QM Sakurai 1.9 - Problem at last step only

[alienskin562]

Homework Statement

http://www.ocf.berkeley.edu/~yayhdapu/postings/sak19.gif

Homework Equations

The Attempt at a Solution

http://www.ocf.berkeley.edu/~yayhdapu/postings/sakurai1.9.pdf"
http://www.ocf.berkeley.edu/~yayhdapu/postings/sakurai1.9.docx"
These are here in this attached PDF. You can see where I'm stuck - how do I solve for a and b?? It just so happens that my BA is in math :(

http://www.ocf.berkeley.edu/~yayhdapu/postings/sak19_2.gif

a and b are related non-algebraically, no?

 

[weejee]

These two equations are actually redundant, as an eigenvector equation is supposed to be.

You can express 'b' in terms of 'a' or vice versa, and an additional constraint comes from the normalization condition |a|^2 + |b|^2 = 1.

 

[alienskin562]

Hey thanks, I'm such a n00b. Cleared up now.

 

[Bill Foster]

How about clearing it up for me?

 

[alienskin562]

Its just this Bill - since the eigenvalue equaiton is redundant, you can go ahead and specify either a or b as long as the normalization condition is still satisfied.

So, since its a free parameter, you set a equal to the expected solution eigenket, a=cos beta

Then you solve for b.

 

[Bill Foster]

If a = cos(β), then how do you get it into the final form given by Sakurai? Because his form has a = cos(β/2)

 

[alienskin562]

I just forgot the /2 :p