Last hour: Matrices in QM

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Presentation transcript:

Last hour: Matrices in QM Use matrix algebra to solve EF/EV problems in QM (most common computational approach in QM programs). Construct matrix representation of an operator Â using ajk = <uj|Â|uk> Resulting matrix A encodes operation of Â on basis functions |uj> Interpretation of matrix elements: Diagonal elements akk = <uk|Â|uk> is <Â> for system in state |uk> Off-diagonal elements ajk = <uj|Â|uk> describe how two states |uk> and |uk> are coupled by Â. If the uj are EFs of the operator Â, the matrix is diagonal. Solve det(A -1) = 0 to obtain EVs . Plug in EVs into A (j) = j (j) to get eigenvectors, which represent the EFs of Â expressed in the basis of the {uj}.

Learning Goals for Chapter 13 – Matrix QM After this chapter, the related homework problems, and reading the relevant parts of the textbook, you should be able to: construct the matrix representation of an operator for a given basis; find the eigenfunctions and eigenvalues of an operator using matrices; evaluate effects of matrix size on the accuracy of the results.

Learning Goals for Chapter 14 – The 2-State-System After this chapter, the related homework problems, and reading the relevant parts of the textbook, you should be able to: interpret the structure of the Hamiltonian and the contributions of its parts to the eigenvalues in a 2-state-system; apply limiting behavior for the coupling of the levels in a 2-state-system where appropriate.