Question

1. (30 points) The Hamiltonian operator, H, corresponds to energy. For some quantum system, the matrix representation of H is H = ? ? 1 -i ? ? i 1 ? where ? is a constant in units of energy. Let say the state ket |?? of the system is represented as |?? = 1/?2 ? -i ? ? 1 ? (a) What values one can obtain when measuring the energy and with what probabilities? (b) Calculate the expectation values ?H? and ?H²?. (c) Calculate the variance in energy, i.e., ?(?H)²?.

          1. (30 points) The Hamiltonian operator, H, corresponds to energy. For some quantum system, the matrix representation of H is
H = ? ? 1 -i ?
    ? i  1 ?
where ? is a constant in units of energy. Let say the state ket |?? of the system is represented as
|?? = 1/?2 ? -i ?
         ?  1 ?
(a) What values one can obtain when measuring the energy and with what probabilities?
(b) Calculate the expectation values ?H? and ?H²?.
(c) Calculate the variance in energy, i.e., ?(?H)²?.

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1. (30 points) The Hamiltonian operator, H, corresponds to energy. For some quantum system, the matrix representation of H is
H = ? ? 1 -i ?
? i 1 ?
where ? is a constant in units of energy. Let say the state ket |?? of the system is represented as
|?? = 1/?2 ? -i ?
? 1 ?
(a) What values one can obtain when measuring the energy and with what probabilities?
(b) Calculate the expectation values ?H? and ?H²?.
(c) Calculate the variance in energy, i.e., ?(?H)²?.

Added by Brandy D.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Madhur L

07/27/2023

Step 1

First, we need to find the eigenvalues of the Hamiltonian matrix H. The eigenvalues correspond to the possible energy values that can be obtained when measuring the energy of the system. The Hamiltonian matrix H is given by: \[H = \begin{pmatrix} a & b \\ b & -a Show more…

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The Hamiltonian operator, H, corresponds to energy. For some quantum system, the matrix representation of H is given by: H = ϵ ⎛ 1 -i ⎞ ⎜ i 1 ⎟ where ϵ is a constant with units of energy. Let's say the state ket |α⟩ of the system is represented as |α⟩. (a) What values can one obtain when measuring the energy, and with what probabilities? (b) Calculate the expectation values and variance in energy.

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