Consider a two-state problem with the Hamiltonian H = Ho+ AV, where A is a small...

  

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Consider a two-state problem with the Hamiltonian H = Ho+ AV, where A is a small parameter to be treated perturbatively. [10)) and |2(0) are the normalized eigenstates of the unperturbed problem where A = 0, and (10)|V|10) = (20)|V]20) = 0. Below we consider V to be time-independent except for part (V). (I) [5 pts] Write down the matrix form of H in the basis of {10), 12(0)}. (II) [5 pts] The off-diagonal elements of H in (I) can be chosen as real without loss of generality in quantum mechanics. Use the Hermiticity of H to explain why this is the case. (III) [5 pts] Is the choice in (II) legitimate in a non-Hermitian problem? If not, give your reason(s). (Hint: The inner product is defined differently in a non-Hermitian problem, e.g., without the complex conjugation of one eigenstate.) (IV) [10 pts] Find the two energy levels E1,2 of H in (I) and show that they do not cross each other as A varies. (Hint: Expand your result to the second order in A.) Consider a two-state problem with the Hamiltonian H = Ho+ AV, where A is a small parameter to be treated perturbatively. [10)) and |2(0) are the normalized eigenstates of the unperturbed problem where A = 0, and (10)|V|10) = (20)|V]20) = 0. Below we consider V to be time-independent except for part (V). (I) [5 pts] Write down the matrix form of H in the basis of {10), 12(0)}. (II) [5 pts] The off-diagonal elements of H in (I) can be chosen as real without loss of generality in quantum mechanics. Use the Hermiticity of H to explain why this is the case. (III) [5 pts] Is the choice in (II) legitimate in a non-Hermitian problem? If not, give your reason(s). (Hint: The inner product is defined differently in a non-Hermitian problem, e.g., without the complex conjugation of one eigenstate.) (IV) [10 pts] Find the two energy levels E1,2 of H in (I) and show that they do not cross each other as A varies. (Hint: Expand your result to the second order in A.)

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1 H Ho XV 10 20 X V10 V200 E1XV10 E2XV... View the full answer