Energy Spectrum of Two-State System

[atomicpedals]

Homework Statement

A two-state system has Hamiltonian

$\sum |i\right\rangle h_i \left\langle i| + Δ (| 1 \right\rangle \left\langle 2| + |2 \right\rangle \left\langle 1 |)$

Where, $\left\langle i | j \right\rangle = \delta_ij$, $h_i$, and Δ are real.

Compute the energy spectrum of this Hamiltonian.

Homework Equations

N/A

The Attempt at a Solution

What is this question asking me to do? What is meant by "energy spectrum"?

Also; tried cleaning up the tex but something's not right and I can't seem to tell what (other than I'm on a different computer than I normally use).

 

[vela]

Homework Statement

A two-state system has Hamiltonian

$\sum | i \rangle h_i \langle i | + \Delta (| 1 \rangle \langle 2 | + | 2 \rangle \langle 1 |)$

Where, $\langle i | j \rangle = \delta_{ij}$, $h_i$, and Δ are real.

Compute the energy spectrum of this Hamiltonian.

Homework Equations

N/A

The Attempt at a Solution

What is this question asking me to do? What is meant by "energy spectrum"?

Also; tried cleaning up the tex but something's not right and I can't seem to tell what (other than I'm on a different computer than I normally use).

The problem wants you to find all possible results if you measure the energy of the system.

 

[atomicpedals]

Thanks!

 

[atomicpedals]

Ok, my hamiltonian here is an hermitian operator plus a laplacian. This also tells me that |1> and <2| are vectors ([1,0] and [0,1] I think). As a painfully basic question of working with bras and kets, what is the operation (if that's the right word) |1><2| telling me to do?

 

[vela]

Δ is just a number, not the Laplacian.

Try calculating the matrix that represents the Hamiltonian in the $\vert 1 \rangle$ and $\vert 2 \rangle$ basis.

 

[atomicpedals]

Ah, ok, Δ being a number makes life a bit easier (I've just gotten use to it being a Laplacian every other time the prof uses it).

I'm probably getting held up on notation (that I don't know what |1><2| means); and I'm not totally sure what you mean by calculating the matrix that represents the Hamiltonian in the $\vert 1 \rangle$ and $\vert 2 \rangle$ basis. Should this result in a diagonalized matrix?

 

[vela]

You really need to go back and learn the basics of how operators and matrices are related. What I'm telling you to do is find the matrix
\begin{bmatrix}
\langle 1 | \hat{H} | 1 \rangle & \langle 1 | \hat{H} | 2 \rangle \\
\langle 2 | \hat{H} | 1 \rangle & \langle 2 | \hat{H} | 2 \rangle
\end{bmatrix}
Surely your textbook goes over Dirac notation.

 

[atomicpedals]

I am, and it does (we're using both Merzbacher and Griffiths, leaves me in a bit of an information over load).

 

[vela]

Let $\hat{A} = \lvert a \rangle\langle b \rvert$. Say you want to calculate $\langle \psi \lvert \hat{A} \rvert \phi \rangle$. You have
$$\langle \psi \lvert \hat{A} \rvert \phi \rangle = \langle \psi \lvert (\lvert a \rangle\langle b \rvert) \rvert \phi \rangle$$It works just like the notation suggests:
$$\langle \psi \lvert \hat{A} \rvert \phi \rangle = \langle \psi \lvert \lvert a \rangle\langle b \rvert \rvert \phi \rangle = \langle \psi \vert a \rangle \langle b \vert \phi \rangle$$