About blackbody and blackbody radiation

• A blackbody is an idealized physical body that absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence.
• At thermal equilibrium, however, it turns out all blackbody emits radiation, which we call blackbody radiation or thermal radiation, so a blackbody is not really “black”.
• The radiation from a wide variety of sources can be approximated as blackbody radiation: Coal, sun, human body (which emits infrared).
• Curiously, the characteristics of the blackbody radiation depends only on its temperature, and not on the materials.
• As an object gets hotter, the predominant wavelength of the radiation decreases (hence the frequency increases).
• As temperature increases: Infrared $\to$ Red $\to$ Yellow $\to$ White.

Difficulties with classical physics

• The material independence of blackbody radiation means something fundamental is at play because the law that describes it must apply to all materials.
• Yet all the calculations done before 1900 gave the wrong prediction at low wavelength (i.e. high frequency).
• Observation showed intensity of the radiation peaks at a wavelength $\lambda \propto \frac{1}{T}$, but classical theory predicts an ever increasing intensity at low wavelength.
• In 1900, Max Planck discovered (almost by accident) the equation that describes the radiation perfectly, when he made the seemingly unreasonable assumption that energy falls into levels of $E_n = n h f$, for integer $n$, at frequency $f$.
• He found his equation matched the observation if $h = 6.626\times 10^{-34}Js$. This number is now known as the Planck's constant, a fundamental constant in physics that appears in all quantum calculations.
• $E_n = n h f$ means larger energy at higher frequency (shorter wavelength). It means it is more difficult for the atoms on a blackbody to absorb / emit radiation at higher frequency, pushing the radiation curve down at short wavelength (instead of blowing up as in classical calculations).
• The implication is that energy is not always continuous, but could sometimes take discrete values. Energy could exist in discrete packages (or quanta).
• Einstein would later re-interpret the equation $E=hf$ as the energy of one photon (a discrete package of light) to explain the photoelectric effect (see below).

What is the photoelectric effect?

• The photoelectric effect is the emission of electrons when electromagnetic radiation hits a material.
• Electrons emitted in this manner are called photoelectrons (but they are really just like any other electrons).
• Einstein won his Nobel Prize for his explanation of the photoelectric effect using the particle nature of light. He showed that light is quantized into discrete packages called photons.

The experiment

• Experiment carried out in a vacuum tube to minimize collisions.
• Light of various frequency and intensity is shone on the sample, usually a metal.
• An adjustable voltage is applied to the electrodes to accelerate or decelerate the electrons.
• An ammeter measures the resulting current.

Number of electrons $N_e$ and cutoff frequency $f_c$

$N_e \propto I$, thus measuring the current $I$ gives the number of electrons released.

Experimental observations:
• When current is non-zero, stronger light (i.e. higher intensity) releases more electrons.
• No electrons are released when $f \lt f_c$ (the cutoff frequency).
• The cutoff frequency depends on the material from which the electrons are released.

Contradictions with classical physics:
• In classical physics, intensity is proportional to the electric field $E$, which is proprotional to the electric force ($F=qE$) on an electron. Increasing intensity means light is "kicking" the electrons harder, so it is strange that no electrons are released when $f \lt f_c$ even at very high intensity.
• In classical physics, the energy of light does not depend on frequency, so it is strange how the release of electrons is highly correlated with $f$.

Kinetic energy of electrons $KE$ and stopping potential $V_{stop}$

• Electrons are released at different speed. We will focus primarily on the fastest moving electrons.
• $KE_{max}$ is the kinetic energy carried by the fastest electrons, but we will often write it simply as $KE$ when confusion is unlikely.
• The applied voltage can be adjusted to decelerate the electrons (reverse bias).
• The magnitude of the voltage that is just enough to stop all electrons (i.e. $I=0$) is called the stopping potential $V_{stop}$.
• For example, if $V=-5V$ can just about stop all electrons, then $V_{stop} = |V| = 5V$. $V_{stop}$ is defined to be a postive number.

$V_{stop}$ stopping all electrons means even the fastest electrons lose all their kinetic energy going against the potential. Therefore we have: where we used the equation for work done by the electric field $W_E = -q \Delta V$ to deduce the right hand side.

Experimental observations:
• $KE_{max}$ is not affected by intensity.
• $KE_{max}$ increases with $f$.

Contradictions with classical physics:
• In classical physics, intensity is proportional to the electric field $E$, which is proprotional to the electric force ($F=qE$) on an electron. Increasing intensity means light is "kicking" the electrons harder, so it is strange that $KE$ is independent of intensity.
• In classical physics, the energy of light does not depend on frequency, so it is strange how the kinetic energy of electrons is highly correlated with $f$.

Summary

Experimental observations at $f \gt f_c$

Action	$KE$	$N_e$
Increase intensity	No effect	Increase
Increase frequency	Increase	Increase

Predictions from classical physics for all $f$

Action	$KE$	$N_e$
Increase intensity	Increase	Increase
Increase frequency	No effect	No effect

Before we give Einstein's full explanation of the mystery surrounding the photoelectric effect, we will first discuss one of his key insights: light consists of a stream of particles, which we call photons nowadays.

Before Einstein made the proposal in 1905, light has been convincingly "proven" to be a wave. Here were the reasonings:

• Light is an electromagnetic wave of electric and magnetic field fluctuations, as described by Maxwell's equations (around 1850).
• The wave theory of light successfully explain the speed of light, and also its energy and momemtum.
• Light unambiguously exhibits interference and diffraction, a distinct characteristic of wave.

Nature, however, is infinitely subtle, but she found her match in Albert Einstein. Despite the seemingly solid proofs that light is a wave, Einstein thought outside the box and realized the photoelectric effect is evidence that light can also behave like particles. We will discuss his explanation in the next section and focus on the photons in this section.

Einstein proposed that each photon at frequency $f$ carries an amount of energy:

Combine with $c = f \lambda$, we can also write it as $E_\gamma = \frac{hc}{\lambda}$. Note that physicists often use the Greek letter "gamma" $\gamma$ to denote photons.

How could light be particles if we have proofs that it is a wave?

• Wave-particle duality: Einstein proposed that light could have both wave and particle characteristics. The idea is later extended to all matter by Louis de Brogile.
• While in our daily life it seems impossible to imagine something being both a wave and a particle, the microscopic (quantum) world can realize possibilities beyond our basic imagination.
• The two pictures are not mutually exclusive. In the modern quantum field theories, both pictures exists consistently in one mathematical formulation, with particles arising as excitations of the fields that describe matter.

Here is how Einstein describe light, its energy and its intensity in terms of photons:

• Light is a stream of photons each carrying discrete amount of energy $E_\gamma= hf$.
• Intensity of light is proportational to the number of photons, i.e. $\text{intensity} \propto N_\gamma$.
• An electron needs to gain a minimum amount of energy $\Phi$ (called the work function) to overcome the attractive force bounding the electron inside the sample.

The ball pit analogy

Ball pit	Photoelectric effect
Little balls in pit	Electrons bound inside sample
Basketballs	Photons
Number of basketballs thrown at the pit	Number of photons $N_\gamma \propto \text{intensity}$
Energy of each basketball	Energy of a photon $E_\gamma = hf$
Height to the rim of the pit, $H$	Energy that must be overcome for an electron to escape, $\Phi$ (work function)
If each basketball carries too little energy, so no balls come out of the pit. You can throw more basketballs, but still no balls come out.	$hf \lt \Phi$, so the photons do not carry enough energy to release the electrons. Increasing intensity (more photons) does not help.
Each basketball must carry enough energy to bring a ball up by $H$ to release the balls.	The minimum frequency to release electron is $hf_c = \Phi$.
When balls come out, some are faster than others. The fastest balls are the ones closest to the surface and only need to climb up by $H$. Those deeper have to climb up more thus come out with less kinetic energy.	Electrons has a range of speed. The fastest ones are the one that are the loosest (closer to the sample surface) who need to overcome energy $\Phi$. The deeper lying electrons need to overcome more so they come out with less kinetic energy. Therefore $KE_{max} = hf - \Phi$.
When the basketballs are energetic enough to release the balls, throwing more basketballs leads to more balls being released. This does not lead to the little balls moving faster when jumping out.	When eletrons are released ($f \gt f_c$), increasing intensity (thus increasing $N_\gamma$) leads to more electrons released. However, the $KE$ of the electrons is unaffected.
The harder you throw the basketballs, the more energetic are the little balls coming out.	Increasing $f$ increases the energy of each photon $E_\gamma = hf$. So an electron absorbing a photon will escape will more kinetic energy.
The harder you throw the basketballs, the more little balls are coming out because even the balls that are close to the bottom of the pit is shaken out by the fast basketballs.	Increasing $f$ increases the energy of each photon $E_\gamma = hf$ so even the more tightly bound electrons may gain enough energy to escape.

Based on Einstein's explanation, the kinetic energy of the fastest electrons can now be understood:

• When one of the loosest electrons absorb a photon with energy $E_\gamma = hf$, it spends $\Phi$ to overcome the attraction of from the bulk of the sample, and thus escapes with energy $hf - \Phi$.
• Electrons that are buried deeper within the sample (more tightly bound) will need to spend more to escape, thus have less kinetic energy than $hf - \Phi$.
• Different materials have different amount of attraction to electrons, thus $\Phi$ is material-dependent.

The work function $\Phi$

• The electrons are trapped within the sample. The work function $\Phi$ is the energy required for an electron to overcome to atrraction with the sample and escape.
• Analogy: you (the electron) are held hostage, and the bad guys (the sample) won't release you unless your mom (the photon) pays them 1000 dollars (i.e. $\Phi = 1000$). If your mom passes you 1200 dollars ($hf$), then you can give 1000 dollars to the bad guys, and you escape with the remaning 200 dollars ($KE = 1200 - 1000 = 200$).

Work Function of some metals

Metal	$\Phi (eV)$
$Na$	2.46
$Al$	4.08
$Zn$	4.31
$Fe$	4.50
$Cu$	4.70

The cutoff frequency

The photon must have energy of at least $\Phi$ to release an electron. Therefore, the lowest such frequency is: $$ \begin{eqnarray} h f_c &=& \Phi \\ \Rightarrow f_c &=& \frac{\Phi}{h} \end{eqnarray} $$ The cutoff wavelength is: $$ \begin{eqnarray} \lambda_c &=& \frac{c}{f_c} \\ &=& \frac{hc}{\Phi} \end{eqnarray} $$

Using $hc = 1242.4eV\cdot nm$

We know the following values:

• $c = 3\times 10^8 m/s$.
• $h = 6.626 \times 10^{-34}Js$.

A quick calculation gives $hc = 1242.4eV\cdot nm$. This value is very convenient in calculations involving photoelectric effects, where energy and wavelength are usually given in $eV$ and $nm$. See the following exercise for an example.

Stopping potential

Recall $KE_{max} = eV_{stop}$. Since $KE_{max}$ and $V_{stop}$ differ only in $e$, they have the same numerical values when expressed in $eV$ and $V$. For example:

• $KE_{max} = 3eV \Rightarrow V_{stop} = 3V$.
• $V_{stop} = 7V \Rightarrow KE_{max} = 7eV$.

The planetary model of the atoms

• In an early model, electrons were believed to orbit around the nucleus like planets orbiting around the sun.
• However, because of centripetal acceleration of the electrons, Maxwell's equations predict the elctrons would quickly loses energy and spiral into the nucleus in a time scale of around $10^{-8} s$.
• We are still alive (and full of atoms and electrons), so clearly this model cannot be taken literally.

Bohr's hydrogen model

• In 1913, Niels Bohr applied Max Planck's ideas on quantized energy to the hydrogen.
• He proposed that electrons must obey certain conditions that resulted in discrete energy levels.
• Electrons can only have energy in one of the (infinitely many) levels $E_n$ for integer $n\gt 0$. Energy values not on the list are forbidden.
• There exists a lowest energy level (the ground state) with energy $E_1$, so electrons cannot have energy below this value, hence cannot fall into the nucleus.
• Bohr's result matches with experimental observations on atomic spectra (light emitted by excited atoms).
• Bohr's work was semi-classical, meaning it contained a mixture of classical mechanics and quantum mechanics. His calculations is no longer considered rigorous by modern standard.
• Using a later development on the wave nature of matter, we can re-interpret Bohr's conditions as requirement that the electron wave must join smoothly over a circle (a periodic boundary condition), as the simulation below shows. This is still only a semi-classical approach.
• Generalization to atoms beyond hydrogen using his approach was not very successful.

Bohr's model of the hydrogen atom gives the following electron energy levels:

where $\hbar = \frac{h}{2\pi}$. Note that $E_\infty = 0eV$, when electron is freed from the nucleus and can escape from the atom.

Terminology

• $n=1$: Ground state.
• $n=2$: First excited state.
• $n=3$: Second excited state, ...etc.
• Ionization energy: Energy required to free an electron.
• Ionization energy is $13.6eV$ for the hydrogen ground state.

Spontaneous emission

• Electrons tend to fall to lower energy levels spontaneously over time.
• When the electron loses energy during a fall from energy $E_i$ to $E_f$, the energy difference is released as a photon.
• Conservation of energy means $E_\gamma = E_i - E_f$.
• Using $E_\gamma = hf$, the frequency of the released photon can be predicted.
• Historically, the detection of light emitted by excited atoms allowed physicists to deduce the energy levels in atoms, and also the confirmation of Bohr's hydrogen model.

Electron transitions

There are three types of electron transitions among energy levels:

• Absorption: an electron moving to a higher level after absorbing a photon.
• Spontaneous emission: an electron releasing a photon after falling to a lower energy level spontaneously.
• Stimulated emission: a photon causing an electron to fall to a lower energy level, emitting an addition photon that is identical to the original photon.

LASER

• LASER = Light Amplification by the Stimulated Emission of Radiation.
• A photon is multiplied into more and more copies using repeated stimulated emissions.
• Einstein first predicted the existence of stimulated emission using a clever argument based on thermodynamics, even though the quantum dynamics of the process was not understood until many years later.
• LASER requires more electrons to be in the higher energy level than in the lower level, so stimulated emissions will outnumber absorptions. Such a configuration is called population inversion.

Electron as a wave inside a hydrogen atom

• After Bohr, physcists developed a more rigorous approach to model the electron in a hydrogen atom using the Schrödinger equation (more on this in a later chapter).
• The electron (as a wave) exists in rather complicated patterns around the nucleus, as is seen in the pictures and the simulation below.
• So the semi-classical assumptions that went into Bohr's original proof such as circular orbits should no longer be taken seriously.
• He still got the right answer because hydrogen has some special symmetries that dictate the energy levels independent of the details of his calculations.
• In modern physics, one could extract the right answer using the mathematical techniques from group theory (a branch specializes in symmetries) without any of the semi-classical assumptions.

Spin

• Spin $s$ is an intrinsic property of all fundamental particles.
• $s$ is either an integer or half-integer.
• A particle with spin $s$ also carries angular momentum. It is as if the particle is spinning about an axis, like a basketball spinning on your finger. Such classical pictures should not be taken literally.
• When angular momentum is measured along a direction (say $z$), the measured angular momentum $s_z \hbar$ (where $\hbar = \frac{h}{2\pi}$) can only take up discrete values, with $s_z = -s, -s+1, \cdots, +s$.
  Examples:
  • $s=0 \Rightarrow s_z = 0$.
  • $s=\frac{1}{2} \Rightarrow s_z = -\frac{1}{2}, +\frac{1}{2}$.
  • $s=1 \Rightarrow s_z = -1, 0, +1$.
• Fermions are particles with half-integral spin.
  Examples:
  • electrons, muons: $s=\frac{1}{2}$.
  • quarks, protons, neutrons: $s=\frac{1}{2}$.
  • omega baryon $\Omega^-$: $s=\frac{3}{2}$.
• Bosons are particles with integral spin.
  Examples:
  • pions: $s=0$.
  • photons: $s=1$.
  • gravitons: $s=2$.

Pauli's exclusion principle

• Only fermions obey Pauli's exclusion principle: two fermions of the same kind cannot be in exactly the same state.
• Bosons do not obey the exclusion principle so many bosons can occupy the same state.
• Electron being a fermion cannot all be in the ground state as a result. Once the lower energy states are occupied, additional electrons must be in the higher energy state.
• Important in understanding the electron configurations (and hence the chemical properties) of elements in the periodic table.

Electrons

• Spin $s=\frac{1}{2}$.
• Spin along the $z$-direction $s_z=\pm\frac{1}{2}$.
• Angular momentum along the $z$-direction $s_z \hbar =\pm\frac{1}{2} \hbar $.
• Spin up means $s_z=+\frac{1}{2}$, spin state denoted $|\uparrow \rangle = | s=\frac{1}{2}, s_z = +\frac{1}{2} \rangle$.
• Spin down means $s_z=-\frac{1}{2}$, spin state denoted $|\downarrow \rangle = | s=\frac{1}{2}, s_z = -\frac{1}{2} \rangle$.

Quantum numbers

• The quantum state of a system is usually described by a collection of numbers called quantum numbers.
• An electron inside an atom is described by 5 quantum numbers:
  • $n \geq 1$: principal quantum number, must be an integer.
  • $l = 0, 1, 2, \cdots, n-1$: orbital quantum number.
  • $m = -l, -l+1, \cdots, +l$: orbital magnetic quantum number.
  • $s$: spin (always $\frac{1}{2}$ for electrons, so it is often omitted)
  • $s_z = \pm \frac{1}{2}$: spin along the $z$-direction.
• The state of an electron is written as $|\psi \rangle = |nlm \rangle |s s_z\rangle$.
• $|nlm \rangle$ is the spatial wave function, describes the spatial distribution of the electron.
• $|s s_z\rangle$ is the spin wave function, describes the intrinsic behavior of the electron. It is either $|\uparrow \rangle$ or $|\downarrow \rangle$.

Terminology

Shells

$n$	1	2	3	4	5
Shell name	K	L	M	N	O

Orbitals

$l$	0	1	2	3	4	5
Orbital name	$s$	$p$	$d$	$f$	$g$	$h$

Historical names: $s$: sharp, $p$: principal, $d$: diffuse, ...

For example, $1s^2 2s^2 2p^4$ means:
• 2 electrons in the $n=1, l=0$ orbital.
• 2 electrons in the $n=2, l=0$ orbital.
• 4 electrons in the $n=2, l=1$ orbital.

See simulation below for more examples.

Name	Symbol	Unit	Meaning
Maximum kinetic energy of photoelectrons	$KE_{max}$	$J$ or $eV$	kinetic energy of the fastest electrons released by photons
Photon energy	$E_\gamma = hf$	$J$ or $eV$	energy of a single photon at frequency $f$
Work function	$\Phi$	$J$ or $eV$	minimum energy required to release an electron from a material
Cutoff frequency	$f_c$	$Hz$	photons below this frequency are not energetic enough to release electrons
Cutoff wavelength	$\lambda_c$	$m$	photons above this wavelength are not energetic enough to release electrons
Stopping potential	$V_{stop}$	$V$	magnitude of the reverse-biased applied voltage that is just enough to stop the fastest photoelectrons