# Energy Spectrum of Two-State System

1. Oct 5, 2011

### atomicpedals

1. The problem statement, all variables and given/known data

A two-state system has Hamiltonian

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

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

Compute the energy spectrum of this Hamiltonian.

2. Relevant equations

N/A

3. 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).

Last edited: Oct 5, 2011
2. Oct 5, 2011

### vela

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

3. Oct 5, 2011

Thanks!

4. Oct 5, 2011

### 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?

5. Oct 5, 2011

### vela

Staff Emeritus
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.

6. Oct 5, 2011

### 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?

7. Oct 5, 2011

### vela

Staff Emeritus
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.

8. Oct 5, 2011

### atomicpedals

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

9. Oct 6, 2011

### vela

Staff Emeritus
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:
$$\langle \psi \lvert \hat{A} \rvert \phi \rangle = \langle \psi \lvert \lvert a \rangle\langle b \rvert \rvert \phi \rangle = \langle \psi \vert a \rangle \langle b \vert \phi \rangle$$