From f2e57dce1285f02f8953d21e1f9ea5e5d2b1e0d0 Mon Sep 17 00:00:00 2001
From: caes <othoulrich@gmail.com>
Date: Tue, 23 Feb 2016 21:54:06 -0500
Subject: [PATCH] finished exam 1 prob 1

---
 lecture_notes/2-12/Time Evolution of a Free Electron | 10 ++++++++++
 solutions/exam1/prob1                                |  8 +++++++-
 2 files changed, 17 insertions(+), 1 deletion(-)
 create mode 100644 lecture_notes/2-12/Time Evolution of a Free Electron

diff --git a/lecture_notes/2-12/Time Evolution of a Free Electron b/lecture_notes/2-12/Time Evolution of a Free Electron
new file mode 100644
index 0000000..bea2f9e
--- /dev/null
+++ b/lecture_notes/2-12/Time Evolution of a Free Electron	
@@ -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
+
diff --git a/solutions/exam1/prob1 b/solutions/exam1/prob1
index 442e2bc..e3e4d8f 100644
--- a/solutions/exam1/prob1
+++ b/solutions/exam1/prob1
@@ -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 ⎞
     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 ⎠
\ No newline at end of file
+       ⎝ 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 ⎠
\ No newline at end of file