Inner product

  • Thread starter Identity
  • Start date
  • #1
Identity
152
0
I was reading "Principles of Quantum Mechanics" - Shankar, and I'm having trouble understanding the inner product. Can someone help me or link me to a site that explains it?

The axioms of the inner product are

1. [tex]\langle V|W\rangle = \langle W|V\rangle^*[/tex]

2. [tex]\langle V|V\rangle \geq 0\ \ \ \ \ 0 \ \ iff\ \ |V\rangle = |0\rangle[/tex]

3. [tex]\langle V|(a|W\rangle +b|Z\rangle ) \equiv \langle V|aW+bZ\rangle = a\langle V|W \rangle +b\langle V|Z \rangle[/tex]

Given that [tex]|V\rangle[/tex] and [tex]|W \rangle[/tex] can be expressed in terms of their basis vectors,

[tex]|V \rangle = \sum_i v_i |i \rangle[/tex]

[tex]|W \rangle = \sum_j w_j|j \rangle[/tex]

Shankar says "we follow the axioms obeyed by the inner product to obtain"

[tex]\langle V|W \rangle = \sum_i \sum_j v_i^*w_j\langle i|j \rangle[/tex]

I don't understand how this comes about?

thanks
 

Answers and Replies

  • #2
Newtime
348
0
I was reading "Principles of Quantum Mechanics" - Shankar, and I'm having trouble understanding the inner product. Can someone help me or link me to a site that explains it?

The axioms of the inner product are

1. [tex]\langle V|W\rangle = \langle W|V\rangle^*[/tex]

2. [tex]\langle V|V\rangle \geq 0\ \ \ \ \ 0 \ \ iff\ \ |V\rangle = |0\rangle[/tex]

3. [tex]\langle V|(a|W\rangle +b|Z\rangle ) \equiv \langle V|aW+bZ\rangle = a\langle V|W \rangle +b\langle V|Z \rangle[/tex]

Given that [tex]|V\rangle[/tex] and [tex]|W \rangle[/tex] can be expressed in terms of their basis vectors,

[tex]|V \rangle = \sum_i v_i |i \rangle[/tex]

[tex]|W \rangle = \sum_j w_j|j \rangle[/tex]

Shankar says "we follow the axioms obeyed by the inner product to obtain"

[tex]\langle V|W \rangle = \sum_i \sum_j v_i^*w_j\langle i|j \rangle[/tex]

I don't understand how this comes about?

thanks

What is your mathematical background? In most basic abstract algebra courses and even some linear algebra courses you will go into inner products and inner product spaces.
 
  • #3
Identity
152
0
Oh I just finished high school and I'm trying to occupying myself in the holidays

(actually nevermind I was able to get it with some help from a friend)
 

Suggested for: Inner product

  • Last Post
Replies
8
Views
609
  • Last Post
Replies
21
Views
834
  • Last Post
Replies
3
Views
1K
  • Last Post
Replies
4
Views
711
Replies
4
Views
374
  • Last Post
Replies
6
Views
917
  • Last Post
Replies
4
Views
1K
  • Last Post
Replies
2
Views
1K
  • Last Post
Replies
1
Views
258
  • Last Post
Replies
1
Views
2K
Top