update week4

Author: mhjensen

doc/pub/week4/html/week4-bs.html

@@ -520,6 +520,9 @@ <h2 id="time-ordering" class="anchor">Time-ordering  </h2>
 $$
 \int_{t_0}^t dt'\int_{t_0}^{t'} dt'' \hat{H}_I(t')\hat{H}_I(t'')=
 \frac{1}{2}\int_{t_0}^t dt'\int_{t_0}^{t'} dt'' \hat{H}_I(t')\hat{H}_I(t'')
+$$
+
+$$
 +\frac{1}{2}\int_{t_0}^t dt''\int_{t''}^{t} dt' \hat{H}_I(t')\hat{H}_I(t'').
 $$
 
@@ -563,7 +566,11 @@ <h2 id="final-expression-for-u" class="anchor">Final expression for \( U \)  </h
 <p>With this definition, we can rewrite the expression for \( \hat{U} \) as </p>
 $$
 \hat{U}(t,t_0)=\sum_{n=0}^{\infty}\left(\frac{-\imath}{\hbar}\right)^n\frac{1}{n!}
-\int_{t_0}^t dt_1\dots \int_{t_0}^t dt_N \hat{T}\left[\hat{H}_I(t_1)\dots\hat{H}_I(t_n)\right]=\hat{T}\exp{\left[\frac{-\imath}{\hbar}
+\int_{t_0}^t dt_1\dots \int_{t_0}^t dt_N \hat{T}\left[\hat{H}_I(t_1)\dots\hat{H}_I(t_n)\right]=
+$$
+
+$$
+\hat{T}\exp{\left[\frac{-\imath}{\hbar}
 \int_{t_0}^t dt' \hat{H}_I(t')\right]}.
 $$
 
@@ -664,7 +671,11 @@ <h2 id="time-evolution" class="anchor">Time evolution </h2>
 <p>with </p>
 $$
 \hat{U}_{\varepsilon}(t,t_0)=\sum_{n=0}^{\infty}\left(\frac{-\imath}{\hbar}\right)^n\frac{1}{n!}
-\int_{t_0}^t dt_1\dots \int_{t_0}^t dt_N \exp{(-\varepsilon(t_1+\dots+t_n)/\hbar)}\hat{T}\left[\hat{H}_I(t_1)\dots\hat{H}_I(t_n)\right]
+\int_{t_0}^t dt_1\dots \int_{t_0}^t dt_N
+$$
+
+$$
+\times \exp{(-\varepsilon(t_1+\dots+t_n)/\hbar)}\hat{T}\left[\hat{H}_I(t_1)\dots\hat{H}_I(t_n)\right]
 $$
 
 

doc/pub/week4/html/week4-reveal.html

@@ -527,6 +527,11 @@ <h2 id="time-ordering">Time-ordering  </h2>
 $$
 \int_{t_0}^t dt'\int_{t_0}^{t'} dt'' \hat{H}_I(t')\hat{H}_I(t'')=
 \frac{1}{2}\int_{t_0}^t dt'\int_{t_0}^{t'} dt'' \hat{H}_I(t')\hat{H}_I(t'')
+$$
+<p>&nbsp;<br>
+
+<p>&nbsp;<br>
+$$
 +\frac{1}{2}\int_{t_0}^t dt''\int_{t''}^{t} dt' \hat{H}_I(t')\hat{H}_I(t'').
 $$
 <p>&nbsp;<br>
@@ -582,7 +587,13 @@ <h2 id="final-expression-for-u">Final expression for \( U \)  </h2>
 <p>&nbsp;<br>
 $$
 \hat{U}(t,t_0)=\sum_{n=0}^{\infty}\left(\frac{-\imath}{\hbar}\right)^n\frac{1}{n!}
-\int_{t_0}^t dt_1\dots \int_{t_0}^t dt_N \hat{T}\left[\hat{H}_I(t_1)\dots\hat{H}_I(t_n)\right]=\hat{T}\exp{\left[\frac{-\imath}{\hbar}
+\int_{t_0}^t dt_1\dots \int_{t_0}^t dt_N \hat{T}\left[\hat{H}_I(t_1)\dots\hat{H}_I(t_n)\right]=
+$$
+<p>&nbsp;<br>
+
+<p>&nbsp;<br>
+$$
+\hat{T}\exp{\left[\frac{-\imath}{\hbar}
 \int_{t_0}^t dt' \hat{H}_I(t')\right]}.
 $$
 <p>&nbsp;<br>
@@ -708,7 +719,13 @@ <h2 id="time-evolution">Time evolution </h2>
 <p>&nbsp;<br>
 $$
 \hat{U}_{\varepsilon}(t,t_0)=\sum_{n=0}^{\infty}\left(\frac{-\imath}{\hbar}\right)^n\frac{1}{n!}
-\int_{t_0}^t dt_1\dots \int_{t_0}^t dt_N \exp{(-\varepsilon(t_1+\dots+t_n)/\hbar)}\hat{T}\left[\hat{H}_I(t_1)\dots\hat{H}_I(t_n)\right]
+\int_{t_0}^t dt_1\dots \int_{t_0}^t dt_N
+$$
+<p>&nbsp;<br>
+
+<p>&nbsp;<br>
+$$
+\times \exp{(-\varepsilon(t_1+\dots+t_n)/\hbar)}\hat{T}\left[\hat{H}_I(t_1)\dots\hat{H}_I(t_n)\right]
 $$
 <p>&nbsp;<br>
 </section>

doc/pub/week4/html/week4-solarized.html

@@ -445,6 +445,9 @@ <h2 id="time-ordering">Time-ordering  </h2>
 $$
 \int_{t_0}^t dt'\int_{t_0}^{t'} dt'' \hat{H}_I(t')\hat{H}_I(t'')=
 \frac{1}{2}\int_{t_0}^t dt'\int_{t_0}^{t'} dt'' \hat{H}_I(t')\hat{H}_I(t'')
+$$
+
+$$
 +\frac{1}{2}\int_{t_0}^t dt''\int_{t''}^{t} dt' \hat{H}_I(t')\hat{H}_I(t'').
 $$
 
@@ -488,7 +491,11 @@ <h2 id="final-expression-for-u">Final expression for \( U \)  </h2>
 <p>With this definition, we can rewrite the expression for \( \hat{U} \) as </p>
 $$
 \hat{U}(t,t_0)=\sum_{n=0}^{\infty}\left(\frac{-\imath}{\hbar}\right)^n\frac{1}{n!}
-\int_{t_0}^t dt_1\dots \int_{t_0}^t dt_N \hat{T}\left[\hat{H}_I(t_1)\dots\hat{H}_I(t_n)\right]=\hat{T}\exp{\left[\frac{-\imath}{\hbar}
+\int_{t_0}^t dt_1\dots \int_{t_0}^t dt_N \hat{T}\left[\hat{H}_I(t_1)\dots\hat{H}_I(t_n)\right]=
+$$
+
+$$
+\hat{T}\exp{\left[\frac{-\imath}{\hbar}
 \int_{t_0}^t dt' \hat{H}_I(t')\right]}.
 $$
 
@@ -589,7 +596,11 @@ <h2 id="time-evolution">Time evolution </h2>
 <p>with </p>
 $$
 \hat{U}_{\varepsilon}(t,t_0)=\sum_{n=0}^{\infty}\left(\frac{-\imath}{\hbar}\right)^n\frac{1}{n!}
-\int_{t_0}^t dt_1\dots \int_{t_0}^t dt_N \exp{(-\varepsilon(t_1+\dots+t_n)/\hbar)}\hat{T}\left[\hat{H}_I(t_1)\dots\hat{H}_I(t_n)\right]
+\int_{t_0}^t dt_1\dots \int_{t_0}^t dt_N
+$$
+
+$$
+\times \exp{(-\varepsilon(t_1+\dots+t_n)/\hbar)}\hat{T}\left[\hat{H}_I(t_1)\dots\hat{H}_I(t_n)\right]
 $$
 
 

doc/pub/week4/html/week4.html

@@ -522,6 +522,9 @@ <h2 id="time-ordering">Time-ordering  </h2>
 $$
 \int_{t_0}^t dt'\int_{t_0}^{t'} dt'' \hat{H}_I(t')\hat{H}_I(t'')=
 \frac{1}{2}\int_{t_0}^t dt'\int_{t_0}^{t'} dt'' \hat{H}_I(t')\hat{H}_I(t'')
+$$
+
+$$
 +\frac{1}{2}\int_{t_0}^t dt''\int_{t''}^{t} dt' \hat{H}_I(t')\hat{H}_I(t'').
 $$
 
@@ -565,7 +568,11 @@ <h2 id="final-expression-for-u">Final expression for \( U \)  </h2>
 <p>With this definition, we can rewrite the expression for \( \hat{U} \) as </p>
 $$
 \hat{U}(t,t_0)=\sum_{n=0}^{\infty}\left(\frac{-\imath}{\hbar}\right)^n\frac{1}{n!}
-\int_{t_0}^t dt_1\dots \int_{t_0}^t dt_N \hat{T}\left[\hat{H}_I(t_1)\dots\hat{H}_I(t_n)\right]=\hat{T}\exp{\left[\frac{-\imath}{\hbar}
+\int_{t_0}^t dt_1\dots \int_{t_0}^t dt_N \hat{T}\left[\hat{H}_I(t_1)\dots\hat{H}_I(t_n)\right]=
+$$
+
+$$
+\hat{T}\exp{\left[\frac{-\imath}{\hbar}
 \int_{t_0}^t dt' \hat{H}_I(t')\right]}.
 $$
 
@@ -666,7 +673,11 @@ <h2 id="time-evolution">Time evolution </h2>
 <p>with </p>
 $$
 \hat{U}_{\varepsilon}(t,t_0)=\sum_{n=0}^{\infty}\left(\frac{-\imath}{\hbar}\right)^n\frac{1}{n!}
-\int_{t_0}^t dt_1\dots \int_{t_0}^t dt_N \exp{(-\varepsilon(t_1+\dots+t_n)/\hbar)}\hat{T}\left[\hat{H}_I(t_1)\dots\hat{H}_I(t_n)\right]
+\int_{t_0}^t dt_1\dots \int_{t_0}^t dt_N
+$$
+
+$$
+\times \exp{(-\varepsilon(t_1+\dots+t_n)/\hbar)}\hat{T}\left[\hat{H}_I(t_1)\dots\hat{H}_I(t_n)\right]
 $$