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finished exam 1 prob 1

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caes 2016-02-23 21:54:06 -05:00
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10
lecture_notes/2-12/Time Evolution of a Free Electron Normal file
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@ -0,0 +1,10 @@
Ĥ = p̂²
2mₑ
∞
|Ψ(t)〉 = exp( -ι Ĥ t) ∫ dp |p〉 〈p|Ψ〉
ħ -∞
〈x|Ψ(t)〉 = exp( -ι Ĥ t ) dp 〈x|p〉 〈p|Ψ〉
ħ

8
solutions/exam1/prob1
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@ -30,4 +30,10 @@ Similarly, S𝓍 (S𝓍 + ħ)(S𝓍 - ħ) ≐
⎛ 0 1 0 ⎞ ⎛ 1 1 0 ⎞ ⎛ -1 1 0 ⎞ ⎛ 1 1 1 ⎞ ⎛ -1 1 0 ⎞ ⎛ 0 1 0 ⎞ ⎛ 0 1 0 ⎞ ⎛ 1 1 0 ⎞ ⎛ -1 1 0 ⎞ ⎛ 1 1 1 ⎞ ⎛ -1 1 0 ⎞ ⎛ 0 1 0 ⎞
ħ³ ⎜ 1 0 1 ⎟ ⎜ 1 1 1 ⎟ ⎜ 1 -1 1 ⎟ = ħ³ ⎜ 1 2 1 ⎟ ⎜ 1 -1 1 ⎟ = ħ³ ⎜ 1 0 1 ⎟. ħ³ ⎜ 1 0 1 ⎟ ⎜ 1 1 1 ⎟ ⎜ 1 -1 1 ⎟ = ħ³ ⎜ 1 2 1 ⎟ ⎜ 1 -1 1 ⎟ = ħ³ ⎜ 1 0 1 ⎟.
⎝ 0 1 0 ⎠ ⎝ 0 1 1 ⎠ ⎝ 0 1 -1 ⎠ ⎝ 1 1 1 ⎠ ⎝ 0 1 -1 ⎠ ⎝ 0 1 0 ⎠ ⎝ 0 1 0 ⎠ ⎝ 0 1 1 ⎠ ⎝ 0 1 -1 ⎠ ⎝ 1 1 1 ⎠ ⎝ 0 1 -1 ⎠ ⎝ 0 1 0 ⎠
This expression results in the non-zero matrix
⎛ 0 1 0 ⎞
ħ³ ⎜ 1 0 1 ⎟.
⎝ 0 1 0 ⎠