6. Matter and Light

Properties of wave functions:

We want to describe a single particle located in a box with infinitely high walls (potential energy) from which it cannot escape. The Box is restricted by the borders 0 and L. The corresponding wave function must be 0 at the borders and continuous in the box. This is achieved with sin functions with λ=L/nπ.

wave function: when squared gives us the probability to find the particle at a certain location

energy: formula gives allowed energy values

• energy of the particle is quantized

• energy of the particle cannot be zero

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_%28Physical_and_Theoretical_Chemistry%29/Quantum_Mechanics/05.5%3A_Particle_in_Boxes

The Particle in a Box model can be used to describe coordinated pi-orbital systems, for example in dyes. Example: cyanine dye

This is similar to the problem of the Particle in a Box (basically you have a “round” box where 0 and L meet). The difference is that the wave has to be continuous where the two “ends” meet, which excludes solutions like n=1 from the Particle in a Box.

This can be used to describe circular molecules, e.g. aromatic systems.

Here, we want to find wave functions that we require to be symmetric around x=0 (because we want to describe harmonic oscillations).

The distance between energy states is equal between all energy states.

Used to describe oscillations, e.g., of bonds (which do not rupture)

• Was soll die Parabel da?

  
  

https://phys.libretexts.org/Bookshelves/University_Physics/University_Physics_(OpenStax)/University_Physics_III_-_Optics_and_Modern_Physics_(OpenStax)/07%3A_Quantum_Mechanics/7.06%3A_The_Quantum_Harmonic_Oscillator

Similar to harmonic oscillator, but it can be used to describe bonds that can rupture.

Used to describe electrons in atoms —> orbital model

When we have a wave function describing a system, we can use it to calculate expectation values (<x>, basically “mean” values) for different properties (e.g., electron density, momentum, …) by squaring the wave function. ^x is the operator of the property we are interested in.

You can also calculate the transition moment (probability of the system to have a transition between state n and m) by using the wave functions from before and after the transition:

(energy, Planck’s constant, frequency)

(frequency, speed of light, wavelength)

(momentum, Planck’s constant, wavelength)

This is the time-independent Schrödinger equation that describes the system without light (only one of the problems described above for which the stationary Schrödinger equation is known).

Time-independence, therefore, separation of variables leads to the given function.

n: quantum number

r: space coordinates

sigma: spin coordinates

When light is added, the hamiltonian of the light is time-dependent and things get more complicated.

A second hamiltonian representing the light is added to the hamiltonian of the system:

H(0): system’s amiltonian

H(1): light’s hamiltonian

The hamiltonian of the light is time-dependent:

Therefore, the resulting Schrödinger equation is also time-dependent:

This has to be solved.

If you have the hamiltonian of light, you can calculate transition moments (for transitions between states caused by the light) from which you can calculate the frequencies of jumps between the states (which is helpful for spectroscopy, but why?).

The equation describes the transition moment between two states of the quantum system k and n (?). It is a result of solving the time-dependent Schrödinger equation of a system interacting with light. For some reason, it is important for spectroscopy.

The transition moment is defined as the space integral of the wave function of state n modified with the light hamiltonian times the wave function of state k (basically wave function squared, which means probability that the particle is there? Where?).

(Why do we want to know the transition moment? What do we need it for?)

(What does the second part of the equation mean, again?

This equation describes the probability to find the system in state k when it was initially in state i.

???

It describes the transition rate, which is the number of transitions between the states i and k over time.

Pk(t): probability to find the system in state k after it was in state i.

H: total energy

wi—>f describes the transition rate, which is the number of transitions between the states i and k over time.

Fermi’s Golden Rule

The spectral intensity is proportional to the transition moment (squared) and to the density of states.

Energetically higher regions have more states in less “distance”, therefore a higher density of states:

We want to calculate H to get a matrix.

(What do we want to find out with it? Where in this thing is even the spectral density? What is the point of the transition rate?)

Matter is interacting with a field of electromagnetic radiation. The radiation is absorbed with a transition rate of:

(Bfi: Einstein coefficient, rho: density of states in the respective energy region)

Emission occurs at the same transition rate:

As this does not fulfill a Boltzmann occupation of energy levels, there is additional spontaneous emission:

Dipole transition moment:

e: quantum numbers defining the electronic state (f: final; i: initial)

v: quantum numbers defining the vibronic state (vibration of the bond)

µ: dipol moment (e*r)

E: energy ? electric field?

In IR spectroscopy, we look at vibrational transitions.

Prerequisite for interaction with IR light: The dipole moment changes with the vibration coordinate.

<vf|µ|vi>: transition rate

dµ/dR: change of the dipol with bond elongation

<vf|dR|vi>: expectation value of the elongation

• used to study vibrational transitions (like IR)

• approach: irradiate with light with higher frequencies/energy than the transitions we want to study

• here, the dipole moment is induced by the electric field of the light, not the molecule itself (as in IR)

• —> we need a change in polarizability with a change in vibration

µ: dipole moment

alpha: polarizability

E: electric field

• when the molecule vibrates, the polarizability oscillates because it depends on the location of the nuclei

• the induced dipol also oscillates because both the polarizability and the electric field of the light oscillate

E0: energy of the light

• and then, somehow, when you do maths to it for reasons that I do not understand, you get this, which looks more complicated but is apparently more useful:

• Because there are three frequencies involved (the initial one from the light, the one from the light plus the vibrational frequency and the one from the light minus the vibrational frequency), you get emission of light in three frequencies

This is about electronic and vibronic transitions of electrons (y-axis). Electronic transitions happen fast, transitions of the nuclei (R, x-axis)are much slower (only move after the electronic transition is done).

Franck-Codon Principle:

The intensity of an electronic-vibronic transition is roughly proportional to the overlap integral between the two vibronic states.

(Which means: A transition from E0 to E1 must take place at a point (R) where both curves are as high as possible, as seen in case of the red arrow. In case of the blue arrow, there is no intensity and therefore no transition).