diff --git a/lecture_notes/1-27/Constant of Motion b/lecture_notes/1-27/Constant of Motion new file mode 100644 index 0000000..88b1a88 --- /dev/null +++ b/lecture_notes/1-27/Constant of Motion @@ -0,0 +1,10 @@ +Compatible Observables occur when commutator is equal to 0. + +[Â,B̂] = ÂB̂ - B̂ + +Constant of Motion +------------------ + if d/dt  = 0 and [Ĥ,Â] = 0, where Ĥ is the Hamiltonian operator. + then  is a constant of motion + + Thm: Two operators such that [Â,B̂] always have common Eigenstates \ No newline at end of file diff --git a/lecture_notes/1-27/Derivative of A Operator.jpg b/lecture_notes/1-27/Derivative of A Operator.jpg new file mode 100644 index 0000000..34e25c3 Binary files /dev/null and b/lecture_notes/1-27/Derivative of A Operator.jpg differ diff --git a/lecture_notes/1-27/Expectation Value Time Dependence b/lecture_notes/1-27/Expectation Value Time Dependence index 1811127..874a1a1 100644 --- a/lecture_notes/1-27/Expectation Value Time Dependence +++ b/lecture_notes/1-27/Expectation Value Time Dependence @@ -1,13 +1,16 @@ -Computed Derivative (pic) d/dt (pic) -Discovered/Introduced Commutator -Cpmpatible Observables occur when commutator is equal to 0. +Computed Derivative d/dt : -[Ĥ,Â] = Ĥ - ÂĤ + d = d 〈Ψ(t)|Â|Ψ(t)〉 + dt dt + + ⎛d 〈Ψ(t)⎞ Â|Ψ(t)〉 + 〈Ψ(t)| ⎛d Ψ(t)〉⎞ + 〈Ψ(t)|d Â|Ψ(t)〉 + ⎝dt ⎠ ⎝dt ⎠ dt + ↓ + ι 〈Ψ(t)|Ĥ Â|Ψ(t)〉 + 〈Ψ(t)| Ĥ -ι|Ψ(t)〉 + 〈Ψ(t)|d Â|Ψ(t)〉 + ħ ħ dt + ↓ + ι 〈Ψ(t)|Ĥ  -  Ĥ|Ψ(t)〉 + 〈Ψ(t)|d Â|Ψ(t)〉 + ħ dt -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/IMG_20160127_131202.jpg b/lecture_notes/1-27/IMG_20160127_131202.jpg new file mode 100644 index 0000000..a399c81 Binary files /dev/null and b/lecture_notes/1-27/IMG_20160127_131202.jpg differ diff --git a/lecture_notes/1-27/IMG_20160127_131402.jpg b/lecture_notes/1-27/IMG_20160127_131402.jpg new file mode 100644 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diff --git a/lecture_notes/1-27/IMG_20160127_134331.jpg b/lecture_notes/1-27/IMG_20160127_134331.jpg new file mode 100644 index 0000000..0aac1bc Binary files /dev/null and b/lecture_notes/1-27/IMG_20160127_134331.jpg differ diff --git a/solutions/exam1/prob1.ps b/solutions/exam1/prob1.ps index 965c9b2..7ad8ea1 100644 --- a/solutions/exam1/prob1.ps +++ b/solutions/exam1/prob1.ps @@ -178,96 +178,24 @@ end /fontdict 1 dict def papsdict begin fontdict begin /AAA { start_ol -3058 3891 m -1732 4386 1262 4930 x -792 5475 792 6395 x -792 7576 1553 8251 x -2315 8928 3646 8928 x -4249 8928 4852 8781 x -5456 8636 6048 8352 x -6048 6696 l -5496 7083 4927 7285 x -4359 7488 3802 7488 x -3181 7488 2850 7239 x -2520 6990 2520 6527 x -2520 6168 2763 5933 x -3006 5699 3780 5415 x -4524 5137 l -5567 4749 6059 4109 x -6552 3469 6552 2494 x -6552 1166 5763 510 x -4974 -144 3379 -144 x -2723 -144 2064 19 x -1406 183 792 504 x -792 2304 l -1496 1794 2154 1544 x -2813 1296 3458 1296 x -4108 1296 4466 1588 x 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5256 x @@ -320,7 +290,7 @@ end_ol 7200 fwd_x end_ol } def -/HAA { start_ol +/GAA { start_ol 2543 1512 m 6203 1512 l 6203 0 l @@ -344,6 +314,26 @@ end_ol 5121 4125 3736 2724 x 2985 1965 2543 1512 x 7200 fwd_x +end_ol + } def +/HAA { start_ol +4104 6854 m +1857 3384 l +4104 3384 l +4104 6854 l +3929 8784 m +5760 8784 l +5760 3384 l +6696 3384 l +6696 1872 l +5760 1872 l +5760 0 l +4104 0 l +4104 1872 l +576 1872 l +576 3560 l +3929 8784 l +7200 fwd_x end_ol } def /IAA { start_ol @@ -394,33 +384,6 @@ end_ol end_ol } def /KAA { start_ol -1152 8784 m -5976 8784 l -5976 7272 l -2592 7272 l -2592 5602 l -2803 5684 3044 5721 x -3285 5760 3555 5760 x -4858 5760 5668 4943 x -6480 4126 6480 2819 x -6480 1449 5596 652 x -4713 -144 3190 -144 x -2623 -144 2044 -54 x -1465 36 864 216 x -864 1800 l -1359 1548 1881 1422 x -2403 1296 2927 1296 x -3857 1296 4340 1684 x -4824 2072 4824 2813 x -4824 3496 4342 3908 x -3861 4320 3060 4320 x -2587 4320 2111 4212 x -1636 4104 1152 3888 x -1152 8784 l -7200 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a/solutions/exam1/prob3 +++ b/solutions/exam1/prob3 @@ -2,5 +2,5 @@ Ĥ = |a〉 δ 〈b| + |b〉 δ 〈a|, with δ a real number. -a) +a) The eigenstates of the Hamiltonian can be determined by