Consider a system of two particles, one with spin 1 and the other with spin 1/2. The particles interact with one another ia the Hamiltonian H = Eo [S(1) · S(2)] + E₁ [S₂(1) + S:(2)], ħ here Eo and E₁ are constants with units of energy, S is the spin operator and the labels (1) and (2) indicate the article on which it operates. (a) Find the energy spectrum of this system, i.e. the eigenvalues of the Hamiltonian H and identify the ground state and first excited state. For numerical purposes assume Eo = 0.2eV and E₁ = 0.1eV. (b) If the two particle system is known to be in a configuration such that the total spin is 1/2, what values a mea- surement of S% of the spin 1 particle would produce and what are the probabilities associated with each one of those values?
Consider a system of two particles, one with spin 1 and the other with spin 1/2. The particles interact with one another ia the Hamiltonian H = Eo [S(1) · S(2)] + E₁ [S₂(1) + S:(2)], ħ here Eo and E₁ are constants with units of energy, S is the spin operator and the labels (1) and (2) indicate the article on which it operates. (a) Find the energy spectrum of this system, i.e. the eigenvalues of the Hamiltonian H and identify the ground state and first excited state. For numerical purposes assume Eo = 0.2eV and E₁ = 0.1eV. (b) If the two particle system is known to be in a configuration such that the total spin is 1/2, what values a mea- surement of S% of the spin 1 particle would produce and what are the probabilities associated with each one of those values?
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![Consider a system of two particles, one with spin 1 and the other with spin 1/2. The particles interact with one another
via the Hamiltonian
H = Eo [S(1). S(2)] + E₁ [ S:(1) + S. (2)]
ħ
"
where Eo and E₁ are constants with units of energy, S is the spin operator and the labels (1) and (2) indicate the
particle on which it operates.
(a) Find the energy spectrum of this system, i.e. the eigenvalues of the Hamiltonian H and identify the ground state
and first excited state. For numerical purposes assume Eo = 0.2eV and E₁ = 0.1eV.
(b) If the two particle system is known to be in a configuration such that the total spin is 1/2, what values a mea-
surement of S% of the spin 1 particle would produce and what are the probabilities associated with each one of
those values?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F18ff836d-3cc8-4a2b-b721-7f7a69fbe250%2Fc09aa412-8d9a-44cf-82bd-74e900e19d66%2Fvi2gorf_processed.png&w=3840&q=75)
Transcribed Image Text:Consider a system of two particles, one with spin 1 and the other with spin 1/2. The particles interact with one another
via the Hamiltonian
H = Eo [S(1). S(2)] + E₁ [ S:(1) + S. (2)]
ħ
"
where Eo and E₁ are constants with units of energy, S is the spin operator and the labels (1) and (2) indicate the
particle on which it operates.
(a) Find the energy spectrum of this system, i.e. the eigenvalues of the Hamiltonian H and identify the ground state
and first excited state. For numerical purposes assume Eo = 0.2eV and E₁ = 0.1eV.
(b) If the two particle system is known to be in a configuration such that the total spin is 1/2, what values a mea-
surement of S% of the spin 1 particle would produce and what are the probabilities associated with each one of
those values?
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