Quantum Mechanics and Light (v1.0)

 

Most of this blog is summarization of lectures by Prof Erica W Carlson of Perdue University.

In high school, we learned that light behaves like a wave. There were many experiments we did to ascertain that. Later in history we also learned that light behaves as a particle too. Some of this blog can be better understood if you first read my blog on the strange world of quantum mechanics and quantum chemistry.

Light is an example of an electro-magnetic wave. Maxwell first showed how electricity and magnetism interacted and could form EM waves. The EM spectrum spreads from radio (a wavelength of about 10**3 m) to gamma (a wavelength of 10**-12 m). Visible light is between 400 to 700 nm – an exceedingly small part of the EM spectrum. The frequency of visible light shows up as assorted colors – purple for the high frequency and red for the low frequency.

The wavelength and frequency are related with Lambda (wavelength) * f (frequency) = c (speed of light). So, if you know the wavelength, you know the frequency. c is about 300 million meters/sec. The reason light is both a wave and a particle are due to quantum mechanics. A particle of light is called a photon.

However, it turns out that the energy of a photon is quantized. E (energy of photon) = h (plank’s constant) * f (frequency). The photon also has momentum = h (planks constant)/Lambda (wavelength). A photon has no mass but has momentum. How is that possible? Although momentum = velocity * mass in Newtonian physics with slower masses, the equation changes when you are talking about extremely high speeds like the speed of light. You can think of a photon as having a "mass" of h (planks constant) * f (frequency) / (c **2) where c is the speed of light.

The photon also has angular momentum due to spin. This can be an integer multiple of h bar (angular momentum quanta). h bar is h (planks constant)/2 * Pi = 1.0546 * 10 ** -34 J s. This is vastly different than an electron which has a ½ integer of an angular momentum quantum due to spin. That is the key difference between bosons and fermions. This results in vastly different properties. For example, while Pauli’s exclusion principle applies to electrons (a fermion), they do not for photons (a boson). Also, while the fermi-Dirac statistics apply to fermions, Bose Einstein statistics apply to bosons.

A laser consists of a stream of photons with the exact same frequency and phase. Because a photon is a boson, it does not follow Pauli’s exclusion principle, so the coherence of lasers are possible.

Quantum mechanics explains the photoelectric effect. When light of a particular frequency is shone on a material, electrons are emitted. When the frequency of the light (which defines the energy of a photon) multiplied by an integer N (which is the number of photons that arrive simultaneously) is equal to the energy it takes for an electron in the outer orbital of an atom to be released from the atom to be free, electrons are emitted. Solar cells work on this principle. The amplitude of the wave is a measure of number of photons. Since photons are bosons, they can all be together.

There are many examples where absorption of certain frequencies (or photons of a certain energy) by atoms or molecules results in changes. Most substances absorb photons of certain frequencies and cause a characteristic absorption spectrum when light goes through it. This is called an absorption spectrum. Alternatively, the same substances emit light at these same discrete frequencies in an emission spectrum.

Let is take hydrogen as an example. It has 2 electrons in 1s orbital which is at -13.6 ev. A jump to 2p orbital to 1s absorbs 10.2 ev because 2p has an energy of -3.4 ev. This energy is obtained from absorbing one or more photons where the photon energy is hf = 1.24/ (lambda um) ev where lambda is wavelength in micrometers. Conversely a jump from 2p to 1s emits photons with the excess energy. This works out to 122 nm wavelength which is in the ultraviolet range. The amount of energy lost or gained by the electron is exactly equal to the number of photons emitted/absorbed times the energy of the photon. Similarly transitions from 2p to higher orbitals absorbs red (656.2 nm), green (486.1 nm), blue (434 nm) and violet (410 nm). These are some of the lines in the hydrogen absorption spectrum. When a photon is absorbed or emitted, both angular momentum and energy should be conserved. For angular momentum to be conserved, the photon has h bar for spin while the electron has 1/2h bar for spin and h bar for orbital angular momentum in 2p versus 0 for 1s so the photons emitted or absorbed has to exactly match the excess angular momentum. The lines may not be extremely crisp because of doppler shifts because of moving atoms/molecules and other effects. Even the slight fuzziness reveals things!! The color of anything we see is due to frequencies absorbed by that object. The remaining frequencies are reflected.

Photo synthesis also absorbs light. The leaves have chlorophyl A (formula C55H72O5N4Mg) which absorbs light at the blue/purple and red areas, but not green. The energy from those photons is used to power the plant. The green is reflected which is why leaves look green.

This plays a role in astronomy. The spectrum from distant stars is analyzed to see the elements in them. The James Webb telescope will be able to see distant planets on other solar systems. The light that just grazes the planet at the edge and reaches the telescope will absorb certain wavelengths based on the chemistry of its atmosphere. This allows us to deduce the elements in its atmosphere. Can it sustain life?

Sunlight hitting our skin creates a chemical called melanin that is darker is color. This is the tan. Also, sunlight hitting our skin results in vitamin D production that is crucial for us.

Florescence occurs when a material absorbs light at a higher frequency and after changes to electron state emits light at a lower frequency. The difference in the energy is absorbed.

The ability of the human eye to detect colors is based on quantum mechanics. Most of us can detect 3 colors only (red, green, and blue) with different cone cells in our eyes. A photon of the right frequency makes a molecule in the cone cell switch between its isomers that results in a signal to the brain. This is called trichromatic. The brain can put together signals from many instances of the three types of cone cells to detect the right shade of color.

How do light sources work? Argon, neon, helium, and krypton and sodium gas lights are based on the Townsend effect. An electric current causes the gas to ionize. The ions then recombine with an electron to emit a photon. The light depends on the emission spectrum of the element. In steady state the rate of ionization matches the rate of recombination. Florescent lights use mercury gas, but in addition to the Townsend effect, a coating on the surface of the blub of one or more elements in the lanthanide series (first row - 4f orbitals like europium and terbium) gives it its effect. This coating converts the UV light emitted by mercury emission spectrum into visible with a state transition of the 4f orbital.

Accurate time is fundamental to our lives. Accurate time is defined as 9192631770 cycles of a photon wave function to transition a cesium 133 atom from two hyperfine states. The cesium 133 consists of all filled orbitals of electrons except the outermost in 6s state with one electron. A S state orbital has a high probability of being remarkably close to the nucleus and interacts with it. The nucleus spin and the electron spin can be aligned or unaligned. The difference in energy is 3.8 * 10 ** -5 ev. A photon with exactly that energy makes the transition happen. The precise way to make that transition happen is engineering. A cesium 133 atomic clock is so accurate that it loses 1 second in 300 million years. The cesium 133 standard state transition is also used to precisely define a meter, a volt, an ampere, and a candela.

The GPS functions by your receiver receiving a message from 3 satellites. These satellites are about 20,000 km away. Each satellite has an atomic clock. Each satellite says the time and its precise location. The message is sent via microwave in 1.2 to 1.6 GHz. The messages take about 67 ms to reach the receiver at speed of light. The receiver also has accurate time. With this information it calculates its longitude, latitude, and height above sea level since travel time = speed of light * distance. A 4th satellite is used to compute the location twice and adjust the receivers’ clock, so the two measurements exactly match up. The measurement is accurate to 5 meters. One complication is that the satellite clock is faster by about 7 micro sec due to general theory of relativity (a clock further away from gravity is faster) and about 45 micro sec slower due to special theory of relativity. Clock in motion relative to you is slower) since the satellite is moving at about 14,000 km/hr. So, some slight adjustment is required.

The James Webb telescope can see as much as 13.6 billion light years away. The universe is 13.8 billion years old. So, it can see back in time to within 200 million years from the big bang. But since the universe is expanding, objects seen at that time are moving extremely rapidly away from us so there is a doppler effect on light that reaches the telescope from there, so the light appears as infra-red. That is why James Webb telescope sees in the infra-red (Hubble sees mostly in visible light). However, the special theory of relativity says the speed of light is constant no matter the relative speed of the observer and the observed and space time warps, so this is always true. Very strange!!! More on relativity in another essay!!

One correlated question is if there is a red shift, then the energy of the photon has decreased. What happened to the missing energy? Is the universe leaking energy as it expands, and the conservation of energy does not apply? That is a bigger discussion beyond this blog.