The system described by the Hamiltonian \( H_0 \) has just two orthogonal energy eigenstates \( |1\rangle \) and \( |2\rangle \), with \( \langle 1|1 \rangle = 1 = \langle 2|2 \rangle \), \( \langle 1|2 \rangle = 0 = \langle 2|1 \rangle \). The two eigenstates have the same energy eigenvalue \( E_0: \, H_0|i\rangle = E_0|i\rangle \), \( i=1,2 \). 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 \( \langle 1|V|2 \rangle = V_{12} = \langle 2|V|1 \rangle \), \( \langle 1|V|1 \rangle = 0 = \langle 2|V|2 \rangle \) where \( V_{12} \) is real. 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 \( 2 \times 2 \) 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 \( \langle 1|1 \rangle = 1 = \langle 2|2 \rangle \), \( \langle 1|2 \rangle = 0 = \langle 2|1 \rangle \). The two eigenstates have the same energy eigenvalue \( E_0: \, H_0|i\rangle = E_0|i\rangle \), \( i=1,2 \). 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 \( \langle 1|V|2 \rangle = V_{12} = \langle 2|V|1 \rangle \), \( \langle 1|V|1 \rangle = 0 = \langle 2|V|2 \rangle \) where \( V_{12} \) is real. 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 \( 2 \times 2 \) matrices and the states as column vectors.
Related questions
Question
100%
Quantum Physics 1 please help asap. I will rate you pls do it fast
![The system described by the Hamiltonian \( H_0 \) has just two orthogonal energy eigenstates \( |1\rangle \) and \( |2\rangle \), with \( \langle 1|1 \rangle = 1 = \langle 2|2 \rangle \), \( \langle 1|2 \rangle = 0 = \langle 2|1 \rangle \).
The two eigenstates have the same energy eigenvalue \( E_0: \, H_0|i\rangle = E_0|i\rangle \), \( i=1,2 \).
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 \( \langle 1|V|2 \rangle = V_{12} = \langle 2|V|1 \rangle \), \( \langle 1|V|1 \rangle = 0 = \langle 2|V|2 \rangle \) where \( V_{12} \) is real.
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 \( 2 \times 2 \) 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%2F8aaef7e1-b3fb-4c31-a69b-a735e0363b3c%2Fk3t2f99_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 \( \langle 1|1 \rangle = 1 = \langle 2|2 \rangle \), \( \langle 1|2 \rangle = 0 = \langle 2|1 \rangle \).
The two eigenstates have the same energy eigenvalue \( E_0: \, H_0|i\rangle = E_0|i\rangle \), \( i=1,2 \).
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 \( \langle 1|V|2 \rangle = V_{12} = \langle 2|V|1 \rangle \), \( \langle 1|V|1 \rangle = 0 = \langle 2|V|2 \rangle \) where \( V_{12} \) is real.
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 \( 2 \times 2 \) matrices and the states as column vectors.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 2 images
