Neon Light

Electric discharge in neon gas. The fast-moving electrons hit atoms, causing them to emit light. But why all at one wavelength?

All the light at 632 nm (red light).

Quantum Mechanics

Classical physics: Energy can be transferred by particles (moving objects) and by waves (e.g. EM wave is oscillating E and B fields) – particles and waves are different.

Quantum Physics: Particle-wave duality: Everything has both particle and wave characteristics.Which set of attributes you observe depend on what you measure. Usually, the wave characteristics are more

manifest when you measure how an object travels and the particle characteristics are more manifest when you measure how the motion starts and ends.

A moving particle has energy and momentum. A traveling wave has a frequency and wavelength. What connects these are the “deBroglie” relationships:

particle wavemomentum p = h/λ (wavelength)

energy E = hf (frequency, text in Ch.13 uses ν)Planck’s constant h = 6.63 x 10-34 J⸱s

Because h is so small, the quantum (wave) nature of objects is not obvious for a macroscopic object, but becomes essential for materials at the atomic level.

Let’s start with electromagnetic waves, with λ = c/f. Classically these can have any energy; i.e. the energy can be arbitrarily small by making the amplitude arbitrarily small. However, quantum mechanics states that the energy and momentum come in bundles (called photons or quanta) with energy E = hf and momentum p = h/λ= hf/c = E/c. Therefore, the amplitude cannot be arbitrarily small – i.e. you cannot have less than one photon.

Consider a bundle of EM radiation with energy E = hf. If the EM wave is very intense, the bundles may completely overlap, and it is not important to consider them as separate photons.

a) Consider a radio receiver picking up a weak radio signal at 1MHz (and period T = 10-6 s) with power P = 0.1 mWatt (= 0.0001 J/s). Each photon has energy Ephoton = hf = (6.6 x 10-34) ⸱ 106 J = 6.6 x 10-28 J. So in each period the receiver picks up P*T/Ephoton = (0.0001 J/s)⸱(10-6 s)/ (6.6 x 10-28 J/photon) = 1.5 x 1017 photons (!!), and they will completely overlap.

In contrast, consider a very weak 0.1 mWatt (= 0.0001 J/s) laser emitting red light with frequency f = 4.6 x 1014 Hz (and period T = 2.2 x 10-15 s). The photon energy Ephoton = 3.0 x 10-19 J, so in each period, the laser emits only 0.7 photons, so they will not overlap – they must be considered as separate photons.

So the higher the frequency of the wave, the more manifest its quantum nature.

Each photon also has momentum p = h/λ = hf/c. That means that if photons hit an object, they can transfer momentum, i.e. they will hit the object with force and therefore pressure (= force/area). This pressure is usually negligible compared to other sources of pressure, e.g. the pressure from sunlight near the surface of the earth is only ≈ 5 x 10-11 atmospheric pressure.

However, this pressure is enough to drive particles away from the sun: the solar wind. (In space, of course, there is no atmospheric pressure.)

Now let’s consider a moving “material object” that we normally treat as a particle with momentum (p) and energy (E). According to quantum mechanics, this object also is associated with a wave with frequency f = E/h and wavelength λ= h/p.

For a photon, the frequency and wavelength correspond to oscillations of the electric and magnetic fields, but what is “oscillating” in a material wave? Each “particle” (electron, proton, flea, bowling ball, person, …) is described by an abstract wavefunction (Ψ); Ψ2 = the probability that the particle is at a given location at a given time. The wavelength approximately gives the smallest distance to which you can locate the particle (i.e. the most precisely you can describe its location) and the period (1/f) is approximately how precisely you can determine the time of your measurement.

However, the oscillations of the wavefunction,Ψ, in time are not directly measurable, unless the particle changes its wavefunctions. If it changes from a wavefunction with frequency f1 (and energy E1 = hf1) to one with frequency f2 (and energy E2 = hf2), it will bounce around (between the spatial distributions described by the two wavefunctions) at frequency f2 – f1 while it is changing. If it is electrically charged, it may emit (if E2 < E1) or absorb (if E2 > E1) a photon at frequency |f2-f1|, thus conserving energy.

These are pictures of two possible wavefunctions of an electron in an atom. If it changes between these, it will oscillate between these two patterns at f2 – f1.

λ= h/p

For macroscopic objects (e.g. bowling balls, people, even fleas), these result in wavelengths too small to measure and have any consequence. For example, consider a flea with m = 1 mg = 10-6 kg walking at 1 mm/s = 10-3 m/s. Its momentum p = mv = 10-9 kg⸱m/s. This corresponds to a wavelength λ = 6.6 x 10-25 m. This means you would not be able to find the location of the flea more precisely than ≈ 6.6 x 10-25 m (10 orders of magnitude smaller than the size of an atomic nucleus!), but any real experimental precision is obviously very much worse than this.

On the other hand, consider an electron (m = 9.1 x 10-31 kg). If the electron accelerates across a voltage difference of only 1.5 V, it will have a velocity ≈ 7 x 105 m/s. Therefore, its wavelength λ = 10-9 m = 1 nm. This is about 3-4 spacings of atoms in a solid or liquid and can be measured with instruments such as electron microscopes.

We will only consider the wavefunctions of electrons confined (i.e. trapped) in a solid (Section 13.3) or in the Coulomb electric field around an atom (Section 13.2). In these cases, it is a (three-dimensional) standing wave and is not traveling, so it does does not have a wavelength, but it still has an energy and frequency. When it changes its wavefunction and energy, it may absorb or emit light, i.e. a photon with frequency f2 – f1. Here are pictures of the wavefunctions (Ψ) of electrons around

atoms, called “orbitals”. They show the probability that the electron is in different places around the nucleus. The colors show the values of the phase = 2πft = 2πt/T at different times: Red: t = 0Yellow: t/T = ¼ (2πft = π/2 = 90o)Green: t/T = ½ (2πft = π = 180o)Blue: t/T = ¾ (2πft = 3π/4 = 270o)

r (atomic units)0 1 2 3 4 5

Y (t

=0)

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1s

2s

The 1s orbital is a filled ball, whereas the 2s orbital has a spherical nodal surface (where Ψ goes through zero and changes sign). The 2p and 3s orbitals are not spherically symmetric, with different lobes pointing in different directions. (These become useful in forming chemical bonds.) They have nodal planes between the lobes.

The orbitals are labeled with a number (called the principle quantum number, n): n-1 = the number of nodes (zeroes) in the wavefunction.

The letter (s,p,d,f,…) describes the shape: s orbitals have no lobes, p have 2 lobes, so a nodal plane through the nucleus, d have 4 lobes and 2 such planes, etc.

Generally, the more nodes, the higher the energy.

In addition to these wavefunctions, describing the probable position of the electron, they have another property – their spin (responsible for magnetism). Another strange rule of quantum mechanics is that once you specify a direction of measurement, an electron can only be spinning clockwise (called “spin-down”) or counter-clockwise (called “spin-up”) around an axis in that direction; it cannot be spinning around a different (e.g. perpendicular) axis.

There are 2 other 2p orbitals with the lobes pointing in other directions.

There are 4 other 3d orbitals with lobes pointing in other directions.

Electrons, protons, and neutrons are all “Fermi” particles and obey a very important rule, the “Pauli Exclusion Principle”: No two indistinguishable Fermi particles ever occupy the same state. For electrons in an atom, it means that at most two can be in the same orbital (spatial wavefunction), one with spin-up and one with spin-down. So in its lowest energy state (its “ground state”), the electrons fill up the orbitals, two at a time, from lowest energy to highest. The number of electrons in a neutral atom (= number of protons in the nucleus) = “atomic number”.

The Pauli Exclusion Principle is responsible for the differences between elements and therefore all of chemistry!

Many of the orbitals are relatively close in energy; such nearby energy levels are called “shells”. Atoms which have their most energetic electrons in the same shell are put in the same row in the periodic table. Elements that have filled shells (He, Ne, Ar, Kr, Xe, Rn) are chemically inactive (don’t want to accept or give away any electrons) and are called “inert”. Atoms with a single electron outside the shell give it away easily so are very reactive; these alkali metals (Li, Na, K, Rb, Cs, Fr) are chemically very similar. Similarly, other columns contain atoms that have similar arrangements of electrons outside of a closed shell and are chemically similar to each other.

Neon has 10 electrons, so in its ground state, two go into the 1s orbital. The remaining eight electrons go into the 2s orbital (which holds two) and 2p orbital (which holds six). Since the 2s and 2p are in the same shell, neon has two filled shells and is an inert gas.

Suppose you create an electric discharge, with fast electrons banging into atoms. Some of these collisions will “excite” the atom, i.e. knock its electrons into higher states. If the collision knocks an electron into the 3p state, it will then want to “fall back down” to its ground state. Often, it does this in steps. If it falls from 3p to 3s, it may omit a photon with energy = 3.15 x 10-19 J (the difference in energy of these levels), so frequency f = E/h = 4.75 x 1014 Hz and wavelength λ = 632 nm (in the red part of the spectrum). (The electron will then fall from the 3s to 2p ground state emitting a much higher energy, ultraviolet photon, since the energy difference between states in different shells is large.)

[Atoms can also fall into the ground state without radiating by losing energy in collisions with other atoms or the walls of the container.]

Sodium (Na) has 11 electrons, with one outside neon’s closed shell, in a 3s orbital. Sodium metal has a low boiling point so can easily be turned into a vapor. In a discharge, its most likely excited state is for the 3s electron to rise into a 3p state. If it falls back down into the 3s-state by emitting a photon, the energy change = 3.4 x 10-19 J, so the frequency = 5.1 x 1014 Hz and wavelength = 590 nm, which is in the yellow/orange part of the spectrum.

Many street-lights use very (energy efficient) sodium lamps, but with sodium vapor at a relatively high pressure. Then the sodium atoms will be closer together – because of the interactions between the atoms, the energy levels change. Since each sodium atom will have somewhat different neighbor distances, each will have different energies, so photons come out with a range of frequencies. In addition, some sodium lamps also contain other elements, which give rise to other colors in the spectrum.

Fluorescent lights rely on a discharge in mercury (Hg) vapor. Here, one of the 6s electrons is excited into a higher state; the lowest energy excited state is 6p. When the electron falls back into the 6s state, it emits a photon with energy 7.8 x 10-19 J, frequency = 1.2 x 1015 Hz, and wavelength = 254 nm. This is in the ultraviolet, so we cannot see it. The fluorescent tube is coated with solid “phosphors”. The phosphor molecules can absorb the uv photon from mercury, going into excited states. They then de-excite by a series of lower energy (i.e. in the visible part of the spectrum) transitions. The phosphors are chosen to give several different wavelengths (colors) so that their sum appears ~ white.

We have seen that many of the energies involved in atomic physics are in the range of 10-19 J. It is simpler to give these energies in a different unit, the electron⸱volt (eV). 1 eV is the energy of an electron (with charge = - 1.6 x 10-19 C) accelerated through a potential of 1 volt. Since ΔE = q ΔV, 1 eV = (1.6 x 10-19 C) (1V) = 1.6 x 10-19 J.

Therefore, the uv photon of mercury has an energy E = (7.8 x 10-19 J) /(1.6 x 10-19 J/eV) = 4.9 eV

Exercises:23.While a sodium atom is in its ground state, it cannot emit light. Why not?24.When a sodium atom is in its lowest energy excited state, it can emit light. Why?25.You expose a gas of argon atoms to light with photon energies that don’t correspond to the energy

difference between any pair of states in the argon atom. Explain what happens to the light.26.A discharge in a mixture of gases is more likely to emit a full white spectrum of light than a discharge in a

single gas. Why?27.If the low-pressure neon vapor in a neon sign were replaced by low-pressure mercury vapor, the sign

would emit almost no visible light. Why not?

Problems:2. If a low-pressure sodium vapor lamp emits 50 W of yellow light with frequency 5.08 x 1014 Hz, how many

photons does it emit each second?3. A particular X-ray has a frequency of 1.2 × 1019 Hz. How much energy does its photon carry?4. A particular light photon carries an energy of 3.8 × 10−19 J. What are the frequency, wavelength, and color

of this light?5.If an AM radio station is emitting 50,000 W in its 880-kHz radio wave, how many photons is it emitting

each second?