Spectroscopy II

02-03

本來第二部分想寫一些跟譜學有關的原理推導，但寫著寫著發現亂七八糟什麼都有，簡直變成了電動力學和量子力學的複習。。。反正是為了qual，複習一下也好。這次大部分是我手推的，下次來複習關於graphene的簡單內容

# Outline

1. Fermi golden rule

2. EM radiation

3. Dielectric response function

4. Selection rules

5. Magnetic moment in magnetic field

# Fermi Golden Rule

See figure below.

# EM radiation

- Retarded potentials

- Far field approximation and multipole expansion

- Electric dipole radiation

- Magnetic dipole radiation

- Magnetic quadruple radiation

- Radiation reaction

畫質好渣哈哈哈哈

# Dielectric response function

- Dielectric permittivity and electric conductivity

- Drude model of electric conduction

- K-K relations

- Model dielectric functions

# Selection rules

- Basically if the transition moment integral $int psi_f^* H psi_i dx$ is zero then the transition is forbidden.

- One doesnt really need to do the integral, a symmetry argument is usually enough.

- If we retain only the distinct results of these operations, we obtain by construction, a representation of the group of the Hamiltonian, which we denote by $Gamma$.

- This representation may be either reducible or irreducible, depending on $H_0$ and on the symmetry of it. If $H$ has the same symmetry as $H_0$, then this procedure generates the identical representation.

- At the other extreme, if $H$ has none of the symmetry properties of $H_0$, then this procedure generates a reducible representation whose dimensionality is equal to the order of the group.

- We know $Hpsi_f$ transforms as the direct product $GammaotimesGamma^{(f)}$.

- Since quantities that transform to different irreducible irreducible representations are orthogonal, we have the Thm: the matrix element vanishes if the irreducible representation $Gamma^{(i)}$ corresponding to $psi_i$ is not included in the direct product of representations $Gammaotimes Gamma^{(f)}$ that correspond to $H$ and $psi_f$ respectively.

- (This selection rule only provides a condition that guarantees that the matrix element will vanish. It does not guarantee that the matrix element will not vanish even if the conditions of the theorem are fulfilled.)

- Example: $H$ transforms like a vector $(x,y,z)$. This is the case of dipole transition and Fermi Golden rule. As discussed in the notes of last week, $langle f|H|irangle=frac{e}{m} langle f|pcdot mathbf{A}$.

- Suppose the group of the Hamiltonian corresponds to the symmetry operations of an equilateral triangle, i.e., $C_{3v}$, the character table for which is

- To determine the dipole selection rules for this system, we must first determine the transformation properties of a vector $r = (x, y, z)$. We take the $x$- and $y$-axes in the plane of the equilateral triangle and the $z$-axis normal to this plane to form a right-handed coordinate system. Applying each symmetry operation to $r$ produces a reducible representation because these operations are either rotations or reflections through vertical planes. Thus, the $z$ coordinate is invariant under every symmetry operation of this group which, together with the fact that an $(x, y)$ basis generates the two-dimensional irreducible representation $E$, yields $Gamma=A_1+E$.

- We must now calculate the characters associated with the direct products of between $Gamma$ and each irreducible representation to determine the allowed final states given the transformation properties of the initial states. The characters for these direct products are shown below

- Using decomposition theorem, we find $A_1otimes Gamma=A_1oplus E$ and $A_2otimes Gamma=A_2oplus E$ and $Eotimes Gamma=A_1oplus A_2oplus 2E$.

- Thus, if the initial state transforms as the identical representation $A_1$, the matrix element vanishes if the final state transforms as $A_2$. If the initial state transforms as the 「parity」 representation $A_2$, the matrix element vanishes if the final state transforms as $A_1$. Finally, there is no symmetry restriction if the initial state transforms as the 「coordinate」 representation $E$.

- In one sentence: transition is forbidden if fusion rule is wrong.

# Magnetic moments

- Magnetic moments

- Zeeman splitting

- Bloch equations

- Lamour frequency

# Irrelevant facts

(By $cm^{-1}$ I mean the wave number.)

- $1~meV$ corresponds to about $8~cm^{?1}$.

- $1~mu m$ wavelength corresponds to $1.24~eV$ and

- $1~K$ corresponds to about $0.7~cm^{?1}$.

- $1~meV$ is around $11.4~K$.

References

[1] Same book as part I

[2] http://www.cmth.ph.ic.ac.uk/people/d.vvedensky/groups/Chapter6.pdf

[3] Wikepedia

[4] Electrodynamics lecture notes by Oleg Starykh

[5] Griffiths, Electrodynamics