZAA+ 443232 188960FAATAA+ 468504 188960TAALAAoAAEAAMAAKAAOAA)paps_exec +(+ 36000 172616KAAFAASAAYAAOAAKAALAA+ 103392 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695624MAAQAAPAA+ 69696 695624dAA+ 86544 695624oAAKAACAAEAACAA+ 137088 695624oAAnAA)paps_exec +()paps_exec +(+ 69696 662936HBA+ 86544 662936fAA+ 103392 662936GAAlAA+ 128664 662936gAAhAA+ 146592 662936wAAhAAjAA)paps_exec +()paps_exec +(+ 36000 630248pAAQAAPAALAAPAAZAATAALAAPAAbAA+ 128664 630248MAAQAAPAA+ 162360 630248mAAKAARAAEAAOAAMAATAAFAAEAAKAAFAA)paps_exec +()paps_exec +(+ 69696 597560HBA+ 86544 597560xAA+ 103392 597560GAAlAA+ 128664 597560gAAhAA+ 146592 597560yAAIAAJAA+ 188712 597560zAA+ 205560 597560HAA+ 230832 597560ABA+ 247680 597560BBA)paps_exec +(+ 187632 581216zAA+ 204480 581216ABA+ 221328 581216GAAHAA+ 246600 581216BBA+ 263448 581216jAA)paps_exec +()paps_exec +(+ 36000 548528pAAQAAPAA+ 69696 548528NAATAARAARAAYAAMAAKAAMAATAALAA+ 162360 548528NAAKAAFAA+ 196056 548528FAATAAcAA+ 229752 548528oAAPAA+ 255024 548528NAATAARAADAAYAAMAAPAAWAAjAA+ 339264 548528pAAQAAEAACAA+ 381384 548528NAATAARAADAAYAAMAAKAAMAAEAATAAFAA+ 482472 548528EAACAA)paps_exec +(+ 36000 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90896BBA)paps_exec +(+ 69696 74552zAA+ 86544 74552HAA+ 111816 74552ABA+ 128664 74552BBA+ 145512 74552zAA+ 162360 74552ABA+ 179208 74552GAAHAA+ 204480 74552BBA+ 255024 74552zAA+ 271872 74552HAA+ 305568 74552ABA+ 322416 74552BBA)paps_exec +()paps_exec +(+ 36000 41864GBAEAAFAANAAPAA+ 86544 41864MAAQAAPAACAAPAA+ 137088 41864RAAKAAMAALAAEAAeAA+ 196056 41864DAALAATAAWAAYAANAAMAACAA+ 271872 41864KAALAAPAA+ 305568 41864FAATAAMAA+ 339264 41864PAAvAAYAAKAAOAAbAA+ 398232 41864MAAQAAPAAEAALAA+ 448776 41864WAAEAAZAAZAAPAALAAPAAFAANAAPAA)paps_exec +paps_eop +showpage +%%Page: 3 3 +paps_bop +(+ 36000 744656EAACAA+ 61272 744656FAATAAMAA+ 94968 744656dAAPAALAATAAjAA+ 145512 744656pAAQAAPAALAAPAAZAATAALAAPAAbAA+ 238176 744656MAAQAAPAACAAPAA+ 288720 744656TAADAAPAALAAKAAMAATAALAACAA+ 372960 744656WAATAA+ 398232 744656FAATAAMAA+ 431928 744656NAATAARAARAAYAAMAAPAAjAA)paps_exec +()paps_exec +(+ 36000 711968LBATAALAA+ 69696 711968HBA+ 86544 711968KAAFAAWAA+ 120240 711968wAAKBAMBA)paps_exec +()paps_exec +(+ 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It is known that the Hamiltonian operator for this sytem commutes with the spin operator in the z 𝑧̂ + + diff --git a/solutions/chap3/prob4/sol.ps b/solutions/chap3/prob4/sol.ps new file mode 100644 index 0000000..b8e15fe --- /dev/null +++ b/solutions/chap3/prob4/sol.ps @@ -0,0 +1,1224 @@ +%!PS-Adobe-3.0 +%%Title: sol +%%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! + /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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aligned with the z-axis, so 𝐁 = B𝓏 𝑧̂ = B𝓏 ẑ. It is known that the Hamiltonian operator for this sytem commutes with the spin component operator in the z direction (z basis?) but not with spin component operators in the x and y directions. The following argument should prove that hypothesis. + +The Hamiltonian Ĥ = +$\mathscr{z}$ +$z$ +\end{document} diff --git a/solutions/chap3/prob5/draft b/solutions/chap3/prob5/draft new file mode 100644 index 0000000..4615e74 --- /dev/null +++ b/solutions/chap3/prob5/draft @@ -0,0 +1,29 @@ +A spin-1/2 particle with a magnetic moment is known to be in the state |Ψ(t=0)〉 = |+〉. + +a) If the observable S𝓍 is measured at t=0, the possible results are ħ/2 and -ħ/2 with an equal probability of measuring either. + +b) The system evolves in a uniform magnetic field 𝐁 = B₀ŷ. What is the state of the system at t=T? + +Since the magnetic field is oriented along the y axis, the energy eigenstates will be associated with that direction. The eigenstates are |±𝓎〉. + +The Hamiltonian for this system is + H = - ω₀ S𝓎, with ω₀ = g q/2mₑ B₀ ≈ e/mₑ B₀. + +The eigenvalue equations in the energy basis are therefore + + Ĥ|E±〉 = ω₀ ±ħ/2 |E±〉 = ω₀ ±ħ/2 |±〉 = E± |±〉 + +The initial state is prepared to |+〉, which means, in the y basis, + + |Ψ(t=0)〉 = 1/√2 |+𝓎〉 - 1/√2 |-𝓎〉 + +This Hamiltonian is time independent, so the time evolution is given by multiplying each eigenstate with the time-pdependent phase factor, with ι the imaginary unit. + + |Ψ(t)〉 = exp(-ι E₊ t/ħ)/√2 |+𝓎〉 - exp(-ι E₋ t/ħ)/√2 |-𝓎〉 + + + +If the time-evolution of the general state is known in the energy basis (aligned with the spin y basis), the time-evolved state at some time t in the z basis can be predicted by projecting the z basis onto the state to determine the coefficients in the z basis. 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6172 x +4288 6144 4438 6067 x +5227 5688 5862 5688 x +6338 5688 6805 5898 x +7272 6109 7776 6552 x +7776 4032 m +7776 2811 l +7265 2403 6778 2209 x +6291 2016 5753 2016 x +5140 2016 4370 2368 x +4219 2437 4145 2472 x +3579 2721 3242 2800 x +2905 2880 2571 2880 x +2040 2880 1560 2671 x +1080 2464 576 2016 x +576 3223 l +1113 3651 1611 3841 x +2108 4032 2667 4032 x +3321 4032 4206 3651 x +4213 3651 l +4288 3616 4438 3548 x +5227 3168 5861 3168 x +6324 3168 6795 3378 x +7265 3589 7776 4032 x +8424 fwd_x +end_ol + } def +/HBA { start_ol +3577 12960 m +4846 12960 l +6264 11160 l +5323 11160 l +4208 12364 l +3100 11160 l +2160 11160 l +3577 12960 l +936 10152 m +2304 10152 l +2304 5976 l +6120 5976 l +6120 10152 l +7488 10152 l +7488 0 l +6120 0 l +6120 4824 l +2304 4824 l +2304 0 l +936 0 l +936 10152 l +8424 fwd_x +end_ol + } def +/IBA { start_ol +1368 10152 m +7416 10152 l +7416 9000 l +2736 9000 l +2736 5976 l +7200 5976 l +7200 4824 l +2736 4824 l +2736 1152 l +7560 1152 l +7560 0 l +1368 0 l +1368 10152 l +8424 fwd_x +end_ol + } def +/JBA { start_ol +732 4990 m +432 5842 l +2405 6509 l +3910 1532 l +7119 11549 l +8136 11549 l +8136 10656 l +7813 10656 l +4335 -288 l +3462 -288 l +1773 5351 l +732 4990 l +8424 fwd_x +end_ol + } def +/KBA { start_ol +4752 7632 m +4752 2650 l +4752 1653 4981 1337 x +5223 1008 5952 1008 x +6552 1008 l +6552 0 l +5814 0 l +4618 0 4072 651 x +3534 1315 3528 2752 x +3528 6624 l +2088 6624 l +2088 7632 l +4752 7632 l +8424 fwd_x +end_ol + } def +/LBA { start_ol +4569 4494 m +4569 2795 l +6478 2795 l +6478 2147 l +4569 2147 l +4569 450 l +3846 450 l +3846 2147 l +1937 2147 l +1937 2795 l +3846 2795 l +3846 4494 l +4569 4494 l +8424 fwd_x +end_ol + } def +/MBA { start_ol +1937 2795 m +6478 2795 l +6478 2147 l +1937 2147 l +1937 2795 l +8424 fwd_x +end_ol + } def +/NBA { start_ol +1512 7632 m +7056 7632 l +7056 6486 l +2670 1008 l +7056 1008 l +7056 0 l +1368 0 l +1368 1155 l +5733 6624 l +1512 6624 l +1512 7632 l +8424 fwd_x +end_ol + } def +/OBA { start_ol +4104 -111 m +4104 6624 l +1944 6624 l +1944 7632 l +5328 7632 l +5328 -111 l +5328 -1441 4726 -2160 x +4124 -2880 3013 -2880 x +1296 -2880 l +1296 -1800 l +2877 -1800 l +3490 -1800 3796 -1378 x +4104 -956 4104 -111 x +4104 10584 m +5328 10584 l +5328 9000 l +4104 9000 l +4104 10584 l +8424 fwd_x +end_ol + } def +/PBA { start_ol +5544 -1433 m +5544 -1587 5438 -1649 x +5332 -1712 5242 -1712 x +5061 -1712 4971 -1559 x +2652 3358 l +4971 8274 l +5061 8428 5242 8428 x +5287 8428 5347 8407 x +5407 8385 5475 8316 x +5544 8246 5544 8148 x +5544 8106 5515 8023 x +3308 3358 l +5515 -1307 l +5544 -1391 5544 -1433 x +8424 fwd_x +end_ol + } def +end end +%%EndPrologue +%%EndPrologue +%%Page: 1 1 +paps_bop +(+ 36000 744656AAA+ 52848 744656CAADAAEAAFAAGAAHAAIAAJAA+ 128664 744656DAAKAALAAMAAEAANAAOAAPAA+ 204480 744656QAAEAAMAARAA+ 246600 744656KAA+ 263448 744656SAAKAATAAFAAPAAMAAEAANAA+ 339264 744656SAAUAASAAPAAFAAMAA+ 398232 744656EAACAA+ 423504 744656VAAFAAUAAQAAFAA+ 474048 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74552aAA)paps_exec +paps_eop +showpage +%%Pages: 1 +%%Trailer +%%EOF