Leonard Susskind Quantum Mechanics

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William Frederick

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Hi , I am reading this book again and am stumped at exercise 3.3... It seems simple but I have tried a few times..Can anyone tell me what is “ the eigenvalue equation” referred to in the question please..
 
Here is the text from page 85
“Little sigma sub n =
( cos theta sin theta
Sin theta - cos theta ) ( matrix)
Calculate the eigenvectors and eigenvalues of little sigma sub n
Hint: assume the eigenvector lambda sub 1 has the form
(cos a
sin a ) (column vector), where a is an unknown parameter.
Plug this vector into the eigenvalue equation and solve for a in terms of theta. Why did we use a single parameter a? Notice that our suggested column vector must have unit length.”
I must have got this last time I read it because I have ticked it , but although I was following the text ok I have hit a brick wall this time ?
If you can help much appreciated
 
William Frederick said:
Here is the text from page 85
“Little sigma sub n =
( cos theta sin theta
Sin theta - cos theta ) ( matrix)
Calculate the eigenvectors and eigenvalues of little sigma sub n
Hint: assume the eigenvector lambda sub 1 has the form
(cos a
sin a ) (column vector), where a is an unknown parameter.
Plug this vector into the eigenvalue equation and solve for a in terms of theta. Why did we use a single parameter a? Notice that our suggested column vector must have unit length.”
I must have got this last time I read it because I have ticked it , but although I was following the text ok I have hit a brick wall this time ?
If you can help much appreciated
Click to expand...
Ps answers are
lambda sub 1=1
|lambda sub 1> =( cos theta/2
sin theta/2 ) column vector

lambda sub 2 =-1
| lambda sub 2> =(-sin theta/2
cos theta/2) column vector
From p86 susskind and friedman quantum mechanics the throretical minimum
 
William Frederick said:
Ps answers are
lambda sub 1=1
|lambda sub 1> =( cos theta/2
sin theta/2 ) column vector

lambda sub 2 =-1
| lambda sub 2> =(-sin theta/2
cos theta/2) column vector
From p86 susskind and friedman quantum mechanics the throretical minimum
Click to expand...
I do not quite "feel" the brick wall !

Do you know the properties of eigenvalues and eigenvectors of a matrix?

Do you know how to calculate those?

Please "brush up" those topics. Exactly where you are getting lost ?
 
William Frederick said:
Here is the text from page 85
“Little sigma sub n =
( cos theta sin theta
Sin theta - cos theta ) ( matrix)
Calculate the eigenvectors and eigenvalues of little sigma sub n
Hint: assume the eigenvector lambda sub 1 has the form
(cos a
sin a ) (column vector), where a is an unknown parameter.
Plug this vector into the eigenvalue equation and solve for a in terms of theta. Why did we use a single parameter a? Notice that our suggested column vector must have unit length.”
I must have got this last time I read it because I have ticked it , but although I was following the text ok I have hit a brick wall this time ?
If you can help much appreciated
Click to expand...
My guess to your question is
[imath]\sigma _n = \left ( \begin{matrix} cos( \theta ) & sin( \theta ) \\ sin( \theta ) & -cos( \theta ) \end{matrix} \right ) \text{ times matrix}[/imath]
which makes no sense.

An eigenvalue equation is [imath]Av = a v[/imath] where A is a matrix, v is a vector, and a is a real number. The set of values of a is called the "eigenspace" of A and the set of v's is called the "eigenvectors" of A. What you wrote is a solution to something but note that the LHS depends on n and the RHS (apparently) does not unless your "matrix" that you multiply by has an n in it.

Is there any way you could take a photo of the problem and post it here?

-Dan
 
Thank you so much for your replies
I have brushed up on eigenvectors and worked it thru .. it involved trig identities too..I see my request was ambiguous.. you are generous people
 
topsquark said:
My guess to your question is
[imath]\sigma _n = \left ( \begin{matrix} cos( \theta ) & sin( \theta ) \\ sin( \theta ) & -cos( \theta ) \end{matrix} \right ) \text{ times matrix}[/imath]
which makes no sense.

An eigenvalue equation is [imath]Av = a v[/imath] where A is a matrix, v is a vector, and a is a real number. The set of values of a is called the "eigenvalues" of A and the set of v's is called the "eigenvectors" of A. What you wrote is a solution to something but note that the LHS depends on n and the RHS (apparently) does not unless your "matrix" that you multiply by has an n in it.

Is there any way you could take a photo of the problem and post it here?

-Dan
Click to expand...
Sorry, a slight correction in boldface.

-Dan
 
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