Phy5646 PerturbationExample1
Posted by student team #5 (Chelsey Morien, Anthony Kuchera, Jeff Klatsky)
Adapted from Zettili Quantum Mechanics - Concepts and Application; Solved Problem 9.6
Consider a system whose Hamiltonian is given by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_0\begin{pmatrix} 1 + \lambda & 0 & 0 & 0 \\ 0 & 8 & 0 & 0 \\ 0 & 0 & 3 & -2\lambda \\ 0 & 0 & -2\lambda & 7 \end{pmatrix} } where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda \ll 1 }
(a) By decomposing the Hamiltonian into Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal H = \mathcal H_0 + \mathcal H' } , find the eigenvalues and eigenvectors of the unperturbed Hamiltonian.
(b) Diagonalize Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal H } to find the exact eigenvalues of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal H } ; expand each eigenvalue to the second power of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda }
(c) Using first and second-order non-degenerate perturbation theory, find the approximate eigenenergies of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal H } and the eigenstates to the first order. Compare these with the exact values obtained in (b).
Solution:
(a) The matrix of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal H } can be separated:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal H = \mathcal H_0 + \mathcal H' = E_0\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 8 & 0 & 0 \\ 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 7 \end{pmatrix} + E_0\begin{pmatrix} \lambda & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -2\lambda \\ 0 & 0 & -2\lambda & 0 \end{pmatrix} }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal H_0 } is already diagonalized, so reading off its eigenvalues and eigenstates are trivial:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_1^{(0)} = E_0,} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_2^{(0)} = 8E_0,} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_3^{(0)} = 3E_0,} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_4^{(0)} = 7E_0}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\psi_1\rangle = \begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \end{pmatrix} ,} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\psi_2\rangle = \begin{pmatrix} 0 \\ 1 \\ 0 \\ 0 \end{pmatrix} ,} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\psi_3\rangle = \begin{pmatrix} 0 \\ 0 \\ 1 \\ 0 \end{pmatrix} ,} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\psi_4\rangle = \begin{pmatrix} 0 \\ 0 \\ 0 \\ 1 \end{pmatrix} }
(b) The diagonalization of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal H } leads to the following equation:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle det\begin{pmatrix} (1 + \lambda)E_0 - E & 0 & 0 & 0 \\ 0 & 8E_0 - E & 0 & 0 \\ 0 & 0 & 3E_0 - E & -2\lambda E_0 \\ 0 & 0 & -2E_0\lambda & 7E_0 - E \end{pmatrix} = 0 }
which is equivalent to:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (E_0 + \lambda E_0 - E)(8E_0 - E)\left[(3E_0 - E)(7E_0 - E) - 4\lambda^2E_0^2 \right] = 0 }
Solving the above equation for E yields the following exact eigenenergies:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_1^{ } =(1+\lambda)E_{0} }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_2^{ } = 8E_{0}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_3 = (5 - 2\sqrt{1+\lambda^2})E_0}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_4 = (5 - 2\sqrt{1+\lambda^2})E_0}
Since we have defined Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda << 1 } , we can expand Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{1+\lambda^2}} , keeping only terms up to second order in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda_{ }^{ } } :
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{1+\lambda^2} \simeq 1 + \frac{\lambda^2}{2}} , which leads to:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_3 \simeq (3 - \lambda^2)E_0}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_4 \simeq (7 + \lambda^2)E_0}
(c) From nondegenerate perturbation theory, we can write the first-order corrections to the energies as:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_1^{(1)} = \langle \psi_1 | \mathcal H' | \psi_1 \rangle }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = E_0 \begin{pmatrix} 1 & 0 & 0 & 0 \end{pmatrix} \begin{pmatrix} \lambda & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -2\lambda \\ 0 & 0 & -2\lambda & 0 \end{pmatrix} \begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \end{pmatrix} = \lambda E_0 }
Similarly for the second, third, and fourth eigenvalues we find that there are no first order corrections:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_2^{(1)} = \langle \psi_2 | \mathcal H' | \psi_2 \rangle = 0}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_3^{(1)} = \langle \psi_3 | \mathcal H' | \psi_3 \rangle = 0}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_4^{(1)} = \langle \psi_4 | \mathcal H' | \psi_4 \rangle = 0}
Now solving for second order corrections to the energy, by non-degenerate perturbation theory we have:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_1^{(2)} = \sum_{i = 2,3,4} \frac{| \langle \psi_m | \mathcal H' | \psi_1 \rangle |^2}{E_1^{(0)} - E_m^{(0)}} = 0} since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle \psi_2 | \mathcal H' | \psi_1 \rangle = \langle \psi_3 | \mathcal H' | \psi_1 \rangle = \langle \psi_4 | \mathcal H' | \psi_1 \rangle = 0 }
Similarly for the other eigenenergies:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_2^{(2)} = \sum_{m = 1,3,4} \frac{| \langle \psi_m | \mathcal H' | \psi_2 \rangle |^2}{E_2^{(0)} - E_m^{(0)}} = 0 }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_3^{(2)} = \sum_{m = 1,2,4} \frac{| \langle \psi_m | \mathcal H' | \psi_3 \rangle |^2}{E_3^{(0)} - E_m^{(0)}} = \frac{| \langle \psi_4 | \mathcal H' | \psi_3 \rangle |^2}{E_3^{(0)} - E_4^{(0)}} = \frac{(-2\lambda E_0)^2}{(3-7)E_0} = -\lambda^2E_0 }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_4^{(2)} = \sum_{m = 1,2,3} \frac{| \langle \psi_m | \mathcal H' | \psi_4 \rangle |^2}{E_4^{(0)} - E_m^{(0)}} = \frac{| \langle \psi_3 | \mathcal H' | \psi_4 \rangle |^2}{E_4^{(0)} - E_3^{(0)}} = \frac{(-2\lambda E_0)^2}{(7-3)E_0} = \lambda^2E_0 }
Adding the unperturbed, first order, and second order energies, we obtain the following approximate values for the eigenenergies of the full Hamiltonian:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_1 = E_1^{(0)} + E_1^{(1)} + E_1^{(2)} = (1+\lambda)E_0 }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_2 = E_2^{(0)} + E_2^{(1)} + E_2^{(2)} = 8E_0 }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_3 = E_3^{(0)} + E_3^{(1)} + E_3^{(2)} = (3 - \lambda ^2)E_0 }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_4 = E_4^{(0)} + E_4^{(1)} + E_4^{(2)} = (7 + \lambda ^2)E_0 }
These values for the energies match identically with the expressions for the exact eigenenergies after they were Taylor expanded and truncated to second order in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda } .
Calculating the first order correction to the eigenstates:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\Psi_1^{(1)}\rangle = \sum_{m = 2,3,4} \frac{ \langle \psi_m | \mathcal H' | \psi_1 \rangle }{E_m^{(0)} - E_1^{(0)}}|\psi_m\rangle = \begin{pmatrix} 0 \\ 0 \\ 0 \\ 0 \end{pmatrix} }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\Psi_2^{(1)}\rangle = \sum_{m = 1,3,4} \frac{ \langle \psi_m | \mathcal H' | \psi_2 \rangle }{E_m^{(0)} - E_2^{(0)}}|\psi_m\rangle = \begin{pmatrix} 0 \\ 0 \\ 0 \\ 0 \end{pmatrix} }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\Psi_3^{(1)}\rangle = \sum_{m = 1,2,4} \frac{ \langle \psi_m | \mathcal H' | \psi_3 \rangle }{E_m^{(0)} - E_3^{(0)}}|\psi_m\rangle = \frac{ \langle \psi_4 | \mathcal H' | \psi_3 \rangle }{E_4^{(0)} - E_3^{(0)}}|\psi_4\rangle \begin{pmatrix} 0 \\ 0 \\ 1 \\ -\lambda/2 \end{pmatrix} }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\Psi_4^{(1)}\rangle = \sum_{m = 1,2,3} \frac{ \langle \psi_m | \mathcal H' | \psi_4 \rangle }{E_m^{(0)} - E_4^{(0)}}|\psi_m\rangle = \frac{ \langle \psi_3 | \mathcal H' | \psi_4 \rangle }{E_3^{(0)} - E_4^{(0)}}|\psi_3\rangle \begin{pmatrix} 0 \\ 0 \\ \lambda/2 \\ 0 \end{pmatrix} }
Adding the unperturbed and first order correction eigenstates gives the approximate expressions for the eigenstates of the full Hamiltonian:
