An electron moves along an infinite chain of potential...

An electron moves along an infinite chain of potential wells. For sufficiently low energies we can assume that the set $\{|n\rangle\}$ is complete, where $|n\rangle$ is the state of definitely being in the $n^{\text {th }}$ well. By analogy with our analysis of the $\mathrm{NH}_{3}$ molecule we assume that for all $n$ the only non-vanishing matrix elements of the Hamiltonian are $\mathcal{E} \equiv\langle n|H| n\rangle$ and $A \equiv\langle n \pm 1|H| n\rangle$. Give physical interpretations of the numbers $A$ and $\mathcal{E}$. Explain why we can write $$ H=\sum_{n=-\infty}^{\infty} \mathcal{E}|n\rangle\langle n|+A(|n\rangle\langle n+1|+| n+1\rangle\langle n|) $$ Writing an energy eigenket $|E\rangle=\sum_{n} a_{n}|n\rangle$ show that $$ a_{m}(E-\mathcal{E})-A\left(a_{m+1}+a_{m-1}\right)=0 $$ Obtain solutions of these equations in which $a_{m} \propto \mathrm{e}^{\mathrm{i} k m}$ and thus find the corresponding energies $E_{k}$. Why is there an upper limit on the values of $k$ that need be considered? Initially the electron is in the state $$ |\psi\rangle=\frac{1}{\sqrt{2}}\left(\left|E_{k}\right\rangle+\left|E_{k+\Delta}\right\rangle\right) $$ where $0<k \ll 1$ and $0<\Delta \ll k$. Describe the electron's subsequent motion in as much detail as you can.

An electron moves along an infinite chain of potential wells. For sufficiently low energies we can assume that the set $\{|n\rangle\}$ is complete, where $|n\rangle$ is the state of definitely being in the $n^{\text {th }}$ well. By analogy with our analysis of the $\mathrm{NH}_{3}$ molecule we assume that for all $n$ the only non-vanishing matrix elements of the Hamiltonian are $\mathcal{E} \equiv\langle n|H| n\rangle$ and $A \equiv\langle n \pm 1|H| n\rangle$. Give physical interpretations of the numbers $A$ and $\mathcal{E}$. Explain why we can write
$$
H=\sum_{n=-\infty}^{\infty} \mathcal{E}|n\rangle\langle n|+A(|n\rangle\langle n+1|+| n+1\rangle\langle n|)
$$
Writing an energy eigenket $|E\rangle=\sum_{n} a_{n}|n\rangle$ show that
$$
a_{m}(E-\mathcal{E})-A\left(a_{m+1}+a_{m-1}\right)=0
$$
Obtain solutions of these equations in which $a_{m} \propto \mathrm{e}^{\mathrm{i} k m}$ and thus find the corresponding energies $E_{k}$. Why is there an upper limit on the values of $k$ that need be considered?
Initially the electron is in the state
$$
|\psi\rangle=\frac{1}{\sqrt{2}}\left(\left|E_{k}\right\rangle+\left|E_{k+\Delta}\right\rangle\right)
$$
where $0<k \ll 1$ and $0<\Delta \ll k$. Describe the electron's subsequent motion in as much detail as you can.

Key Concepts

Wave Packet Propagation

When an electron is prepared as a superposition of two nearby energy eigenstates, it forms a wave packet that evolves in time. The interference between the slightly different wavevector components causes the packet to propagate along the lattice with a group velocity. This describes how the initial state, localized in energy space, exhibits dynamical motion reflecting both the quantum mechanical interference effects and the dispersion of the medium.

Brillouin Zone

Due to the discrete translational symmetry of the lattice, the wavevector k is defined modulo reciprocal lattice vectors, leading to a finite, periodic interval known as the Brillouin zone. Only k-values within the first Brillouin zone are unique, which is why there is an upper limit on the values of k that need to be considered.

Energy Dispersion Relation

The dispersion relation describes how the energy of an electron in the lattice varies with its wavevector k. In the tight-binding model, this relation typically takes a cosine form, indicating a continuous energy band whose curvature reflects the effective mass and mobility of the electron.

Plane Wave Solutions

By expanding the energy eigenstates in terms of the localized basis states and choosing coefficients proportional to plane waves (e^(ikm)), the tight-binding Hamiltonian becomes diagonal in the momentum basis. This approach leads to an eigenvalue equation where the energy can be expressed as a function of the wavevector k.

Hopping Term (Transfer Integral)

Represented by A in the problem, the hopping term quantifies the amplitude for an electron to move from one site to a neighboring one. It arises from the overlap between the electron wavefunctions of adjacent wells and is responsible for coupling the localized states, leading to the spread of electronic states into energy bands.

On-site Energy

This refers to the energy ? associated with an electron when it is localized at a specific lattice site. In the tight-binding model, the on-site energy represents the inherent energy level of the state at a single well, serving as the diagonal element in the Hamiltonian matrix.

Tight-binding Model

This is a simplified model for electrons in a periodic potential, where the electron is assumed to be localized in lattice sites (or potential wells) and is allowed to tunnel (hop) to adjacent sites. It provides an intuitive framework for understanding energy band formation in crystalline solids by considering interference between localized electron wavefunctions.

Hamiltonian Construction

The Hamiltonian is built by considering contributions from both the on-site energies and the hopping processes. The on-site energies contribute diagonal terms (?|n??n|), while the hopping between neighboring sites contributes off-diagonal terms (A(|n??n+1| + |n+1??n|)). This construction captures the essential physics of electron propagation in a lattice.

Transcript

00:01 Okay, so we have the hamiltonian matrix for their two -state system.

00:04 So, h equals e10, lambda -delta, lambda, and e -20.

00:12 Okay, dokey.

00:14 So we have an unperturbed eigenstate whenever lambda equals zero is unperturbed.

00:22 So there's no external system affecting our functions.

00:28 And when lambda equals zero, we have our states, phi -1 of zero, which equals 1 -0, and 5 .2 of 0, which equals 0 .1.

00:42 So how do we solve your eigenvalue problem? well, remember, hsi equals e -sci.

00:50 So your hamiltonian operator acts on your wave function.

00:53 It spits out your energy and your wave function itself.

00:56 So to determine the eigenvalues of h, this is what you do.

01:00 You take the determinant of the matrix minus the energy times the identity matrix and set equal to zero.

01:10 The values that you get out of this are your eigenvalues.

01:13 So when we do this, we have our matrix, e10, minus e, lambda delta, lambda, e20, minus e, and we set that equal to zero.

01:26 So now we compute the determinant, and we get e10, minus e, times e20, minus e, minus lambda delta squared, equals zero.

01:40 So then we rearrange this and simplify.

01:43 And we get e squared minus e times e10 plus e20 plus e10 minus a lambda delta squared equals 0 this is a quadratic equation so to find the two eigenvalues which are going to be the energies of this system e1 and 2 this equals e1 0 plus e2 over 2 plus or minus square root of e10 minus e20 over 2 quantity squared plus lambda delta squared.

02:29 And these are exact eigenvalues giving the energy for this system.

02:36 Now the eigen functions.

02:39 So for your functions, you're going to let si equals a, b, just arbitrary, and plug them into the eigenvalue equation.

02:51 So we have e10, lambda, lambda delta e2 0, a, b, and this equals eab.

03:04 And this is just your h -si equals e -sy.

03:08 We get two equations from this when we see the e -1 -0a plus lambda delta -b equals ea, and lambda - delta a plus e2 -0, b equals e -b.

03:23 Two equations, two unknowns, and we can get these guys...

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