Planck's quantum theory

Point a thermometer-like spectrometer at a red-hot iron rod and something odd shows up: no classical theory of light could predict the exact colour and brightness spread of the glow it emits. In 1900 Max Planck fixed this by proposing something radical — that a hot object cannot radiate energy in a smooth, continuous trickle. It must emit and absorb energy in discrete parcels, or quanta, each worth hνh\nu. Five years later Einstein used the very same idea to crack an even stranger puzzle, the photoelectric effect, where light knocks electrons out of a metal instantly, or not at all, depending only on colour, never on brightness. Together these two results dismantled the classical picture of light as a smooth wave carrying any amount of energy, and opened the door to the quantised, particle-like view of energy that Bohr would soon use to explain atomic spectra.

The core insight

Radiant energy is not continuous but comes in discrete packets of size E=hνE = h\nu, and this single idea explains why light below a threshold frequency cannot eject electrons from a metal no matter how intense it is.

The core idea

Wave description of light. Electromagnetic radiation travels through vacuum at speed c=3×108 m s−1c = 3\times10^8\ \text{m s}^{-1} and is described by its wavelength λ\lambda and frequency ν\nu, related by

c=νλc = \nu\lambda

where: cc is the speed of light, ν\nu is frequency (in hertz, cycles per second) and λ\lambda is wavelength (in metres). Spectroscopists often use the wavenumber νˉ=1/λ\bar{\nu} = 1/\lambda, the number of wavelengths per unit length, instead of frequency.

Planck's quantum hypothesis (1900). Studying the spectrum of radiation from a hot glowing object (black-body radiation), classical physics predicted an ever-rising intensity at short wavelengths — an absurd result nicknamed the "ultraviolet catastrophe" that was never observed. Planck resolved this by proposing that a body does not emit or absorb energy continuously but only in whole-number multiples of a smallest packet, a quantum. The energy of one quantum of radiation of frequency ν\nu is

E=hνE = h\nu

where: EE is the quantum's energy (joules), ν\nu is the frequency (hertz), and hh is Planck's constant, h=6.626×10−34 J sh = 6.626\times10^{-34}\ \text{J s} — an extremely small number, which is why energy quantisation is invisible in everyday, macroscopic events but decisive at the atomic scale. More generally, a body can only exchange energy in amounts E=nhνE = nh\nu, where n=1,2,3,…n = 1, 2, 3, \ldots is a positive integer; for light, the quantum of energy is later given the name photon.

Figure
Diagram comparing continuous classical energy absorption with Planck quantised energy levels spaced by h nu
Continuous versus quantised energy classical picture (wrong) energy absorbed smoothly, any amount Planck picture (correct) n=1 n=2 n=3 n=4 energy jumps only in steps of h nu

The photoelectric effect. Shine light of high enough frequency on a metal surface and electrons are ejected instantaneously. Three experimental facts about it defeated classical wave theory: electrons appear only above a minimum threshold frequency ν0\nu_0 characteristic of the metal, regardless of intensity; below ν0\nu_0, no electrons are ever ejected, however bright the light; and above ν0\nu_0, the ejected electrons' kinetic energy rises linearly with frequency, while brighter light only ejects more electrons per second, not faster ones. Einstein explained all three at once (1905) by treating light itself as a stream of photons, each carrying hνh\nu, and applying conservation of energy to a single photon-electron collision.

Derivation AEinstein's photoelectric equation from energy conservation

The goal is to show, starting only from Planck's idea that light energy arrives as discrete photons and from ordinary conservation of energy for one photon striking one electron, that the maximum kinetic energy of an ejected electron must be KEmax=hν−hν0KE_{max} = h\nu - h\nu_0, and that this single equation explains the threshold frequency, the intensity-independence of KEmaxKE_{max}, and the instantaneous nature of the effect that classical wave theory could never account for.

Step 1. Picture one photon of frequency ν\nu, carrying energy E=hνE = h\nu (Planck's quantum relation), striking one electron bound inside the metal. Einstein's key assumption is that this is a one-to-one event: a single photon transfers its entire energy to a single electron, not a slow accumulation from many photons.

Step 2. Freeing that electron from the metal costs a fixed minimum energy, the work function hν0h\nu_0, the energy needed to overcome the attractive forces holding the electron inside the metal. This quantity is a property of the metal alone and does not depend on the incident light.

Step 3. Apply conservation of energy to the collision: the photon's energy hνh\nu is split into two parts — the energy hν0h\nu_0 used to free the electron, and whatever energy remains, which becomes the electron's kinetic energy after escape. Writing this as an energy balance, hν=hν0+KEmaxh\nu = h\nu_0 + KE_{max}, and rearranging for the kinetic energy of the most weakly bound (least strongly held) electrons gives

KEmax=hν−hν0KE_{max} = h\nu - h\nu_0

Step 4. Read off the three experimental facts from this one equation. If ν<ν0\nu < \nu_0, the right-hand side is negative, which is impossible for a kinetic energy — so no electron can be ejected below the threshold frequency, however intense the beam, because intensity only changes the number of photons arriving per second, not the energy hνh\nu carried by each one. If ν>ν0\nu > \nu_0, KEmaxKE_{max} grows linearly with ν\nu, exactly as observed. And because each electron absorbs one photon's energy in a single instantaneous event rather than accumulating energy gradually, emission (when it happens at all) is immediate, with no measurable time lag.

Worked example

Question. The work function of sodium metal is 2.3 eV2.3\ \text{eV} (1 eV=1.602×10−19 J1\ \text{eV} = 1.602\times10^{-19}\ \text{J}). Light of wavelength 450 nm450\ \text{nm} (blue) falls on sodium. Find the threshold frequency of sodium, confirm whether this light can eject electrons, and find the maximum kinetic energy of the ejected electrons in both joules and electronvolts.

Step 1. Convert the work function to joules: hν0=2.3×1.602×10−19=3.685×10−19 Jh\nu_0 = 2.3 \times 1.602\times10^{-19} = 3.685\times10^{-19}\ \text{J}.

Step 2. Find the threshold frequency: ν0=hν0h=3.685×10−196.626×10−34=5.56×1014 Hz\nu_0 = \dfrac{h\nu_0}{h} = \dfrac{3.685\times10^{-19}}{6.626\times10^{-34}} = 5.56\times10^{14}\ \text{Hz}.

Step 3. Find the frequency of the incident 450 nm light: ν=cλ=3×108450×10−9=6.67×1014 Hz\nu = \dfrac{c}{\lambda} = \dfrac{3\times10^8}{450\times10^{-9}} = 6.67\times10^{14}\ \text{Hz}.

Step 4. Compare: ν=6.67×1014 Hz\nu = 6.67\times10^{14}\ \text{Hz} is greater than ν0=5.56×1014 Hz\nu_0 = 5.56\times10^{14}\ \text{Hz}, so yes, this blue light can eject electrons from sodium.

Step 5. Find KEmaxKE_{max}: KEmax=hν−hν0=(6.626×10−34)(6.67×1014)−3.685×10−19=4.42×10−19−3.685×10−19=7.4×10−20 JKE_{max} = h\nu - h\nu_0 = (6.626\times10^{-34})(6.67\times10^{14}) - 3.685\times10^{-19} = 4.42\times10^{-19} - 3.685\times10^{-19} = 7.4\times10^{-20}\ \text{J}. Converting to electronvolts, KEmax=7.4×10−20/1.602×10−19=0.46 eVKE_{max} = 7.4\times10^{-20}/1.602\times10^{-19} = 0.46\ \text{eV}. The ejected electrons leave sodium with a maximum kinetic energy of about 7.4×10−20 J7.4\times10^{-20}\ \text{J}, roughly 0.46 eV0.46\ \text{eV}.

Competitive stretch

If the same sodium surface is instead lit with red light of wavelength 700 nm, is emission still possible? The frequency is ν=c/λ=(3×108)/(700×10−9)=4.29×1014 Hz\nu = c/\lambda = (3\times10^8)/(700\times10^{-9}) = 4.29\times10^{14}\ \text{Hz}, which is less than ν0=5.56×1014 Hz\nu_0 = 5.56\times10^{14}\ \text{Hz}. Since ν<ν0\nu < \nu_0, KEmaxKE_{max} would be negative — impossible — so no electrons are ejected, no matter how bright the red beam is made. This is the sharp frequency cutoff that intensity alone can never overcome.

Common traps

  • Plugging wavelength directly into E=hνE = h\nu instead of converting it to frequency first via ν=c/λ\nu = c/\lambda.
  • Assuming that increasing the intensity of light below ν0\nu_0 will eventually eject electrons — it will not; intensity changes only the number of photons per second, never the energy per photon.
  • Confusing the work function hν0h\nu_0 (fixed property of the metal) with the incident photon energy hνh\nu (depends on the light source); emission needs hνh\nu greater than hν0h\nu_0, never the reverse.
  • Mixing units — forgetting to convert electronvolts to joules (or vice versa) consistently before substituting into E=hνE = h\nu or KEmax=hν−hν0KE_{max} = h\nu - h\nu_0.

Ask the tutor: if you want to see how the photoelectric equation extends to finding stopping potential or how it leads into Bohr's model of the atom, ask the AI tutor to walk through a few more examples with you.

Why does classical wave theory fail to explain the photoelectric effect?

Classical theory treats light as a continuous wave whose energy depends only on intensity (amplitude), spread evenly over the wavefront. It predicts that even dim light, given enough time, should build up enough energy to eject an electron at any frequency, and that brighter light should eject electrons with more kinetic energy. Neither happens: emission is instantaneous or absent depending only on frequency, and KEmaxKE_{max} depends only on frequency too. Only treating light as photons, each carrying a fixed energy hνh\nu, matches these facts.

Is the threshold frequency the same as the work function?

No, they describe the same physical fact in different units. The work function hν0h\nu_0 is an energy (joules or electronvolts), the minimum energy needed to remove an electron from a particular metal. The threshold frequency ν0\nu_0 is that same quantity expressed as a frequency, obtained by dividing the work function by Planck's constant, ν0=(hν0)/h\nu_0 = (h\nu_0)/h. Different metals have different work functions and therefore different threshold frequencies; alkali metals like sodium have low work functions, so even visible light can eject electrons from them.

Does every photon above the threshold frequency eject an electron?

Not necessarily every single photon, but the theory concerns what is possible in principle: any photon with ν>ν0\nu > \nu_0 carries enough energy to free an electron, so emission becomes possible and, in practice, happens essentially instantaneously and continuously as long as light of that frequency keeps arriving. The maximum kinetic energy formula KEmax=hν−hν0KE_{max} = h\nu - h\nu_0 describes the most energetic electrons, those least tightly bound; electrons bound more strongly inside the metal emerge with less kinetic energy, but none exceed KEmaxKE_{max} for a given frequency.

Before this

Where this leads