Perturbation of a Magnetic Field

[Taylor_1989]

Homework Statement

Could someone please see if my working are correct for this question, I have never actually done a question of this nature before, and after reading up about the derivation on the perturbation I thought I give ago and apply, my final answer dose not seem correct, as I believe the energy is suppose to decrease not increase when split.

I have attached a picture of the question due to the large content

Homework Equations

The Attempt at a Solution

My working as follows

a) Ground Energy Level

Unperturbed Energy level
$$E_0^{(0)}=-h\bot $$
First Order Perturbed Ground Energy Level
$$E_0^{(1)}=\langle 0|\hat{H}|0\rangle = \frac{1}{\sqrt{2}}\begin{pmatrix}1&-1\end{pmatrix}\begin{pmatrix}h\bot &0\\ 0&-h\bot \end{pmatrix}\:\frac{1}{\sqrt{2}}\begin{pmatrix}1\\ 1\end{pmatrix}=\frac{1}{2}\left(h\bot -h\bot \right)=0 $$

First Excited Energy level

perturbed Energy level
$$E_1^{0}=h\bot $$
First Order Perturbed First Energy Level
$$E_1^{(1)}=\langle 1|\hat{H}|1\rangle = \frac{1}{\sqrt{2}}\begin{pmatrix}1&1\end{pmatrix}\begin{pmatrix}h\bot \:&0\\ \:0&-h\bot \:\end{pmatrix}\:\frac{1}{\sqrt{2}}\begin{pmatrix}1\\ \:1\end{pmatrix}=\frac{1}{2}\left(h\bot -h\bot \right)=0$$

$$E_0\approx -h\bot +\left(\frac{h\parallel }{h\bot }\right)^2E_0^{(2)} $$

$$E_1\approx h\bot +\left(\frac{h\parallel }{h\bot }\right)^2E_1^{(2)}$$

b)
Ground state second order perturbed
$$E_0^{\left(2\right)}=\frac{\left|\langle1^{0}|\hat{H_1}|0^{(0)}\rangle\right|^2}{E_0^{\left(0\right)}-E_1^{\left(0\right)}}$$

$$\langle 1^{(0)}|\hat{H_1}|0^{(0)}\rangle= \frac{1}{\sqrt{2}}\left(1,1\right)\begin{pmatrix}h\bot &0\\ 0&-h\bot \end{pmatrix}\:\frac{1}{\sqrt{2}}\begin{pmatrix}1\\ -1\end{pmatrix}=\frac{1}{2}\left(h\bot -h\bot \right)=0$$

$$E_0^{(2)}=0$$

First Excited state second order perturbed

$$E_0^{\left(2\right)}=\frac{\left|\langle0^{0}|\hat{H_1}|1^{(0)}\rangle\right|^2}{E_1^{\left(0\right)}-E_0^{\left(0\right)}} $$

$$ \langle 1^{(0)}|\hat{H_1}|0^{(0)}\rangle =\frac{1}{\sqrt{2}}\left(1,-1\right)\begin{pmatrix}h\bot &0\\ 0&-h\bot \end{pmatrix}\:\frac{1}{\sqrt{2}}\begin{pmatrix}1\\ 1\end{pmatrix}=h\bot$$

$$E_1^{\left(2\right)}=\frac{h\bot }{2}$$

$$E_1=h\bot+\left(\frac{h\parallel }{h\bot }\right)^2\frac{h\bot }{2}$$

c)
$$\Delta E=E_1-E_0$$

$$\Delta E=h\bot +\left(\frac{h\parallel }{h\bot }\right)^2\:\frac{h\bot }{2}+h\bot =2h\bot +\left(\frac{h\parallel \:}{h\bot \:}\right)^2\:\frac{h\bot \:}{2}$$

$$\Delta E=2T+\left(\frac{0.1}{1}\right)^2\cdot \frac{1}{2\:}=2.005$$

As mention I do not believe this to be correct , If possible could someone please point out my mistake.

 

[TSny]

b)
Ground state second order perturbed
$$\langle 1^{(0)}|\hat{H_1}|0^{(0)}\rangle= \frac{1}{\sqrt{2}}\left(1,1\right)\begin{pmatrix}h\bot &0\\ 0&-h\bot \end{pmatrix}\:\frac{1}{\sqrt{2}}\begin{pmatrix}1\\ -1\end{pmatrix}=\frac{1}{2}\left(h\bot -h\bot \right)=0$$

Check this. Looks like a sign error in the calculation. Otherwise, it generally looks good.

 

[Taylor_1989]

Check this. Looks like a sign error in the calculation. Otherwise, it generally looks good.

Ah, thank you.