- #1
Xyius
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So I know this might be a lot to read but I am having a very hard time understanding how to use the formulas in degenerate perturbation theory. Here is the problem I am on.
A system of two spin-1/2 particles is described by the following Hamiltonian.
[tex]\hat{H}=\hat{\vec{S}}_1 \cdot \hat{\vec{S}}_2 + B \hat{S}_1^z[/tex]
For simplicity, let [itex]\hbar =1 [/itex] and the Hamiltonian be dimensionless.
(i) Compute the exact energy eigenvalues and eigenvectors.
(ii) Treating the term proportional to B as a perturbation, compute the corrections to the unperturbed energies through second order in B and to the eignkets through first order in B.
(iii) How do your results in (ii) compare to the Taylor expansion of the exact results for the energy eigenvalues and eigenvectors in (i)?
In the following equations, superscripts represent orders of approximation (first order, second order, ect.). [itex]|n^0,l_n>[/itex] represents a ket in the degenerate subspace. S_n is the degenerate subspace. [itex]|k^0>[/itex] represents a ket that is NOT in the degenerate subspace.
(1): First order Energy Correction
[tex]\Delta^1_{n,l_n}=<n^0,l_n|\hat{V}|n^0,l_n>[/tex]
(2): Second order Energy Correction
[tex]\Delta^2_{n,l_n}= \sum_{k \notin S_n}\frac{|<k^0|\hat{V}|n^0,l_n>|^2}{E_n^0-E_k^0}[/tex]
(3): First order approximation for the eigenkets in the degenerate subspace
[tex]|n^1,l_n>=\sum_{l_n \ne l_n'}\frac{|n^0,l_n>}{\Delta_{n,l_n}^1-\Delta_{n,l_n'}^1}\sum_{k \notin S_n}<n^0,l_n|\hat{V}|k^0>\frac{1}{E_n^0-E_k^0}<k^0|\hat{V}|n^0,l_n>[/tex]
(i) So for this one, I first re-wrote the Hamiltonian by computing the dot product and writing the [itex]S_x[/itex] and [itex]S_y[/itex] terms in terms of the ladder operators. I then wrote all the possible kets I can have in a system with two spin 1/2 particles, then using clebch-gordon coefficients, re-wrote them in the [itex]|s_1,s_2,m_1,m_2>[/itex] basis. I then computed the entire matrix and calculated the eigenvalues. There was a LOT of room for error here but I got..
[tex]E_1=-\frac{1}{4}[/tex]
[tex]E_2=\frac{1}{4}-\frac{1}{2}B[/tex]
[tex]E_{3,4}=\frac{1}{3}B \pm \sqrt{\frac{1}{9}B^2+\frac{5}{6}}[/tex]
First question: How is it possible I got a negative energy? I then plugged these into the matrix and found the eigevectors.
(ii) This is the part I am really stuck on. First I computed the matrix form of the perturbation V. Which was NOT diagonal, but only had TWO terms survive. I then diagonalized it and found the eigenvalues to be..
[tex]\text{Eigenvalues}=0,-\frac{B}{2}[/tex]
Meaning that, these are the first order corrections to the energies via equation (1) above. This makes sense considering that in the first two energies I listen above for the total Hamiltonian, the term [itex]-\frac{1}{4}[/itex] is not being added to anything (eigenvalue of 0 in the perturbation) and the term [itex]\frac{1}{4}-\frac{B}{2}[/itex], has the original energy of 1/4, plus the perturbation of B/2. I then wanted to find the eigenenergies of the unperturbed Hamiltonian to confirm this, so I did and indeed the eigenenergies are [itex]\pm \frac{1}{4}[/itex] and [itex]\pm \frac{1}{2}\sqrt{\frac{5}{2}}[/itex].
So here is my second question: Once again I am getting negative energies, why? Also, is this system degenerate? I have four distinct eigenenergies in the unperturbed Hamiltonian, doesn't that mean it is non-degenerate and I can use non-degenerate perturbation theory?
So now my issue comes with calculating the second order energy level shift and the first order eigenkets. For both equations (2) and (3) above, I don't know how to evaluate the following term in the summation.
[tex]\frac{1}{E_n^0-E_k^0}[/tex]
Both of the energy terms in the denominator represent unperturbed energies. But I am just having a really hard time figuring out how to evaluate the summations. I don't know where to go here and I keep trying to look up examples in second order but everything I find only goes to FIRST order.
Can anyone help??
Homework Statement
A system of two spin-1/2 particles is described by the following Hamiltonian.
[tex]\hat{H}=\hat{\vec{S}}_1 \cdot \hat{\vec{S}}_2 + B \hat{S}_1^z[/tex]
For simplicity, let [itex]\hbar =1 [/itex] and the Hamiltonian be dimensionless.
(i) Compute the exact energy eigenvalues and eigenvectors.
(ii) Treating the term proportional to B as a perturbation, compute the corrections to the unperturbed energies through second order in B and to the eignkets through first order in B.
(iii) How do your results in (ii) compare to the Taylor expansion of the exact results for the energy eigenvalues and eigenvectors in (i)?
Homework Equations
In the following equations, superscripts represent orders of approximation (first order, second order, ect.). [itex]|n^0,l_n>[/itex] represents a ket in the degenerate subspace. S_n is the degenerate subspace. [itex]|k^0>[/itex] represents a ket that is NOT in the degenerate subspace.
(1): First order Energy Correction
[tex]\Delta^1_{n,l_n}=<n^0,l_n|\hat{V}|n^0,l_n>[/tex]
(2): Second order Energy Correction
[tex]\Delta^2_{n,l_n}= \sum_{k \notin S_n}\frac{|<k^0|\hat{V}|n^0,l_n>|^2}{E_n^0-E_k^0}[/tex]
(3): First order approximation for the eigenkets in the degenerate subspace
[tex]|n^1,l_n>=\sum_{l_n \ne l_n'}\frac{|n^0,l_n>}{\Delta_{n,l_n}^1-\Delta_{n,l_n'}^1}\sum_{k \notin S_n}<n^0,l_n|\hat{V}|k^0>\frac{1}{E_n^0-E_k^0}<k^0|\hat{V}|n^0,l_n>[/tex]
The Attempt at a Solution
(i) So for this one, I first re-wrote the Hamiltonian by computing the dot product and writing the [itex]S_x[/itex] and [itex]S_y[/itex] terms in terms of the ladder operators. I then wrote all the possible kets I can have in a system with two spin 1/2 particles, then using clebch-gordon coefficients, re-wrote them in the [itex]|s_1,s_2,m_1,m_2>[/itex] basis. I then computed the entire matrix and calculated the eigenvalues. There was a LOT of room for error here but I got..
[tex]E_1=-\frac{1}{4}[/tex]
[tex]E_2=\frac{1}{4}-\frac{1}{2}B[/tex]
[tex]E_{3,4}=\frac{1}{3}B \pm \sqrt{\frac{1}{9}B^2+\frac{5}{6}}[/tex]
First question: How is it possible I got a negative energy? I then plugged these into the matrix and found the eigevectors.
(ii) This is the part I am really stuck on. First I computed the matrix form of the perturbation V. Which was NOT diagonal, but only had TWO terms survive. I then diagonalized it and found the eigenvalues to be..
[tex]\text{Eigenvalues}=0,-\frac{B}{2}[/tex]
Meaning that, these are the first order corrections to the energies via equation (1) above. This makes sense considering that in the first two energies I listen above for the total Hamiltonian, the term [itex]-\frac{1}{4}[/itex] is not being added to anything (eigenvalue of 0 in the perturbation) and the term [itex]\frac{1}{4}-\frac{B}{2}[/itex], has the original energy of 1/4, plus the perturbation of B/2. I then wanted to find the eigenenergies of the unperturbed Hamiltonian to confirm this, so I did and indeed the eigenenergies are [itex]\pm \frac{1}{4}[/itex] and [itex]\pm \frac{1}{2}\sqrt{\frac{5}{2}}[/itex].
So here is my second question: Once again I am getting negative energies, why? Also, is this system degenerate? I have four distinct eigenenergies in the unperturbed Hamiltonian, doesn't that mean it is non-degenerate and I can use non-degenerate perturbation theory?
So now my issue comes with calculating the second order energy level shift and the first order eigenkets. For both equations (2) and (3) above, I don't know how to evaluate the following term in the summation.
[tex]\frac{1}{E_n^0-E_k^0}[/tex]
Both of the energy terms in the denominator represent unperturbed energies. But I am just having a really hard time figuring out how to evaluate the summations. I don't know where to go here and I keep trying to look up examples in second order but everything I find only goes to FIRST order.
Can anyone help??