Expanding a given vector into another orthonormal basis

  • #1
Pushoam
890
39
upload_2018-8-19_18-29-6.png
upload_2018-8-19_19-51-2.png
upload_2018-8-19_19-55-44.png
upload_2018-8-19_19-56-16.png
upload_2018-8-19_19-57-23.png
upload_2018-8-19_19-57-59.png




Equation 9.2.25 defines the inner product of two vectors in terms of their components in the same basis.
In equation 9.2.32, the basis of ## |V \rangle## is not given.
## |1 \rangle ## and ## |2 \rangle ## themselves form basis vectors. Then how can one calculate ## \langle 1| V \rangle ## ?
Do we have to assume that all of these vectors are given in the orthogonal basis with vectors given by
## \begin{bmatrix}
1
\\ 0
\end{bmatrix} ## and ## \begin{bmatrix}
0
\\ 1
\end{bmatrix} ## ?
 

Attachments

  • upload_2018-8-19_18-29-6.png
    upload_2018-8-19_18-29-6.png
    20.1 KB · Views: 1,290
  • upload_2018-8-19_18-29-42.png
    upload_2018-8-19_18-29-42.png
    35.6 KB · Views: 474
  • upload_2018-8-19_18-31-30.png
    upload_2018-8-19_18-31-30.png
    40.5 KB · Views: 518
  • upload_2018-8-19_18-32-10.png
    upload_2018-8-19_18-32-10.png
    34.4 KB · Views: 480
  • upload_2018-8-19_18-35-13.png
    upload_2018-8-19_18-35-13.png
    110.7 KB · Views: 518
  • upload_2018-8-19_19-51-2.png
    upload_2018-8-19_19-51-2.png
    25.3 KB · Views: 1,212
  • upload_2018-8-19_19-55-44.png
    upload_2018-8-19_19-55-44.png
    44.6 KB · Views: 4,343
  • upload_2018-8-19_19-56-16.png
    upload_2018-8-19_19-56-16.png
    32.8 KB · Views: 988
  • upload_2018-8-19_19-57-23.png
    upload_2018-8-19_19-57-23.png
    11.2 KB · Views: 983
  • upload_2018-8-19_19-57-59.png
    upload_2018-8-19_19-57-59.png
    51.3 KB · Views: 1,201
Last edited:

Answers and Replies

  • #2
jasonRF
Science Advisor
Gold Member
1,501
567
The author is assuming that the components of all of the vectors in 9.2.32 and 9.2.33 are with respect to the same basis. They are being somewhat sloppy in their writing, which is probably the source of your confusion.

jason
 

Suggested for: Expanding a given vector into another orthonormal basis

  • Last Post
Replies
4
Views
162
Replies
2
Views
833
  • Last Post
Replies
4
Views
779
Replies
7
Views
519
Replies
3
Views
304
  • Last Post
2
Replies
38
Views
4K
  • Last Post
Replies
24
Views
772
  • Last Post
Replies
2
Views
644
Top