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Homework Statement
I am having a issue with how my lecture has normalised the energy state in this question.
I will post my working and I will print screen his solution to the given question below, we have the same answer but I am unsure to why he has used the ratio method.
Q4. a), b), c)
Homework Equations
The Attempt at a Solution
My solution for a), b), c)
a)
$$\hat HE> = EE> ;[1]$$
$$\hat HE>  EE>=0 ;[2]$$
$$\hat HE>  IEE>=0; [3]$$
$$(\hat HIE)=0 ;[4]$$
$$\begin{pmatrix}0&\Omega \\ \:\Omega &\frac{2\Omega }{\sqrt{3}}\end{pmatrix}E\begin{pmatrix}1&0\\ \:0&1\end{pmatrix}=0 ;[5] $$
So expanding out and finding the determinate of the following matrix
$$\begin{pmatrix}E&\Omega \\ \:\Omega &\frac{2\Omega }{\sqrt{3}}E\end{pmatrix}=0 ; [6]$$
$$\begin{pmatrix}E&\Omega \\ \:\Omega &\frac{2\Omega }{\sqrt{3}}E\end{pmatrix}= E^2\frac{2E\Omega }{\sqrt{3}}\Omega ^2=0 ;[7]$$
$$\left(E\frac{3\Omega }{\sqrt{3}}\right)\left(E+\frac{\Omega }{\sqrt{3}}\right)=0 [8]$$
So solving gives me the following for ##E_{}## and ##E_{+}##
$$E_{}=\frac{3\Omega }{\sqrt{3}} ; [9]$$
$$E_{+}=\frac{\Omega }{\sqrt{3}} ; [10]$$
b) I solved the eignvector for ground state in the following way
$$\begin{pmatrix}0&\Omega \\ \Omega &\frac{2\Omega }{\sqrt{3}}\end{pmatrix}\begin{pmatrix}A\\ B\end{pmatrix}=\frac{\Omega }{\sqrt{3}}\begin{pmatrix}A\\ B\end{pmatrix} ; [11] $$
$$\Omega B= \frac{\Omega }{\sqrt{3}}A ; [12]$$
$$\Omega A=\frac{3}{\sqrt{3}}B ; [13]$$
So solving for both of these give ##A=\sqrt 3## so therfore:
$$E_{}= 0> + \sqrt 3 1> ; [14]$$
To normalize [14] I found the normalizing constant in the following way:
$$< E_{}  E_{} > = 1+3=4 ; [15]$$
$$N^2 < E_{}  E_{} > = 1 ; [16]$$
$$N^2=\frac{1}{4}; [17]$$
$$N= \frac{1}{2}; [18]$$
$$E_{}> = \frac{1}{2} 0> + \frac{\sqrt 3}{2}  1 > ; [19]$$
c) $$P(1)=\left(\frac{\sqrt{3}}{2}\right)^2=\frac{3}{4}=75\% [20]$$
Here is my lecture solution:
a)
b)
last part of b) and all of c)
I just dont understand the ratio method, it is a quicker method than mine or is it used more in more difficult matrices
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