• Let's consider an electron confined to a two-atom system, as seen below:

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The function representation of this system's Hamiltonian will look like:

$$H=-\frac{\hbar^{2}}{2m}\frac{\partial^{2}}{\partial x^{2}} + V_{atom}\left(x-a\right) + V_{atom}\left(x-2a\right) \; \; . $$

• Referring to the coupled oscillator, we solved for the motion of each mass and found the eigenfrequencies of the possible envelope functions. For the two-well electron system, the process will be very similar, only now we will solve for the eigenstates of the electron and find the eigenenergies of each state. So,

Our goal: Find the eigenstates $\vert \psi_{n} \rangle$ of this system's Hamiltonian $H$, and find the eigenenergy of each state.

• Before we can describe the possible eigenstates of the electron, we will need a basis to describe the eigenstates. If students have experience with the Stern-Gerlach Experiment, refer to making quantum state vectors as a superposition of the $\vert + \rangle$ or $\vert - \rangle$ states. Let's use the ground states for a single well as our basis, where

$$\vert 1 \rangle \; \dot{=} \; \phi^{g}(x-a) \; \; $$

and

$$\vert 2 \rangle \; \dot{=} \; \phi^{g} (x-2a) \; \; .$$

Notice that the basis wave functions are now centered at their respective atoms. Working in the ground state basis, the eigenstates must have the form

$$\vert \psi \rangle \; = \; c_{1} \vert 1 \rangle + c_{2} \vert 2 \rangle \; \; . $$

• Now that we have our basis, we want to use the energy eigenvalue equation to find the coefficients $c_{1}$ and $c_{2}$, where

$$H \, \vert \psi_{n} \rangle \; = \; E_{n} \, \vert \psi_{n} \rangle \; \; .$$

• Trying to solve for the constants using wave function notation is going to be a challenging task. Let's instead work with a general matrix representation of the Hamiltonian. Since there are only two atomic states possible for the 2-atom chain, the Hamiltonian for the system should only be a 2×2 matrix. The general matrix will have the form

$$H \; = \; \left[\begin{array}{cc} \alpha & \beta \\ \beta & \alpha \\ \end{array}\right] \; \; . $$

NOTE: If students are puzzled by why you chose this form, remind them that the Hamiltonian is an observable. Because of this, the matrix must be Hermitian.

Alternatively, the Hermitian matrix can also be represented as

$$H \; = \; \left[\begin{array}{cc} \langle 1 \vert H \vert 1 \rangle & \langle 1 \vert H \vert 2 \rangle \\ \langle 2 \vert H \vert 1 \rangle & \langle 2 \vert H \vert 2 \rangle \\ \end{array}\right] \; \; , $$

where

$$\alpha=H_{11}=H_{22}=\langle 1 \vert H \vert 1 \rangle=\langle 2 \vert H \vert 2 \rangle \; \; ,$$ $$\beta=H_{12}=H_{21}=\langle 1 \vert H \vert 2 \rangle=\langle 2 \vert H \vert 1 \rangle \; \; .$$

Notice that $\alpha$ is just computing the expectation value for either the first or second basis state, while $\beta$ is called an overlap evaluation. For the two-well system, $\beta$ is actually a negative value associated with the probability of an electron to move between wells.

If you choose, this is a good opportunity to discuss solving for $\alpha$ and $\beta$.

NOTE: If students are puzzled by the Hamiltonian representations, remind them that $\vert 1 \rangle$ and $\vert 2 \rangle$ are the basis states, and can be represented in matrix notation as

$$\vert 1 \rangle \; \dot{=} \; \left[\begin{array}{c} 1\\ 0\\ \end{array}\right] \; \; \; \; \; \; \; \text{and} \; \; \; \; \; \; \; \vert 2 \rangle \; \dot{=} \; \left[\begin{array}{c} 0\\ 1\\ \end{array}\right] \; \; . $$

Performing the matrix operations will show that

$$\langle 1 \vert H \vert 1 \rangle = \left[\begin{array}{cc} 1 & 0 \\ \end{array}\right] \left[\begin{array}{cc} \alpha & \beta \\ \beta & \alpha \\ \end{array}\right] \left[\begin{array}{c} 1\\ 0\\ \end{array}\right] = \alpha \; \; . $$

• With our Hamiltonian, we can now try to find our eigenvectors (which are the eigenstates) and eigenvalues (which are the energies of each eigenstate). Have the students show that two vectors are eigenstates in the 2-well system. After completing the activity, the class will find that

$$\vert \psi_{1} \rangle \; \dot{=} \; \left[\begin{array}{c} 1\\ 1\\ \end{array}\right] \; \dot{=} \; \vert 1 \rangle + \vert 2 \rangle \; \; \; \; \; \; and \; \; \; \; \; \; E_{1}=\alpha+\beta \; \; ,$$

while

$$\vert \psi_{2} \rangle \; \dot{=} \; \left[\begin{array}{c} 1\\ -1\\ \end{array}\right] \; \dot{=} \; \vert 1 \rangle - \vert 2 \rangle \; \; \; \; \; \; and \; \; \; \; \; \; E_{2}=\alpha-\beta \; \; .$$

Ask the class if these states are properly normalized. How would they properly normalize the eigenstates?

• Let's represent these wave states graphically.

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Does this look familiar? Graphically representing the eigenstates yields envelope functions nearly identical to those we saw in the coupled oscillator. This motivates us to introduce envelope functions to make a function representation of our final wave states.

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The function representation of each eigenstate is equivalent to the bra-ket notation, but now we can explicitly show how the wave vector of the envelope function and location of the well will affect each coefficient in the eigenstates.