From 031785514fd0caff35419e29cdfd322de8a09c0b Mon Sep 17 00:00:00 2001
From: Maciej Topyla <m.m.topyla@student.tudelft.nl>
Date: Sun, 4 Sep 2022 18:50:51 +0000
Subject: [PATCH] Update src/4_vector_spaces_QM.md
---
src/4_vector_spaces_QM.md | 34 +++++++++++++++-------------------
1 file changed, 15 insertions(+), 19 deletions(-)
diff --git a/src/4_vector_spaces_QM.md b/src/4_vector_spaces_QM.md
index 72265c9..996b9d8 100644
--- a/src/4_vector_spaces_QM.md
+++ b/src/4_vector_spaces_QM.md
@@ -105,25 +105,21 @@ The inner product in quantum mechanics is the analog of the usual scalar product
The inner product (probability amplitude) $\langle \psi | \phi \rangle$ exhibits the following properties:
-!!! info "Complex conjugate:"
- $$\langle \psi | \phi \rangle=\langle \phi | \psi \rangle^*$$
-
-!!! info "Distributivity and associativity"
- $$\langle \psi |\{c_1 |\phi_1\rangle+c_2 |\phi_2 \rangle\}=c_1\langle \psi | \phi_1\rangle+c_2\langle \psi | \phi_2\rangle$$
-
-!!! info "Positivity"
- $$\langle \psi | \psi \rangle\geq0$$. If $\langle \psi | \psi \rangle = 0$$
- then this implies that the state vector $|\psi\rangle=0$ is the null element of the Hilbert space.
-
-!!! info "Orthogonality"
- Two states $|\psi \rangle$ and $|\phi \rangle$ are said to be *orthogonal* if
- $$\langle \psi | \phi\rangle=0 \, .$$
- By analogy with regular vector spaces, we can think of these two state vectors $|\psi \rangle$ and $|\phi \rangle$ as being *perpendicular* to each other. Note that for a quantum system occupying a certain state, there is a vanishing probability of it being observed in a state orthogonal to it.
-
-!!! info "Norm:"
- The quantity $\sqrt{\langle \psi | \psi \rangle}$ is known as the *length* or the *norm* of the state vector $|\psi\rangle$. You can see from the properties of complex algebra that this length must be a real number.
-
-A physically valid state $|\psi \rangle$ must be normalized to unity, that is $\langle \psi | \psi \rangle=1$. Note that a state that cannot be normalized to unity does not represent a physically acceptable state.
+!!! info
+ 1. Complex conjugate:
+ $$\langle \psi | \phi \rangle=\langle \phi | \psi \rangle^*$$
+ 2. Distributivity and associativity:
+ $$\langle \psi |\{c_1 |\phi_1\rangle+c_2 |\phi_2 \rangle\}=c_1\langle \psi | \phi_1\rangle+c_2\langle \psi | \phi_2\rangle$$
+ 3. Positivity:
+ $$\langle \psi | \psi \rangle\geq0 \, .$$
+ If $\langle \psi | \psi \rangle = 0$ then, this implies that the state vector $|\psi\rangle=0$ is the null element of the Hilbert space.
+ 4. Orthogonality:
+ Two states $|\psi \rangle$ and $|\phi \rangle$ are said to be *orthogonal* if
+ $$\langle \psi | \phi\rangle=0 \, .$$
+ By analogy with regular vector spaces, we can think of these two state vectors $|\psi \rangle$ and $|\phi \rangle$ as being *perpendicular* to each other. Note that for a quantum system occupying a certain state, there is a vanishing probability of it being observed in a state orthogonal to it.
+ 5. Norm:
+ The quantity $\sqrt{\langle \psi | \psi \rangle}$ is known as the *length* or the *norm* of the state vector $|\psi\rangle$.
+ You can see from the properties of complex algebra that this length must be a real number. A physically valid state $|\psi \rangle$ must be normalized to unity, that is $\langle \psi | \psi \rangle=1$. Note that a state that cannot be normalized to unity does not represent a physically acceptable state.
A set of orthonormal basis vectors $\{|\psi_i\rangle\text{;}\; i=1,2,3,...,n\}$ will have the property $\langle \psi_i |\psi_j \rangle=\delta_{ij}$ where $\delta_{ij}$ is a mathematical symbol known as the *Kronecker delta*, which equals unity if $i=j$ and zero if $i\neq j$.
--
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