From 437023a7611752ed52918b6fe4bcda6b87f3303b Mon Sep 17 00:00:00 2001
From: Michael Wimmer <m.t.wimmer@tudelft.nl>
Date: Tue, 15 Sep 2020 22:14:53 +0200
Subject: [PATCH] fix more math
---
src/8_differential_equations_2.md | 10 +++++-----
1 file changed, 5 insertions(+), 5 deletions(-)
diff --git a/src/8_differential_equations_2.md b/src/8_differential_equations_2.md
index b1034f9..22f6d3c 100644
--- a/src/8_differential_equations_2.md
+++ b/src/8_differential_equations_2.md
@@ -204,7 +204,7 @@ $$f(x) = e^{\lambda_1 x}, \ x e^{\lambda_1 x} , \ \cdots, \ x^{m_{1}-1} e^{\lamb
since they are linear combinations of $f_1$ and $f_2$ which remain linearly
independent,
- $$\tilde{f_1}(x)=\cos(kx), \tilde{f_2}(x)=sin{kx}.$$
+ $$\tilde{f_1}(x)=\cos(kx), \tilde{f_2}(x)=\sin(kx).$$
**Case 2: $E<0$**
This time, define $E=-k^2$, for constant $k$. The characteristic polynomial
@@ -487,9 +487,9 @@ the eigenfunctions of $L$.
!!! info "Connection to quantum states"
- Recall that q quantum state $\ket{\phi}$ can be written in an orthonormal
- basis $\{ \ket{u_n} \}$ as
- $$\ket{\phi} = \underset{n}{\Sigma} \bra{u_n} \ket{\phi} \ket{u_n}.$$
+ Recall that q quantum state $|\phi\rangle$ can be written in an orthonormal
+ basis $\{ |u_n\rangle \}$ as
+ $$\|\phi\rangle = \underset{n}{\Sigma} \langle u_n | \phi \rangle\, |u_n\rangle.$$
In terms of hermitian operators and their eigenfunctions, the eigenfunctions
play the role of the orthonormal basis. In reference to our running example,
@@ -500,7 +500,7 @@ To close our running example, consider the initial condition
$\psi(x,o) = \psi_{0}(x)$. Since the eigenfunctions $\sin(\frac{n \pi x}{L})$
form a basis, we can now write the general solution to the problem as
-$$\psi(x,t) = \overset{\infinity}{\underset{n}{\Sigma}} c_n e^{-i \frac{\lambda_n t}{\hbar}} \sin(\frac{n \pi x}{L}),$$
+$$\psi(x,t) = \overset{\infty}{\underset{n}{\Sigma}} c_n e^{-i \frac{\lambda_n t}{\hbar}} \sin(\frac{n \pi x}{L}),$$
where in the above we have defined the coefficients using the Fourier
coefficient,
--
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