From e29457d9d3608570c7140289883a92eef9c661d1 Mon Sep 17 00:00:00 2001
From: Maciej Topyla <m.m.topyla@student.tudelft.nl>
Date: Sun, 4 Sep 2022 18:34:01 +0000
Subject: [PATCH] Update src/4_vector_spaces_QM.md
---
src/4_vector_spaces_QM.md | 15 ++++++---------
1 file changed, 6 insertions(+), 9 deletions(-)
diff --git a/src/4_vector_spaces_QM.md b/src/4_vector_spaces_QM.md
index 931820d..72265c9 100644
--- a/src/4_vector_spaces_QM.md
+++ b/src/4_vector_spaces_QM.md
@@ -95,17 +95,14 @@ between them, $\langle{\psi}|{\phi}\rangle$, as follows.
The inner product in quantum mechanics is the analog of the usual scalar product that one encounters in vector spaces, and which we reviewed in the previous lecture. As in usual vector spaces, the inner product of two state vectors is a *scalar* and in this case a complex number in general.
-!!! tip ""
- 1. The value of the inner product $\langle{\psi}|{\phi}\rangle$ indicates the **probability amplitude** (not the probability) of measuring a system, which characterised by the state $|{\phi}\rangle$, to be in the state $|{\psi}\rangle$.
-
-!!! tip ""
- 2. This inner product can also be understood as measuring the **overlap** between the state vectors $|{\psi}\rangle$ and $|{\phi}\rangle$.
-
-!!! tip ""
- 3. Then the **probability of observing the system to be in the state $|\psi\rangle$** given that it is in the state $|\phi\rangle$ will be given by $$|\langle \psi | \phi \rangle|^2$$.
- Since the latter quantity is a probability, we know that it should satisfy the condition that
+!!! tip "Meaning of the inner product in quantum mechanics"
+ 1. The value of the inner product $\langle{\psi}|{\phi}\rangle$ indicates the **probability amplitude** (not the probability) of measuring a system, which characterised by the state $|{\phi}\rangle$, to be in the state $|{\psi}\rangle$.
+ 2. This inner product can also be understood as measuring the **overlap** between the state vectors $|{\psi}\rangle$ and $|{\phi}\rangle$.
+ 3. Then the **probability of observing the system to be in the state $|\psi\rangle$** given that it is in the state $|\phi\rangle$ will be given by $$|\langle \psi | \phi \rangle|^2$$. Since the latter quantity is a probability, we know that it should satisfy the condition that
$$0 \le |\langle \psi | \phi \rangle|^2 \le 1 \, .$$
+### Properties of the inner product
+
The inner product (probability amplitude) $\langle \psi | \phi \rangle$ exhibits the following properties:
!!! info "Complex conjugate:"
--
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