Science > Physics > Little Question about Matrix Representation of Quantum Operator
| Topic: |
Science > Physics |
| User: |
"Calvin Fong" |
| Date: |
06 Sep 2003 11:55:48 PM |
| Object: |
Little Question about Matrix Representation of Quantum Operator |
Dear all,
I came across a question while I'm studying modern quantum mechanics.
I'm using the text book "Modern Quantum Mechanics" written by J.J.
Sakurai. While I'm working on the exercises, I got stuck in the matrix
representation of operator. Here it is:
Suppose a 2x2 matrix X (not necessarily Hermitian, nor unitary) is
written as
X = a0 + b . a
where b & a are vectors and a0, a1, a2, a3 are numbers
From my understanding, a1, a2 & a3 are the magitude of vector a in
different direction.
My real question, how can X be written as X = a0 + b.a
The RHS of the above formula are one dimension. But X should be a 2x2
matrix.
Can anyone give me some hints in understanding this problem. Thank you.
Best Regards,
Calvin FONG
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| User: "Bastian" |
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| Title: Re: Little Question about Matrix Representation of Quantum Operator |
09 Sep 2003 04:06:24 PM |
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If this is one of the questions of the first chapter the answer is
quite easy. The entries auf b are the Pauli Matrizes. a.b is a short
form of a_1*sigma_x+a_2*sigma_y+a_3*sigma_z. There is on little error
in the book - a_0 should be replaced by 1*a_0, and 1 is the 2x2
diag(1,1)-Matrix. The Spin-Matrices and the 1 are all orthogonal to
each other. So you have 4 linear independet Matrices which form a
basis of the 4-dimensional vectorspace of the 2x2 Matrices.
Calvin Fong <hoiwai930@hotmail.com> wrote in message news:<bjed53$s6q$1@news.ctimail.com>...
Dear all,
I came across a question while I'm studying modern quantum mechanics.
I'm using the text book "Modern Quantum Mechanics" written by J.J.
Sakurai. While I'm working on the exercises, I got stuck in the matrix
representation of operator. Here it is:
Suppose a 2x2 matrix X (not necessarily Hermitian, nor unitary) is
written as
X = a0 + b . a
where b & a are vectors and a0, a1, a2, a3 are numbers
From my understanding, a1, a2 & a3 are the magitude of vector a in
different direction.
My real question, how can X be written as X = a0 + b.a
The RHS of the above formula are one dimension. But X should be a 2x2
matrix.
Can anyone give me some hints in understanding this problem. Thank you.
Best Regards,
Calvin FONG
.
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| User: "Laurel Amberdine" |
|
| Title: Re: Little Question about Matrix Representation of Quantum Operator |
09 Sep 2003 02:59:53 PM |
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On Sun, 07 Sep 2003 12:55:48 +0800, Calvin Fong <hoiwai930@hotmail.com> wrote:
Dear all,
I came across a question while I'm studying modern quantum mechanics.
I'm using the text book "Modern Quantum Mechanics" written by J.J.
Sakurai. While I'm working on the exercises, I got stuck in the matrix
representation of operator. Here it is:
Suppose a 2x2 matrix X (not necessarily Hermitian, nor unitary) is
written as
X = a0 + b . a
where b & a are vectors and a0, a1, a2, a3 are numbers
From my understanding, a1, a2 & a3 are the magitude of vector a in
different direction.
My real question, how can X be written as X = a0 + b.a
The RHS of the above formula are one dimension. But X should be a 2x2
matrix.
Can anyone give me some hints in understanding this problem. Thank you.
I am a completely ignorant newbie. Perhap someone more knowledgable will
help. My best guess (I repeat *GUESS*) is that X = a0 + b.a means that
each element of X would be called a0, a1, a2, etc. as in:
[a0 a1]
X= [a2 a3]
And that you compute the value of each element by taking the dot (inner)
product of the relevant a and b and adding it (??) to whatever is already
at that element.
Like I said, this is a complete guess, which I am only bothering to share
because no one else answered. I looked through my QM textbook and the
approach is too different for me to be sure.
(Someday, I *will* be able to usefully answer questions!! Mutter. I'll
go hang around a Linux group for a while to cheer myself up.)
-Laurel
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