Phy5646 PerturbationExample1: Difference between revisions

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(New page: 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 Ha...)
 
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<math>
<math>
\begin{pmatrix}
det\begin{pmatrix}
  (1 + \lambda)E_0 - E & 0 & 0 & 0 \\
  (1 + \lambda)E_0 - E & 0 & 0 & 0 \\
  0 & 8E_0 - E & 0 & 0 \\
  0 & 8E_0 - E & 0 & 0 \\
  0 & 0 & 3E_0 - E & -2\lambda E_0 \\
  0 & 0 & 3E_0 - E & -2\lambda E_0 \\
  0 & 0 & -2E_0\lambda & 7E_0 - E  
  0 & 0 & -2E_0\lambda & 7E_0 - E  
\end{pmatrix}
\end{pmatrix} = 0
</math>
</math>
which is equivalent to:
<math> (E_0 + \lambda E_0 - E)(8E_0 - E)\left[(3E_0 - E)(7E_0 - E) - 4\lambda^2E_0^2 \right] = 0 </math>
Solving the above equation for E yields the following '''''exact''''' eigenenergies:
<math>E_1^{ } =(1+\lambda)E_{0} </math>
<math>E_2^{ } = 8E_{0}</math>
<math>E_3 = (5 - 2\sqrt{1+\lambda^2})E_0</math>
<math>E_4 = (5 - 2\sqrt{1+\lambda^2})E_0</math>
Since we have defined <math>\lambda << 1 </math>, we can expand <math>\sqrt{1+\lambda^2}</math>, keeping only terms up to second order in <math> \lambda_{ }^{ } </math>:
<math>\sqrt{1+\lambda^2} \simeq 1 + \frac{\lambda^2}{2}</math>, which leads to:
<math>E_3 \simeq (3 - \lambda^2)E_0</math>
<math>E_4 \simeq (7 + \lambda^2)E_0</math>
(c)

Revision as of 16:56, 5 February 2010

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 \\ 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_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)

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