Quantum Physics
Early quantum theory.
When an electron jumps from an initial state to a final state, which uses hc/(λ), if the final principle quantum number is smaller than the initial principal quantum number, then the electron moves closer to the nucleus, the change in energy is a negative number, and therefore the atom is releasing energy. Wavelength and frequency are always reported as positive numbers.
Some constants to know are Planck’s constant, h = 6.626 * 10-34 Js, Rydberg’s constant RH = 1.096776 * 107/m, and then, a formula combined by Niels Bohr, E = (-hcRH)(1/n2). Simplifying, we get (-2.179 * 10-18J)(1/n2) (n is the principal quantum number). This formula is used when calculating how much Joules is used when an electron jumps from 1 orbit to another, in the hydrogen atom. Orbits are part of the Bohr model, while orbitals are part of the quantum mechanical model.
If we multiply the 1st 3 constants again (hcRH, all as positive numbers), with Avogadro’s number, we get 1312 kJ/mol, which is the 1st ionization energy of hydrogen gas per mole (or 13.6 electron volts, which is what physicists use). 1 eV = 96.4853 kJ/mol.
When n = 1, the radius is 52.93 pm, or 5.293 * 10-11 m. This can be found from the equation rn = E0h2n2/πmee2. Here, E0 = permittivity of a vacuum, 8.84 * 10-12 F/m, me = mass of an electron at rest, e = charge on an electron, 1.602 * 10-19 C, and n is the orbit.
Frequency (f).
We can use frequency (sometimes referred to as nu) to determine the maximum wavelength of light used to break a bond. The bond energy for O for example, is 495 kJ/mol, and divide Avogadro’s number, gives per molecule, so 8.22 * 10-19 J/molecule (E). Using Planck’s formula E = hf, then f = E/h, which is 1.24 * 1015/s. Plugging in λ = c/f, * (109 nm/1 m) = 241.667 nm. And since photon energy increases as wavelength decreases, then any photon with <241 nm will be strong enough to break the bond. 241 is in the UV-C region. N2 gas, which has a bond energy of 941 kJ/mol, would require 127 nm photon to break it. For C-C at 348 kJ/mol, would require 343 nm, which is in UV-A.
E = hf = hc/λ also tells us that longer radiation carries less energy and shorter radiation carry more energy.
Emission and absorption diagrams.
When atoms get excited, electrons jump to higher orbitals, which can be the 20th, 25th, 100th, etc. What happen is that when they jump to lower orbitals from very high orbitals, the photons produced are not visible light, and they are useless to make the emission and absorption diagram. It is not difficult to figure out up to what orbital can produce visible light photons for any atom. In the hydrogen atom, only the electrons jumping to the 2nd energy level from higher orbitals can produce visible light photons.
The visibility of light emitted or absorbed by an atom depends on the energy differences between electron orbitals. For practical purposes, only transitions involving relatively low orbitals (1st 20 to 25) produce visible photons, as higher orbital transitions result in UV or X-ray emissions. Therefore, for creating emission and absorption spectrum diagrams, considering transitions within these lower orbitals is sufficient to capture the visible spectrum of an atom like oxygen.
λ = hc/Ehigh-Elow
Atomic orbitals.
The s, p, d, and f are known as l. l is known as the 2nd quantum number, and as the angular momentum quantum number. Like in Java and C++, they start at 0, so s = 0, p = 1, d = 2, and f = 3.
But the principal quantum number, is n, and starts at 1.
The 3rd quantum number, also known as the magnetic quantum number, is ml, and ranges from -l to l.
The collection of orbitals with the same value of n are electron shells, and the set of orbitals that have the same value of n and l, are subshells, and are designated by the values "nl." So if n = 3 and l = 2, the subshells is "3d."
The 4th quantum number, also known as the spin magnetic quantum number, is ms, and the only 2 possible values are 1/2 and -1/2, which indicate as 2 opposite directions of "spin." It does not literally mean which direction an electron spins, but a spinning charge does produce a magnetic field.
Removing and adding electrons:
So if remove 2 electrons from Fe, the result is.
[Ar]4s23d6 -> [Ar]3d6.
If we remove a 3rd electron, then the 3d6 becomes 3d5.
If there is more than 1 occupied subshell for a given value of n, the electrons are 1st removed frin the orbital with the highest l. So if we remove an electron from tin:
[Kr]5s24d105p2 -> [Kr]5s24d10.
When electrons are added to form an anion, they're added to the empty or partially empty orbital having the lowest value of n.
Spin.
Electron spin is a quantum property, like angular momenum, which 2 possible states: up or down. Normally, in non-magnetic materials, electron spin are random, with no preference for spin direction.
Chirality-induced spin selectivity is a quantum mechanical phenomenon in which chiral molecules preferentially allow electrons with 1 specific spin orientation to pass through them, effectively acting like a spin filter, without needing a magnetic field. This means right-handed molecules may favor spin-up electrons, and left-handed molecules favor spin-down electrons. DNA and proteins are both chiral.
Spin-orbit coupling.
Spin-orbit coupling is where a particle's spin (intrinsic angular momentum) interacts with its orbital motion. It's like the way a spinning top moves along a circular path (orbit) and the 2 motions influence each other. When electrons move in orbits around the nucleus, the nucleus is moving and creates a magnetic field. The nucleus's magnetic field interacts with the electron's magnetic moment caused by its spin, leading to an energy shift.
Spin-orbit coupling itself does not have a unit, but the energy associated with it does (the energy splitting it causes), expressed in eV in solid-state physics, in Hartree in quantum chemistry, and as /cm in spectroscopy.
Which non-radioactive element has the highest spin-orbit coupling? #83 bismuth, and other honorable mentions are #82 lead, #79 gold, and #81 thallium (but more toxic and less stable in compounds). Spin-orbit coupling is part of why gold looks yellow.
Hydrogen has the lowest spin-orbit coupling, at .000045 eV (~10.97 GHz), and for bismuth, depends on which energy split. Between bismuth's 6p 1/2 and 6p 3/2 orbitals, the splitting in the 6p level is around 1.5 eV.
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Note: c is the speed of light, in m/s, which cancels out the s in frequency. The frequency is found by using the wavelength / Avogadro's #, then / h. That is at nm kJ / mol s, dividing from the speed of light, cancels out the nm and s, getting kJ/mol.
Table of bonds UV light can break. Green = UV-A can break, blue = UV-B can break, red = end of UV-B range can break.
NO2, which is neither a N-O or N=O bond, requires 302 kJ to dissociate (1974).
Absorption and emissions.
When the electrons are in opposite spins, the electronic state is singlet and referred to as “ground state.” Electrons with the same spin are “triplet state,” but if singlet molecules absorb energy without changing spin, the molecule is in an “excited singlet state.”
Photons have to originate from a non-ground state in order to do anti-Stokes. (If a photon originated from a ground state, then it can never do anti-Stokes.).
Molecules in general have 2 ways of storing energy: electronic, vibrational, as well as rotational, and translational. Translational energy levels are so small we don’t really worry about them. Rotational modes absorb in the microwave region, we’re not going to worry about them either. Electronic is where the orbit of an electron gets bigger due to increased energy, and vibrational is where the nucleus of an atom jiggles and spins around in a small area. Heat generally refers to the vibrational energy each atom has. Electronic energy levels are much farther apart than vibrational levels, so think of them as the floors of a building, and the vibrational levels are like a ladder on each floor. So your total energy is what floor you’re on, combined with how high up the ladder you are, on that floor.
Vibrational modes generally absorb in the IR, and electronic generally absorbs in the UV-vis region. But this is general, exceptions exist. Vibrations are the molecule stretching and bending. That takes energy, and usually corresponds to a photon with an IR wavelength. Electronic modes are electrons being excited to higher energy states, and that energy gap is usually in the UV or visible range.
As stated, the energy gap between electronic states is much larger than between vibrational states. So each electronic state has vibrational states within it. So you have the ground electronic state with a ground vibrational state, then ground electronic and 1st vibrational excited state, and so on. Then you have the 1st electronic excited state with its vibrational levels (and to go deeper each vibrational state has a set of rotational levels, but that’s not important right now).
So when you electronically excite an electron it goes up an electronic energy level. But it doesn’t necessarily go into the same vibrational level it came from. This is because the excited state will have the same inter-nuclear distance but a slightly different energy landscape so the same vibration isn’t available. So it goes into a higher vibrational level. Then the electron can relax non-radiatively into lower vibrational levels before de-exciting back to the ground electronic state. And again it may not drop back to the same vibrational state for the same reason as above.
So the result is due to vibrational relaxation the de-excitation energy gap is smaller than the excitation and so you get a lower energy photon. You can also de-excite to a lower vibrational level and so have a shorter wavelength but this is rarer because you’d have to start in a non-ground vibrational state which is statistically less probable from the Boltzmann distribution.
As for what molecules absorb in what modes, that gets into selection rules. There has to be a net change in dipole moment for the transition to occur. If you want to get into that you’ll need a spectroscopy text or chemistry table.
Selection rules only state whether the transition can occur, then it’s down to population. Basically if a transition is allowed then both emission and absorption can occur. Stokes and anti-Stokes has to do with population. Anti-Stokes needs the initial excitation to occur from an excited vibrational level so the emission can drop down to the ground state (or a lower vibrational level), hence the lower wavelength. However, since most population is in the ground state due to the Boltzmann distribution this occurs much less frequently. That’s why anti-Stokes lines aren’t really observed outside more specialized techniques like Raman spectroscopy.
You do not need to absorb in order to emit.
Vibrational states are quantized. If only the 1st excited state is considered, then the shifts are equal and opposite (lose energy into the excited state, or gain the energy from the excited state).
Stokes shifts:
Internal conversion: a non-radiative transition between 2 electronic states of the same spin multiplicity.
Fluorescence.
A radiative transition between 2 electronic states of the same spin multiplicity.
The emission of photons from the S1 -> S0 radiative transition is known as fluorescence which occurs on a timescale of 10-10 to 10-7 s. As a consequence of the rapid vibrational relaxation and internal conversion processes the fluorescence occurs, with some exceptions, from the lowest vibrational level of the 1st electronic excited singlet state to the singlet ground state. This energy loss prior to fluorescence is the physical origin of the famous Kasha’s Rule which states that: “luminescence (fluorescence or phosphorescence) only occurs with appreciable yield from the lowest excited state of a given multiplicity.”
Phosphorescence.
A radiative transition between 2 electronic states of different spin multiplicity.
The emission of photons from the T1 -> S0 transition is known as phosphorescence, similar to intersystem crossing. Phosphorescence is in principle a forbidden transition but is weakly allowed through spin-orbit coupling. A consequence of being a forbidden transition is that the phosphorescence rate constant is very low and phosphorescence therefore occurs on a much longer timescale than fluorescence, with typical phosphorescence lifetimes being in the 10-6 to 10 s range.
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Fluorescence: most molecules have a ground state “singlet” where all their electrons are paired together so their up and down spins cancel out. So there’s a “single” way to arrange the spins. When you excite a molecule, it usually stays as a singlet, and it can relax quickly because the spins are still paired.
Phosphorescence: sometimes when you excite a molecule, an electron can flip its spin. Now you have, say, 2 “up” electrons, and you can have “triple” the amount of arrangements. You can have both up, both down, or both kinda sideways. This is the “triplet” state. Since the spins are no longer paired, it’s hard for the molecule to relax, and it stays excited for longer while it waits for its spins to pair back up.
When molecules absorb energy, we consider it positive kJ, and emission as negative kJ (or whatever units you use).
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Photon conversion.
Singlet fission: the conversion of 1 high-energy singlet excited state into 2 lower-energy triplet excited states. So this process splits the energy from 1 photon into 2, effectively increasing the number of excitons, which are electron-hole pairs, generated per absorbed photon.
Singlet fission can be used to improve the quantum efficiency of solar cells.
Photon upconversion.
Photon upconversion is where 2 or more photons are absorbed and emitted as light where the wavelength is shorter than the absorbed wavelength.
An example of this is 2nd harmonic generation, where 2 photons of the same frequency interact in a nonlinear crystal, combining to form a single photon with twice the frequency and half the wavelength. An example of this are green laser pointers, where 1064 nm light is emitted as 532 nm green light. However, the 1st 2nd harmonic generation experiment was in 1961 by Peter Franken and colleagues at the University of Michigan, where 694.3 nm ruby laser was generated into 347.15 nm UV light. Green laser pointers were later discovered in the 1990s.
For inorganic compounds, the 3 main types are energy transfer upconversion, excited-state absorption, and photon avalanche.
Triplet-triplet annihilation (upconversion):
The low-energy photons --> intersystem crossing --> triplet energy transfer --> triplet-triplet annihilation --> become high-energy photons.
These systems require 1 absorbing species, called the sensitizer, and 1 emitting species, called the emitter or annihilator. Another way is the donor, which is the low-energy absorber, and the acceptor, which is the annihilator and high-energy emitter. Emitters are typically polyaromatic chromophores such as anthracene, perylene, and the o/m/p-terphenyl.
Note that triplet-triplet annihilation is not fluorescence or phosphorescence, it is 1 of 2 types of delayed fluorescence. It was 1st discovered in pyrene.
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Singlet sensitizers and triplet sensitizers.
While not al chemicals can fluoresce or phosphoresce, some can it given a singlet sensitizer (for fluorescence) or triplet sensitizer (for phosphorescence).
A triplet sensitizer is a molecule that helps "transfer" energy to another molecule to allow it to enter its triplet state (T1). This sensitizer absorbs energy or light, transition to an excited singlet state, undergoes intersystem crossing to its own triplet state, and then transfers this triplet energy to the target molecule. This enables the target molecule to reach its triplet state, which is necessary for phosphorescence to occur. This can be done in the solid state, as in the liquid or gas phase, molecules are farther apart, so energy transfer from a sensitizer would be less efficient.
Examples of triplet sensitizers include heavy metal complexes (such as Pt(II) and Pd(II)), organic molecules (such as anthraquinone, thioxanthone, and benzophenone), and porphyrins (such as zinc porphyrin). The use of heavy atoms (like in metal complexes) enhances spin-orbit coupling, which accelerates intersystem crossing, making these molecules particularly good triplet sensitizers.
Examples of singlet sensitizers include fluorescent dyes (fluorescein, rhodamines, coumarins, cyanines, and Nile Red), pyrene, and anthracene.
Can triplet sensitizers themselves do phosphorescence? Yes, some can, such as benzophenone in the solid state or in low-temperature solutions.
Can there be triplet sensitizers that themselves cannot phosphoresce?
Yes, this can happen if non-radiative decay pathways dominate, if environmental quenching occurs, if spin-orbit coupling is weak, or if energy transfer to other molecules is more efficient than radiative decay.
Benzophenone can be another example again, in certain environments such as at room temperature or in solution with oxygen, it may undergo rapid non-radiative decay or energy transfer, leading to little or no observed phosphorescence.
Can a molecule be both a singlet sensitizer and triplet sensitizer?
Yes, depending on whether it transfers energy while in its singlet excited state (before ISC) or in its triplet excited state (after ISC). Those compounds tend to be complex, such as Ru(2,2'-bipyridine)32+.
Singlet sensitization:
The same molecule could also directly transfer energy to another molecule's singlet excited state (S1) of an acceptor via Förster Resonance Energy Transfer (FRET) or Dexter Energy Transfer. This occurs before the molecule undergoes intersystem crossing to the triplet state, during its singlet state lifetime.
Magnetic 2D materials.
2d materials are another term for single-layer materials. Magnetic 2D materials is a field that generally started in 2016, with the exception of the discovery of graphene in 2004. The 1st few-layered van der Waals magnetism was reported in 2017, for Cr2Ge2Te6, and CrI3. Then this has become 1 of the most active fields in condensed matter physics. Magnetic van der Waals materials is a new addition to the growing list of 2D materials. The special feature of these new materials is that they exhibit a magnetic ground state, either antiferromagnetic or ferromagnetic, when they are thinned down to very few sheets or even 1 layer of materials.
Quantum information science.
Quantum information science (QIS) investigates how to exploit quantum behavior and its ability to encode, sense, process, and transmit information.
Coherence is a quantum property that describes the relationship between 2 waves that are well defined (i.e., having the same phase and amplitude). Decoherence occurs when the excited-state, ground-state, and interference wave functions are out of phase. The process of losing coherence is known as decoherence. In general, quantum technologies require slower or smaller decoherence rates to be considered useful. Bits do decoherence. Decoherence results in the loss and erasure of quantum information due to strong interactions between quantum bits and the uncontrolled degrees of freedom around them.
Molecules offer 3 key attributes useful for QIS applications: atomic precision, reproducibility, and tunability. The ability to design a molecule, position atoms, tune various properties, and subsequently create arrays or integrated systems is unique to molecular systems. Designing molecules that feature quantum properties inherently involves synthetic control of coherence. The dominant contributions to decoherence are spin–spin coupling, either through electron spin–nuclear spin coupling or hyperfine interactions.
Electron spins are good qubits because their 2 spin states constitute the quintessential 2-level quantum system, in which the 2 states can exist in a superposition. In addition, coupling 2 or more spins via the spin spin exchange (J) and/or dipolar (D) interactions results in rich spin physics that allow for quantum entanglement as well as implementation of 2-qubit gates essential for quantum gate operations. Electronic spins have larger magnetic moment, and faster gate times, while nuclear spins have longer memory. Longer memory correlates to longer coherence times.
The 2 approaches to quantum information processing are unitary quantum computing and dissipative quantum computing.
Quantum mechanics.
Dirac notation (Bra-Ket) is a shorthand for describing quantum states and operations on them, like 〈ψ∣ (bra) and ∣ψ〉 (ket).
The time-independent version of the Schrödinger equation, H∣ψ〉 = E∣ψ〉, is an eigenvalue problem, where H (the Hamiltonian) is a Hermitian operator.
Eigenvalues are the allowed energy levels, while eigenstates are the corresponding wavefunctions.
In the formula H(p)∣n(p)〉 = En(p)∣n(p)〉, which is the parameter-dependent version of the time-independent version of the Schrödinger equation, this version of the formula is often used when the Hamiltonian depends on some parameter, such as in systems with varying external fields.
The symbol ∣n(p)〉 is the state vector, or eigenstate, of the system associated with and labeled by a quantum number n and possibly dependent on the parameter p. The 〉 is just part of the Dirac notation.
H(p) is the Hamiltonian operator, which represents the total energy of the system and may depend on momentum p.
∣n(p)〉 is the state vector (or eigenstate) of the system associated with the quantum number n and parameter p.
En(p) is the eigenvalue corresponding to the eigenstate ∣n(p)〉, representing the energy of that state.
The equation says that when the Hamiltonian H(p) acts on the state ∣n(p)〉, it returns the same state multiplied by the scalar energy En(p). This is a typical eigenvalue equation in quantum mechanics.
Fermi's Golden Rule gives the transition rate between quantum states due to a weak perturbation (time-dependent interaction).
In the harmonic world, everything's nice and symmetric. But in real systems, anharmonicity is everywhere, and it's what gives molecules their complexity and chemical reactivity. A potential energy surface describes how the energy of a molecule changes as a functin of its nuclear coordinates (like bond lengths and angles). A harmonic potential energy surface is the idealized, quadratic potential, V(x) = (1/2)kx2. (Like a perfect spring.). But real bonds aren't perfect springs, as they can break, have asymmetric vibrations, and exhibit coupling between modes.
Qubits.
A qubit is a superposition of 2 energy levels: ∣0〉 and ∣1〉 (instead of ψ1 and ψ2)
Measuring a qubit ∣ψ〉 = α∣0〉 + β∣1〉 we obtain ∣0〉 with probability ∣α∣2 and we obtain ∣1〉 with probability ∣β∣2
As photons (quanta of light) can be linearly polarized, we assign ∣0〉 to horizontal polarization and ∣1〉 to vertical polarization.
A 45 degree linear polarization can be seen as a superposition ∣D〉 = (1/√2)∣H〉 +
(1/√2)∣V〉 where D, H, and V stand for diagonal, horizontal, and vertical. A 135 degree is called rectilinear.
Entanglement: for a 2-particle system, the solution of the Schrödinger equation could be the product ∣ψ1〉 = ∣α〉A∣β〉B and ∣ψ2〉 = ∣c〉A∣d〉B is another solution, then their superposition is a solution as well ∣ψ〉 = c1∣a〉A∣b〉B + c2∣c〉A∣d〉B
Qubits in modern quantum processors.
There are 4 types: trapped ions, spins of electrons, photons, and superconducting circuits. Furthermore, transmon qubits and fluxonium qubits are 2 types of superconducting circuits. Out of these 4, superconducting qubits make the most noise, but are also the fastest.
There are 2N basis states (complex amplitude) and for storage, there are 2N * 128 bits.
So for N = 65 qubits, the memory or disk space is 590 billion GB.
Harmonic ocillators are not a great qubit, but if you used a Josephson-junction (which is basically a non-linear inductor) the energy levels are no longer equidistant, making this a good thing.
2 types of decoherence:
-By environment (before coin toss, unintentional), where there is qubit relaxation / dephasing.
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Classical light version of the quantum-limit bit.
ψ(polarization) = α1∣h〉 + α2∣v〉
a quantum description of a 2-degree-of-freedom photonic qubit: 1 in polarization (∣h〉,∣v〉 for horizontal/vertical) and 1 in spatial mode (∣u〉,∣l〉 for “upper”/“lower” path).
W2×2i = αi∣a1〉 + βi∣a2〉
a single 2-dimensional quantum system (a qubit, here labeled α) to a 4-dimensional composite system (2 qubits, here α and β), with coefficients labeling the amplitudes of each basis state.
W2×2 is just a label for the “effective state” in a 2-dimensional Hilbert space.
W4x4i = ∣Ψ〉 = ci1∣a1〉∣b1〉 + ci2∣a2〉∣b1〉 + ci3∣a1〉∣b2〉 + ci4∣a2〉∣b2〉
The dimension grows as 2×2=4 hence the “4×4” Hilbert space notation (though the actual state vector is 4×1, not 4×4).
Classical light analog:
If you treat ∣a1〉,∣a2〉 as polarization basis vectors (e.g., h,v) and ∣b1〉,∣b2〉 as spatial mode basis vectors (e.g., upper/lower beam), this is just the most general coherent superposition of those polarization+mode combinations - basically a classical Jones vector in a 4-component basis.
History of quantum mechanics.
In 1834-35, William R. Hamilton published 2 papers that introduced the Hamilton-Jacobi equation, Hamilton's equations of motion, and the principle of least action. These 3 formulations of classical mechanics became the 3 forerunners of quantum mechanics. However, none of these is what Hamilton was looking for, he was looking for a magical function, which is the principal function S(q1,q2,t), from which the entire trajectory history can be obtained just by differentiation. All of the above formulations of classical mechanics can be derived just from assuming that S(q1,q2,t) is additive, with no input of physics.
From Schrödinger's introduction, in his 2nd paper on wave mechanics, Annalen der Physik (1926):
"The inner connection between Hamilton's theory and the process of wave propagation was not only well known to Hamilton, but it also served him as the starting point for his theory of mechanics.."
When electrons are removed to form a cation, they are always removed 1st from the occupied orbital having the largest principal quantum number n.
Wavelength f λ (c/f) λ (c/f) Start of UV-A
380 nm 9.523*1011/s .000315 315 nm Start of UV-B
320 nm 8.020*1011/s .000374 374 nm Start of UV-C
290 nm 7.268*1011/s .000412 412 nm End of UV-B
280 nm 7.017*1011/s .000427 427 nm
C-H
413
N-H
391
O-H
463
F-F
155
C-C
348
N-N
163
O-O
146
C-N
293
N-O
201
O-F
190
Cl-F
253
C-O
358
N-F
272
O-Cl
203
Cl-Cl
242
C-F
485
N-Cl
200
O-I
234
C-Cl
328
N-Br
243
Br-F
237
C-Br
276
S-H
339
Br-Cl
218
C-I
240
H-H
436
S-F
327
Br-Br
193
C-S
259
H-F
567
S-Cl
253
H-Cl
431
S-Br
218
I-Cl
208
Si-H
323
H-Br
366
S-S
266
I-Br
175
Si-Si
226
H-I
299
I-I
151
Si-C
301
Si-O
368
Si-Cl
464
C=C
614
N=N
418
O=O
495
C=N
615
N=O
607
S=O
523
C=O
799
S=S
425
C=C
839
N=N
941
C=N
891
C=O
1072
Also note that all absorbing dense-matter warmer than 0K emits, without the need to absorb 1st (this is called black body radiation, but doesn't apply to gases).
An incoming photon can always lose energy to excite a vibration state in a molecule. The only way to gain energy is if there is an excited vibrational state. You increase the anti-Stokes to Stokes ratio of a particular molecule by increasing the temperature, and you will get a higher ratio at a given temperature for molecules with lower energy vibrational states.
Intersystem crossing: a non-radiative transition between 2 isoenergetic vibrational levels belonging to electronic states of different spin multiplicity.
For organic compounds, the 2 main types are sensitized triplet-triplet annihilation and energy pooling. There is a singlet-singlet annihilation but is much less efficient. Triplet-triplet annihilation is the most commonly studied and utilized, where 2 low-frequency photons become 1 high-frequency photon.
∣ψ〉 = α∣0〉 + β∣1〉, ∣α∣2 + ∣β∣ = 12
α and β are complex magnitudes.
-By measurement (after coin toss, intentional) but there are quantum limits on measurement sensitivity.
ϕ(superposition) = β1∣u〉 + β2∣l〉
∣a1〉 and ∣a2〉 are basis states for qubit a (e.g., ∣0〉,∣1〉).
αi and βi are the complex probability amplitudes.
Classical mechanics
Quantum mechanics
Principle of least action
The path taken by a system is the 1 for which the action is minimized (or stationary).
Underlying Feynman's path integral formulation, a particle takes all possible paths, but the classical path (least action) is where the contributions construtively interfere). However, quantum mechanics allows non-classical paths.
Hamilton's equations
Describes time evolution using generalized coorinates and momenta.
The Hamilton becomes an operator Ĥ that generates time evolution via the Schrödinger equation.
Canonical transformations
Preserve the form of Hamilton's equations, like a kind of symmetry in phase space.
Survives only in the form of unitary transformations on quantum states and operators.