The system described by the Hamiltonian \( H_0 \) has just two orthogonal energy eigenstates \( |1\rangle \) and \( |2\rangle \), with \[ \langle 1|1 \rangle = 1 = \langle 2|2 \rangle, \quad \langle 1|2 \rangle = 0 = \langle 2|1 \rangle \] The two eigenstates have the same energy eigenvalue \( E_0 \): \[ H_0 |i\rangle = E_0 |i\rangle, \quad i = 1, 2 \] Now suppose the Hamiltonian for the system is changed by the addition of the term \( V \), giving \[ H = H_0 + V \] The matrix elements of \( V \) are \( \langle 1|V|2 \rangle = V_{12} = \langle 2|V|1 \rangle \), \[ \langle 1|V|1 \rangle = 0 = \langle 2|V|2 \rangle \quad \text{where } V_{12} \text{ is real.} \] **Tasks:** a. Find the eigenvalues of the new Hamiltonian, \( H \), in terms of the above quantities. b. Find the normalized eigenstates of \( H \) in terms of \( |1\rangle, |2\rangle \) and the other given expressions. Hint: Write \( H_0, V \) and \( H \) as 2x2 matrices and the states as column vectors.
The system described by the Hamiltonian \( H_0 \) has just two orthogonal energy eigenstates \( |1\rangle \) and \( |2\rangle \), with \[ \langle 1|1 \rangle = 1 = \langle 2|2 \rangle, \quad \langle 1|2 \rangle = 0 = \langle 2|1 \rangle \] The two eigenstates have the same energy eigenvalue \( E_0 \): \[ H_0 |i\rangle = E_0 |i\rangle, \quad i = 1, 2 \] Now suppose the Hamiltonian for the system is changed by the addition of the term \( V \), giving \[ H = H_0 + V \] The matrix elements of \( V \) are \( \langle 1|V|2 \rangle = V_{12} = \langle 2|V|1 \rangle \), \[ \langle 1|V|1 \rangle = 0 = \langle 2|V|2 \rangle \quad \text{where } V_{12} \text{ is real.} \] **Tasks:** a. Find the eigenvalues of the new Hamiltonian, \( H \), in terms of the above quantities. b. Find the normalized eigenstates of \( H \) in terms of \( |1\rangle, |2\rangle \) and the other given expressions. Hint: Write \( H_0, V \) and \( H \) as 2x2 matrices and the states as column vectors.
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![The system described by the Hamiltonian \( H_0 \) has just two orthogonal energy eigenstates \( |1\rangle \) and \( |2\rangle \), with
\[
\langle 1|1 \rangle = 1 = \langle 2|2 \rangle, \quad \langle 1|2 \rangle = 0 = \langle 2|1 \rangle
\]
The two eigenstates have the same energy eigenvalue \( E_0 \):
\[
H_0 |i\rangle = E_0 |i\rangle, \quad i = 1, 2
\]
Now suppose the Hamiltonian for the system is changed by the addition of the term \( V \), giving
\[
H = H_0 + V
\]
The matrix elements of \( V \) are \( \langle 1|V|2 \rangle = V_{12} = \langle 2|V|1 \rangle \),
\[
\langle 1|V|1 \rangle = 0 = \langle 2|V|2 \rangle \quad \text{where } V_{12} \text{ is real.}
\]
**Tasks:**
a. Find the eigenvalues of the new Hamiltonian, \( H \), in terms of the above quantities.
b. Find the normalized eigenstates of \( H \) in terms of \( |1\rangle, |2\rangle \) and the other given expressions. Hint: Write \( H_0, V \) and \( H \) as 2x2 matrices and the states as column vectors.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa51fe8c9-754c-4d41-9e28-7c8620c80147%2Fc1606bfe-54b0-4461-854e-e303858ea5da%2Fiuc120d_processed.png&w=3840&q=75)
Transcribed Image Text:The system described by the Hamiltonian \( H_0 \) has just two orthogonal energy eigenstates \( |1\rangle \) and \( |2\rangle \), with
\[
\langle 1|1 \rangle = 1 = \langle 2|2 \rangle, \quad \langle 1|2 \rangle = 0 = \langle 2|1 \rangle
\]
The two eigenstates have the same energy eigenvalue \( E_0 \):
\[
H_0 |i\rangle = E_0 |i\rangle, \quad i = 1, 2
\]
Now suppose the Hamiltonian for the system is changed by the addition of the term \( V \), giving
\[
H = H_0 + V
\]
The matrix elements of \( V \) are \( \langle 1|V|2 \rangle = V_{12} = \langle 2|V|1 \rangle \),
\[
\langle 1|V|1 \rangle = 0 = \langle 2|V|2 \rangle \quad \text{where } V_{12} \text{ is real.}
\]
**Tasks:**
a. Find the eigenvalues of the new Hamiltonian, \( H \), in terms of the above quantities.
b. Find the normalized eigenstates of \( H \) in terms of \( |1\rangle, |2\rangle \) and the other given expressions. Hint: Write \( H_0, V \) and \( H \) as 2x2 matrices and the states as column vectors.
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