From 4eb811263c19fa511950a5d4a02fc3ce37f8fe64 Mon Sep 17 00:00:00 2001 From: caes Date: Fri, 12 Feb 2016 12:26:39 -0500 Subject: [PATCH] Initialization --- Postulates | 14 + book_notes/chap2/S² operator | 10 + book_notes/chap2/commutators | 5 + book_notes/chap2/operators | 35 + book_notes/chap2/spin-1 system | 17 + book_notes/chap3/Bohr Frequency | 14 + .../chap3/Time-Independent Hamiltonian Recipe | 7 + lecture_notes/1-11/Determing Coefficients.jpg | Bin 0 -> 434677 bytes lecture_notes/1-11/Eigenstates.jpg | Bin 0 -> 413716 bytes lecture_notes/1-11/Heisenberg | 29 + lecture_notes/1-11/Hero-Villian Basis.jpg | Bin 0 -> 421117 bytes lecture_notes/1-11/Operators.jpg | Bin 0 -> 450159 bytes lecture_notes/1-13/Basis | 14 + lecture_notes/1-13/Stern-Gerlach | 10 + lecture_notes/1-15/Applications | 19 + lecture_notes/1-22/IMG_20160122_130917.jpg | Bin 0 -> 621900 bytes lecture_notes/1-22/IMG_20160122_132503.jpg | Bin 0 -> 481769 bytes lecture_notes/1-22/IMG_20160122_132757.jpg | Bin 0 -> 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solutions/chap3/prob4/tex/draft.tex | 19 + solutions/chap3/prob5/draft | 29 + solutions/chap3/prob5/draft.ps | 1710 ++++++++++++++++ 40 files changed, 6655 insertions(+) create mode 100644 Postulates create mode 100644 book_notes/chap2/S² operator create mode 100644 book_notes/chap2/commutators create mode 100644 book_notes/chap2/operators create mode 100644 book_notes/chap2/spin-1 system create mode 100644 book_notes/chap3/Bohr Frequency create mode 100644 book_notes/chap3/Time-Independent Hamiltonian Recipe create mode 100644 lecture_notes/1-11/Determing Coefficients.jpg create mode 100644 lecture_notes/1-11/Eigenstates.jpg create mode 100644 lecture_notes/1-11/Heisenberg create mode 100644 lecture_notes/1-11/Hero-Villian Basis.jpg create mode 100644 lecture_notes/1-11/Operators.jpg create mode 100644 lecture_notes/1-13/Basis create mode 100644 lecture_notes/1-13/Stern-Gerlach create mode 100644 lecture_notes/1-15/Applications create mode 100644 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solutions/chap3/prob4/sol.ps create mode 100644 solutions/chap3/prob4/tex/draft.aux create mode 100644 solutions/chap3/prob4/tex/draft.log create mode 100644 solutions/chap3/prob4/tex/draft.pdf create mode 100644 solutions/chap3/prob4/tex/draft.ps create mode 100644 solutions/chap3/prob4/tex/draft.tex create mode 100644 solutions/chap3/prob5/draft create mode 100644 solutions/chap3/prob5/draft.ps 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 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UOd2ukKmqGXns?HW4{8Aa*)pA`R{#J2 literal 0 HcmV?d00001 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. 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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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9864 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 +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 +/JBA { start_ol +5749 4829 m +6055 4829 6249 4637 x +6442 4446 6442 4134 x +6442 3578 5869 3578 x +5563 3578 5429 3758 x +5296 3938 5276 4118 x +5256 4298 5212 4298 x +5052 4298 4638 3701 x +4224 3103 4020 2678 x +4020 2193 3846 1369 x +3673 544 3673 447 x +3673 142 3900 142 x +4124 142 4362 276 x +4601 410 4951 790 x +5302 1172 5434 1334 x +5568 1496 6003 2059 x +6109 2197 6162 2265 x +6336 2155 l 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x +7851 2099 l +7348 1444 6907 1005 x +6467 567 6062 329 x +5657 93 5468 9 x +5280 -74 4903 -185 x +3395 -2988 1438 -2988 x +1065 -2988 784 -2794 x +504 -2602 504 -2298 x +504 -1982 751 -1694 x +999 -1406 1417 -1172 x +1837 -939 2256 -753 x +2676 -569 3143 -376 x +3297 -321 3367 -294 x +4889 2795 l +3758 1248 3073 552 x +2388 -144 1843 -144 x +1438 -144 1186 168 x +936 482 936 970 x +936 1527 1229 2154 x +1523 2781 1872 3219 x +2220 3658 2514 4006 x +2808 4354 2808 4424 x +2808 4536 2668 4536 x +2054 4536 168 2016 x +-27 2140 l +2012 4815 2779 4815 x +3143 4815 3387 4507 x +3632 4201 3632 3769 x +937 -2408 m +937 -2572 1064 -2682 x +1191 -2792 1369 -2792 x +2235 -2792 3255 -555 x +937 -1625 937 -2408 x +7632 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 +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 +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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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. Load and exec + /name ps pos 3 getinterval def + papsdict begin name draw_char end + /pos 3 pos add def + } ifelse + } ifelse + } ifelse + } ifelse + } loop + end +} def +/fontdict 1 dict def +papsdict begin fontdict begin +/AAA { start_ol +3632 7736 m +2376 3240 l +4890 3240 l +3632 7736 l +2912 8784 m +4359 8784 l +7056 0 l +5823 0 l +5173 2304 l +2086 2304 l +1449 0 l +216 0 l +2912 8784 l +7200 fwd_x +end_ol + } def +/CAA { start_ol +5688 6264 m +5688 5184 l +5229 5472 4764 5616 x +4299 5760 3817 5760 x +3090 5760 2732 5526 x +2376 5293 2376 4813 x +2376 4381 2646 4168 x +2917 3954 3992 3753 x +4426 3675 l +5225 3524 5636 3073 x +6048 2622 6048 1899 x +6048 938 5361 397 x +4675 -144 3453 -144 x +2971 -144 2441 -36 x +1912 72 1296 288 x +1296 1440 l +1895 1116 2442 954 x +2989 792 3478 792 x +4188 792 4577 1075 x +4968 1360 4968 1870 x +4968 2605 3528 2886 x +3480 2898 l +3075 2976 l +2147 3156 1721 3583 x +1296 4009 1296 4746 x +1296 5680 1929 6187 x +2562 6696 3736 6696 x +4259 6696 4740 6588 x +5223 6480 5688 6264 x +7200 fwd_x +end_ol + } def +/DAA { start_ol +2232 873 m +2232 -2520 l +1152 -2520 l +1152 6552 l +2232 6552 l +2232 5679 l +2503 6175 2954 6435 x +3405 6696 3993 6696 x +5190 6696 5871 5782 x +6552 4870 6552 3252 x +6552 1665 5868 760 x +5185 -144 3993 -144 x +3393 -144 2941 115 x +2491 376 2232 873 x +5400 3276 m +5400 4506 5004 5133 x +4608 5760 3827 5760 x +3042 5760 2637 5130 x +2232 4501 2232 3276 x +2232 2056 2637 1424 x +3042 792 3827 792 x +4608 792 5004 1418 x +5400 2045 5400 3276 x +7200 fwd_x +end_ol + } def +/EAA { start_ol +1512 6552 m +4248 6552 l +4248 864 l +6408 864 l +6408 0 l +1008 0 l +1008 864 l +3168 864 l +3168 5688 l +1512 5688 l +1512 6552 l +3168 9144 m +4248 9144 l +4248 7776 l +3168 7776 l +3168 9144 l +7200 fwd_x +end_ol + } def +/FAA { start_ol +6192 4050 m +6192 0 l +5112 0 l +5112 4050 l +5112 4930 4798 5344 x +4485 5760 3817 5760 x +3054 5760 2642 5225 x +2232 4692 2232 3694 x +2232 0 l +1152 0 l +1152 6552 l 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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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4836 5133 x +4417 5760 3596 5760 x +3596 6696 m +4961 6696 5684 5817 x +6408 4939 6408 3276 x +6408 1606 5686 731 x +4966 -144 3596 -144 x +2233 -144 1512 731 x +792 1606 792 3276 x +792 4939 1512 5817 x +2233 6696 3596 6696 x +7200 fwd_x +end_ol + } def +/DAA { start_ol +6264 360 m +5827 108 5364 -18 x +4900 -144 4416 -144 x +2881 -144 2016 762 x +1152 1670 1152 3276 x +1152 4881 2016 5788 x +2881 6696 4416 6696 x +4894 6696 5349 6572 x +5803 6449 6264 6192 x +6264 5040 l +5834 5420 5401 5590 x +4968 5760 4419 5760 x +3399 5760 2851 5115 x +2304 4471 2304 3276 x +2304 2085 2855 1438 x +3406 792 4419 792 x +4984 792 5432 967 x +5881 1143 6264 1512 x +6264 360 l +7200 fwd_x +end_ol + } def +/EAA { start_ol +1152 2502 m +1152 6552 l +2232 6552 l +2232 2502 l +2232 1621 2548 1206 x +2864 792 3526 792 x +4296 792 4703 1325 x +5112 1859 5112 2857 x +5112 6552 l +6192 6552 l +6192 0 l +5112 0 l +5112 1008 l +4822 441 4323 148 x +3825 -144 3159 -144 x +2143 -144 1647 513 x +1152 1170 1152 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l +1296 -1080 l +7200 fwd_x +end_ol + } def +/QAA { start_ol +2232 873 m +2232 -2520 l +1152 -2520 l +1152 6552 l +2232 6552 l +2232 5679 l +2503 6175 2954 6435 x +3405 6696 3993 6696 x +5190 6696 5871 5782 x +6552 4870 6552 3252 x +6552 1665 5868 760 x +5185 -144 3993 -144 x +3393 -144 2941 115 x +2491 376 2232 873 x +5400 3276 m +5400 4506 5004 5133 x +4608 5760 3827 5760 x +3042 5760 2637 5130 x +2232 4501 2232 3276 x +2232 2056 2637 1424 x +3042 792 3827 792 x +4608 792 5004 1418 x +5400 2045 5400 3276 x +7200 fwd_x +end_ol + } def +/RAA { start_ol +1368 9144 m +2448 9144 l +2448 3831 l +5316 6552 l +6646 6552 l +4027 4083 l +7056 0 l +5719 0 l +3261 3381 l +2448 2626 l +2448 0 l +1368 0 l +1368 9144 l +7200 fwd_x +end_ol + } def +/SAA { start_ol +5040 3281 m +5040 4497 4644 5128 x +4249 5760 3494 5760 x +2703 5760 2287 5128 x +1872 4497 1872 3281 x +1872 2066 2290 1428 x +2709 792 3505 792 x +4249 792 4644 1432 x +5040 2072 5040 3281 x +6120 380 m +6120 -1047 5418 -1783 x +4717 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5112 4680 5477 x +5040 5843 5040 6540 x +5040 7248 4682 7619 x +4325 7992 3632 7992 x +2952 7992 2592 7617 x +2232 7242 2232 6540 x +7200 fwd_x +end_ol + } def +/ZAA { start_ol +6552 6552 m +4191 3416 l +6768 0 l +5519 0 l +3596 2626 l +1680 0 l +432 0 l +3008 3416 l +648 6552 l +1848 6552 l +3596 4182 l +5333 6552 l +6552 6552 l +7200 fwd_x +end_ol + } def +/aAA { start_ol +4536 9144 m +4536 -1584 l +2016 -1584 l +2016 -720 l +3456 -720 l +3456 8280 l +2016 8280 l +2016 9144 l +4536 9144 l +7200 fwd_x +end_ol + } def +/bAA { start_ol +2578 1008 m +4072 1008 4664 1739 x +5256 2471 5256 4383 x +5256 6312 4666 7043 x +4078 7776 2578 7776 x +2016 7776 l +2016 1008 l +2578 1008 l +2602 8784 m +4593 8784 5536 7713 x +6480 6642 6480 4383 x +6480 2135 5536 1067 x +4593 0 2602 0 x +792 0 l +792 8784 l +2602 8784 l +7200 fwd_x +end_ol + } def +/cAA { start_ol +3528 -126 m +3528 5688 l +1656 5688 l +1656 6552 l +4608 6552 l +4608 -126 l +4608 -1275 4080 -1897 x +3553 -2520 2583 -2520 x +1080 -2520 l +1080 -1584 l +2464 -1584 l +2997 -1584 3262 -1219 x +3528 -855 3528 -126 x +3528 9144 m +4608 9144 l +4608 7776 l +3528 7776 l +3528 9144 l +7200 fwd_x +end_ol + } def +/dAA { start_ol +3632 1000 m +5688 8784 l +6912 8784 l +4353 0 l +2918 0 l +360 8784 l +1584 8784 l +3632 1000 l +7200 fwd_x +end_ol + } def +/eAA { 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 +/fAA { start_ol +504 8784 m 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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. In mathematical terms, + + 〈+|Ψ(t)〉|+〉 = + 〈-|Ψ(t)〉|-〉 = ( \ No newline at end of file diff --git a/solutions/chap3/prob5/draft.ps b/solutions/chap3/prob5/draft.ps new file mode 100644 index 0000000..3bc452f --- /dev/null +++ b/solutions/chap3/prob5/draft.ps @@ -0,0 +1,1710 @@ +%!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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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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def +/WAA { start_ol +6264 3816 m +6264 5278 5792 6023 x +5322 6768 4402 6768 x +3475 6768 2997 6019 x +2520 5271 2520 3816 x +2520 2367 2997 1615 x +3475 864 4402 864 x +5322 864 5792 1608 x +6264 2353 6264 3816 x +2520 6672 m +2821 7239 3352 7543 x +3884 7848 4584 7848 x +5969 7848 6764 6781 x +7560 5715 7560 3843 x +7560 1944 6761 864 x +5962 -216 4569 -216 x +3884 -216 3359 84 x +2835 385 2520 959 x +2520 0 l +1296 0 l +1296 10584 l +2520 10584 l +2520 6672 l +8424 fwd_x +end_ol + } def +/YAA { start_ol +4752 10584 m +4752 -3384 l +3600 -3384 l +3600 10584 l +4752 10584 l +8424 fwd_x +end_ol + } def +/ZAA { start_ol +4896 10152 m +4896 3816 l +5411 4046 5735 4637 x +6264 5622 6264 7822 x +6264 10152 l +7632 10152 l +7632 7814 l +7632 5141 6784 3839 x +6108 2810 4896 2592 x +4896 1152 l +5904 1152 l +5904 0 l +2520 0 l +2520 1152 l +3528 1152 l +3528 2592 l +2315 2810 1639 3839 x +792 5141 792 7814 x +792 10152 l +2160 10152 l +2160 7822 l +2160 5622 2695 4637 x +3012 4046 3528 3816 x +3528 10152 l +4896 10152 l +8424 fwd_x +end_ol + } def +/aAA { 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 +/bAA { 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 +/cAA { start_ol +3312 5072 m +3312 5440 3570 5708 x +3830 5976 4195 5976 x +4573 5976 4842 5708 x +5112 5440 5112 5072 x +5112 4698 4845 4437 x +4579 4176 4195 4176 x +3817 4176 3564 4430 x +3312 4684 3312 5072 x +4244 9288 m +3267 9288 2785 8246 x +2304 7204 2304 5072 x +2304 2947 2785 1905 x +3267 864 4244 864 x +5229 864 5710 1905 x +6192 2947 6192 5072 x +6192 7204 5710 8246 x +5229 9288 4244 9288 x +4244 10368 m +5881 10368 6720 9028 x +7560 7689 7560 5072 x +7560 2462 6720 1122 x +5881 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+3312 0 l +3312 2088 l +8424 fwd_x +end_ol + } def +/hAA { 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 +/iAA { start_ol +7200 10584 m +7200 9576 l +5792 9576 l +5126 9576 4866 9297 x +4608 9019 4608 8311 x +4608 7632 l +7200 7632 l +7200 6624 l +4608 6624 l +4608 0 l +3384 0 l +3384 6624 l +1368 6624 l +1368 7632 l +3384 7632 l +3384 8167 l +3384 9409 3948 9996 x +4514 10584 5711 10584 x +7200 10584 l +8424 fwd_x +end_ol + } def +/jAA { start_ol +648 7632 m +1953 7632 l +4172 1226 l +6399 7632 l +7704 7632 l +4984 0 l +3367 0 l +648 7632 l +8424 fwd_x +end_ol + } def +/kAA { start_ol +6912 9864 m +6912 8496 l +6278 8852 5641 9033 x +5005 9216 4358 9216 x +3375 9216 2803 8771 x +2232 8327 2232 7571 x +2232 6909 2608 6562 x +2985 6214 4015 5980 x +4747 5815 l +6179 5477 6833 4755 x +7488 4034 7488 2788 x +7488 1324 6588 553 x +5688 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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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+============================================= + + - 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. 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