| Topic: |
Science > Physics |
| User: |
"" |
| Date: |
01 Aug 2007 03:45:57 PM |
| Object: |
Quantum Mechanics Question |
I am trying to work through Thomas Jordan's "Quantum Mechanics in
Simple Matrix Form". I got through the first 6 chapters OK but #7 (2
pages) has me scratching my head.
First of all it give the equation
QP - PQ = ih / 2pi
Where Q is a position matrix and P is a momentum matrix. The problem I
have is that it looks like the left side of the equation is a matrix
and the right side is a (complex) scalar. Any explanation? (I have
found many places in the book where this mixing of matrices on one
side and apparent scalars on the other is confusing.)
In the problems 7-1 Two quantities are represented by the matrices
M=
{ 3 0 -1 }
{ 0 1 0 }
{ 1 0 3 }
N=
{ 3 0 2i }
{ 0 7 0 }
{-2i 0 3 }
The possible values of the quantity represented by M are 1, 2 an"d
4.What are the possible values of the quantity represented by N? How
do you know that?"
Could anyone explain this? (If so, I can then hit you with my
questions about Pauli spin matrices in the next chapter.)
.
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| User: "Eric Gisse" |
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| Title: Re: Quantum Mechanics Question |
01 Aug 2007 07:49:24 PM |
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On Aug 1, 12:45 pm, "faceman28...@yahoo.com" <faceman28...@yahoo.com>
wrote:
I am trying to work through Thomas Jordan's "Quantum Mechanics in
Simple Matrix Form". I got through the first 6 chapters OK but #7 (2
pages) has me scratching my head.
First of all it give the equation
QP - PQ = ih / 2pi
Where Q is a position matrix and P is a momentum matrix. The problem I
have is that it looks like the left side of the equation is a matrix
and the right side is a (complex) scalar. Any explanation? (I have
found many places in the book where this mixing of matrices on one
side and apparent scalars on the other is confusing.)
Understandable mistake. This was something that was hammered into us
repeatedly - these are _operators_ and not just functions. They
operate on _other_ functions. It isn't really an equation between a
matrix and a scalar for that reason.
Specifically, the left hand side is the commutator of Q and P. [Q,P] =
QP - PQ.
In the problems 7-1 Two quantities are represented by the matrices
M=
{ 3 0 -1 }
{ 0 1 0 }
{ 1 0 3 }
N=
{ 3 0 2i }
{ 0 7 0 }
{-2i 0 3 }
The possible values of the quantity represented by M are 1, 2 an"d
4.What are the possible values of the quantity represented by N? How
do you know that?"
It would help if you write down the _whole_ problem.
Could anyone explain this? (If so, I can then hit you with my
questions about Pauli spin matrices in the next chapter.)
.
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| User: "" |
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| Title: Re: Quantum Mechanics Question |
05 Aug 2007 02:00:35 AM |
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On Aug 1, 8:49 pm, Eric Gisse <jowr...@gmail.com> wrote:
In the problems 7-1 Two quantities are represented by the matrices
M=
{ 3 0 -1 }
{ 0 1 0 }
{ 1 0 3 }
N=
{ 3 0 2i }
{ 0 7 0 }
{-2i 0 3 }
The possible values of the quantity represented by M are 1, 2 an"d
4.What are the possible values of the quantity represented by N? How
do you know that?"
It would help if you write down the _whole_ problem.
That is actually the whole problem. I guess you can see why I am
mystified.
.
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| User: "Ray Vickson" |
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| Title: Re: Quantum Mechanics Question |
01 Aug 2007 09:17:53 PM |
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On Aug 1, 1:45 pm, "faceman28...@yahoo.com" <faceman28...@yahoo.com>
wrote:
I am trying to work through Thomas Jordan's "Quantum Mechanics in
Simple Matrix Form". I got through the first 6 chapters OK but #7 (2
pages) has me scratching my head.
First of all it give the equation
QP - PQ = ih / 2pi
Where Q is a position matrix and P is a momentum matrix. The problem I
have is that it looks like the left side of the equation is a matrix
and the right side is a (complex) scalar. Any explanation? (I have
found many places in the book where this mixing of matrices on one
side and apparent scalars on the other is confusing.)
Basically, regard the scalar r as the matrix rI, where I is the
identity matrix. This says that the action of (QP - PQ) on a vector x
is the same as multiplying x by the scalar ih/2pi.
In the problems 7-1 Two quantities are represented by the matrices
M=
{ 3 0 -1 }
{ 0 1 0 }
{ 1 0 3 }
N=
{ 3 0 2i }
{ 0 7 0 }
{-2i 0 3 }
The possible values of the quantity represented by M are 1, 2 an"d
4.
I don't have the book, but I guess they mean "eigenvalues". If so, I
disagree: I get the eigenvalues of M as 1, 3 + i and 3 - i where i =
sqrt(-1). Note: the eigenvalues are the roots r of the characteristic
polynomial det(M - r*I) = det[[3-r,0,-1],[0,1-r,0],[1,0,3-r]] = (3-
r)^2*(1-r) + (1-r) = (1-r)*[(3-r)^2 + 1].
R.G. Vickson
What are the possible values of the quantity represented by N? How
do you know that?"
Could anyone explain this? (If so, I can then hit you with my
questions about Pauli spin matrices in the next chapter.)
.
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| User: "Ben Rudiak-Gould" |
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| Title: Re: Quantum Mechanics Question |
01 Aug 2007 09:36:02 PM |
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Ray Vickson wrote:
M=
{ 3 0 -1 }
{ 0 1 0 }
{ 1 0 3 }
N=
{ 3 0 2i }
{ 0 7 0 }
{-2i 0 3 }
The possible values of the quantity represented by M are 1, 2 an"d
4.
I don't have the book, but I guess they mean "eigenvalues". If so, I
disagree: I get the eigenvalues of M as 1, 3 + i and 3 - i where i =
sqrt(-1).
I think M was supposed to be Hermitian like N. If you negate the lower left
or upper right entries then you get eigenvalues of 1, 2, and 4.
-- Ben
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| User: "" |
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| Title: Re: Quantum Mechanics Question |
05 Aug 2007 02:01:56 AM |
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On Aug 1, 10:36 pm, Ben Rudiak-Gould <br276delet...@cam.ac.uk> wrote:
Ray Vickson wrote:
I think M was supposed to be Hermitian like N. If you negate the lower left
or upper right entries then you get eigenvalues of 1, 2, and 4.
Interestingly, at this point in the book, neither eigenvectors/values
nor the concept of a Hermitian matrix has been introduced.
.
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