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unscientific
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Homework Statement
Part (a): Explain origin of each term in Hamiltonian. What does n, l, m mean?
Part (b): Identify which matrix elements are non-zero
Part (c): Applying small perturbation, find non-zero matrix elements
Part (d): Find combinations of n=2 states and calculate change in energies. Sketch energies before and after perturbation.
Homework Equations
The Attempt at a Solution
Part (a)
First term is KE, second term is PE.
n: energy levels, l: eigenvalues of L2, m: eigenvalues of Lz.
For n =2, 0 ≤ l ≤ 1 and m ≤|l|.
Part (b)
Electric dipole selection rules:
##\Delta l = \pm 1## and ##\Delta m = 0, \pm 1## for l' and m'.
Thus non-zero elements are: ##<2,0,0|z|2,1,0>## and ##<2,1,0|z|2,0,0>##, ##<2,0,0|z|2,1,1>## and ##<2,1,1|z|2,0,0>## and finally ##<2,1,-1|z|2,0,0>##.
You can see that l' on the bra vectors differ by l in the ket by ##\pm1##.
Part(c)
The perturbation is ##eEz##.
I have found that ##<2,0,0|z|2,1,0> = <2,1,0|z|2,0,0> = -3a_z##.
But, the rest give zero values, simply by observing the factor in ##e^{i\phi} d\phi##.
[tex]<2,0,0|z|2,1,1> = \frac{1}{8a_z^4} \frac{-1}{\pi \sqrt{8}} \int_0^{\infty}r^4\left(1 - \frac{r}{2a_z}\right)e^{-\frac{r}{a_z}} dr \int_0^{\pi} sin^2 \theta cos \theta d\theta \int_0^{2\pi} e^{i\phi} d\phi[/tex]
Which is zero since ##\int_0^{2\pi} e^{i\phi} d\phi = 0##.
Same with finding ##<2,1,-1|z|2,1,1>##.
Part(d)
What do they mean linear combination of n=2 states? By perturbation theory, the first order correction to perturbed state ##|E_2'> = |E_2> + \beta |b>##.
Then by comparing the powers of ##\beta## we get:
[tex]|b> = eE \sum_{m\neq 2} \frac{<E_m|z|E_2>}{E_2 - E_m} |E_m>[/tex]