Outer product problem of derac notation

In summary, the conversation discusses how the operator |a><b| expresses projection and how it can be represented in matrix format. It is explained that the ket represents a column vector and the bra represents a row vector, and their multiplication results in a square matrix. This matrix performs a projection from the state |b> onto the part that is parallel to |a>. The conversation also discusses the importance of choosing a basis of representation for the kets and bras in order to understand the multiplication of the matrices.
  • #1
hasanhabibul
31
0
why |a><b| expresses the projection...how can it be possible on matrix..if we multiply a ket a with a bra b ...we get a product of two matrix(one is a column matrix,an0ther is row matrix)..from where nothing can be realized very clearly..how this multiplication of matrix can give a projection..??
 
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  • #2
hasanhabibul said:
why |a><b| expresses the projection...how can it be possible on matrix..if we multiply a ket a with a bra b ...we get a product of two matrix(one is a column matrix,an0ther is row matrix)..from where nothing can be realized very clearly..how this multiplication of matrix can give a projection..??
Think of the ket as a column vector, and the bra as a row vector.
So braket gives a scalar, but ketbra gives a square matrix.
 
  • #3
I know ket bra is a square matrix...but what is the especiality of this squar matrix??how it expresses projection.
 
  • #4
(|a><a|)*|b> = <a|b>|a>

It projects from the state |b> the part of it which is parallel to |a>
 
  • #5
pleasez i know those..but i want the matrix format..if |A>=a
b
c
and <B|={d,e,f}. now perform a matrix multiplication this two matrix..and it will give u a result.and how this odd looking matrix express the projection..that was my question.
 
  • #6
In what basis do you want to represent the operator |a><b| ?

(|a><a|)*|b> = <a|b>|a>

IS a matrix multiplying the ket |b> !
 
  • #7
Look at it in a basis where |a> is a basis state
[tex]|a\rangle =\begin{pmatrix} 1\\0\\0\end{pmatrix}[/tex]
then the matrix |a><a| will be
[tex] |a\rangle\langle a|=\begin{pmatrix} 1 & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & 0\end{pmatrix}[/tex]
acting with this on any vector
[tex]|b\rangle=\begin{pmatrix} c\\ d\\ e\end{pmatrix}
[/tex]
will give you just

[tex](|a\rangle\langle a|)|b\rangle=\begin{pmatrix} c \\ 0 \\ 0 \end{pmatrix}=\langle a| b\rangle |a\rangle[/tex]

Does this help?
 
  • #8
u must choose basis of representation for your kets and bras as shown by jensa.
 

FAQ: Outer product problem of derac notation

What is the Outer Product Problem of Derac Notation?

The Outer Product Problem of Derac Notation is a mathematical problem that arises in the context of matrix multiplication. It involves finding the outer product of two matrices, which is a matrix formed by taking the product of each element of one matrix with each element of the other matrix.

Why is the Outer Product Problem of Derac Notation important?

The Outer Product Problem of Derac Notation is important because it is a fundamental operation in linear algebra and is used in many applications, such as image processing, machine learning, and data analysis.

What are some strategies for solving the Outer Product Problem of Derac Notation?

There are several strategies for solving the Outer Product Problem of Derac Notation, including using the distributive property, converting the matrices to vectors, and using the Kronecker product.

What are some common misconceptions about the Outer Product Problem of Derac Notation?

One common misconception about the Outer Product Problem of Derac Notation is that it is the same as matrix multiplication. While the two operations are related, they are not equivalent.

How can I practice and improve my skills in solving the Outer Product Problem of Derac Notation?

You can practice and improve your skills in solving the Outer Product Problem of Derac Notation by working through practice problems, using online resources and tutorials, and seeking help from a math tutor or teacher.

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