Black-body radiation and Planck's quantum hypothesis E = hν: Question bank

A few symbols reused from the parent, in plain words first:

$ν$ (Greek "nu") — frequency, how many wave-crests pass per second.
$λ$ (lambda) — wavelength, distance between crests. They trade off through $ν = c / λ$ .
$c$ — the speed of light in vacuum, a fixed constant ( $c \approx 3.0 \times 1 0^{8} m/s$ ). Because every light wave travels at this same speed, a shorter wavelength must fit more crests per second, so $ν$ and $λ$ are locked together by $ν = c / λ$ .
$h$ — Planck's constant, the tiny number that sets the size of one energy packet.
$k T$ — a thermal energy budget: $k$ (Boltzmann's constant) times temperature $T$ ; roughly how much energy random jiggling can hand a mode.
A mode — one standing-wave pattern the cavity can hold, like one specific note a guitar string can sing.
$u (ν)$ — the spectral energy density: how much radiation energy sits in the cavity per unit volume, per small slice of frequency around $ν$ . It is the height of the black-body curve at frequency $ν$ .

Read these figures first

The whole topic is one picture: energy per mode versus frequency. Two ideas combine — how many modes there are, and how much energy each one gets.

How the $ν^{2}$ mode count arises (why so many high- $ν$ modes): a cubic cavity of side $L$ only fits standing waves whose half-wavelengths pack whole numbers along each edge — allowed patterns sit on a grid of points $(n_{x}, n_{y}, n_{z})$ . Counting how many grid points have frequency near $ν$ is like counting points on a spherical shell of radius $\propto ν$ ; a shell's area grows as (radius) $^{2}$ , so the number of modes per frequency slice grows as $ν^{2}$ . That is the origin of the $ν^{2}$ factor — see the shell in the first figure.

Figure — Black-body radiation and Planck's quantum hypothesis E = hν

Where $⟨ E ⟩$ comes from (the partition-function sum): the average uses Boltzmann weights over the discrete rungs $E_{n} = nh ν$ . The denominator $Z = \sum_{n} e^{- nh ν / k T}$ is a geometric series (each term is the previous times $x = e^{- h ν / k T}$ ), which sums to $1/ (1 - x)$ ; the numerator $\sum_{n} nh ν e^{- nh ν / k T}$ sums to $h ν x / (1 - x)^{2}$ . Dividing gives the famous denominator: $⟨ E ⟩ = \frac{h ν x / ( 1 - x ) ^{2}}{1/ ( 1 - x )} = \frac{h ν x}{1 - x} = \frac{h ν}{e ^{h ν / k T} - 1} .$ The second figure shows those shrinking Boltzmann bars and where the " $- 1$ " is born.

Two curves, two shapes — the $u (ν)$ vs $u (λ)$ trap: the third figure overlays the classical Rayleigh–Jeans blow-up, Planck's finite hump, and the two limiting slopes. It also marks Wien's peak $λ_{ma x} T = b$ and reminds you the area under the curve is Stefan–Boltzmann ( $\propto T^{4}$ ).

True or false — justify

Black-body emission depends on what the body is made of.

False. By definition a black body's spectrum depends only on temperature; material and shape drop out, which is exactly why the curve is called universal.

A perfect black body looks pitch-black at every temperature.

False. "Black" means it absorbs all incident light; when heated it emits strongly and can glow red, white, even blue. Black is about absorption, not appearance when hot.

The Rayleigh–Jeans law is simply wrong everywhere.

False. It is correct at low frequency / long wavelength (where

h ν ≪ k T

); it only fails catastrophically toward the ultraviolet.

Planck's law reduces to Rayleigh–Jeans in a limiting case.

True. For

h ν ≪ k T

e^{h ν / k T} \approx 1 + h ν / k T

, so

⟨ E ⟩ \to k T

— the classical answer is a special case of Planck's.

Energy of a single oscillator can take any value you like.

False. Planck restricts it to

E = nh ν

with

n = 0, 1, 2, \dots

— a staircase, not a ramp. Only whole steps are allowed.

Doubling the frequency doubles the total energy the body radiates.

False. Each quantum

h ν

doubles, but high-

ν

modes are rarely excited (Boltzmann suppression

e^{- h ν / k T}

), so total radiated energy does not scale that simply.

The ultraviolet catastrophe was actually observed in experiments.

False. It was a prediction of classical theory; experiment always showed the curve falling to zero at short wavelength. The catastrophe is a failure of the theory, not of nature.

Hotter black bodies peak at longer wavelengths.

False. Wien's law

λ_{ma x} T = b

(a constant) means hotter → shorter

λ_{ma x}

(red-hot to blue-hot), because raising

T

must lower the product's partner.

At absolute zero a black body still emits a black-body spectrum.

False (degenerate case). With

T \to 0

k T \to 0

, no mode can afford even one quantum, and the total (

\propto T^{4}

, by Stefan–Boltzmann) vanishes — no thermal emission.

The frequency-spectrum peak

ν_{ma x}

and wavelength-spectrum peak

λ_{ma x}

satisfy

λ_{ma x} = c / ν_{ma x}

False. Because

u_{λ} = u_{ν} c / λ^{2}

reshapes the curve, the two peaks are different points; converting one peak directly into the other is a classic error.

Spot the error

E = h ν

, so if I know the wavelength I just plug

λ

in for

ν

Error:

ν

and

λ

are different quantities related by

ν = c / λ

. Convert first, or use

E = h c / λ

"Planck freezes out high-frequency modes because

h ν

is negative there."

Error:

h ν

is always positive. They freeze because a whole packet

h ν

exceeds the thermal budget

k T

, so the Boltzmann factor

e^{- h ν / k T}

makes excitation vanishingly rare.

"Equipartition gives

k T

per mode, and since gases obey it, black-body radiation must too."

Error: gases have finitely many modes, but a cavity has infinitely many high-

ν

modes (the

ν^{2}

shell count); handing

k T

to each diverges. Quantization is what rescues the count.

"The area under the spectral energy-density curve

u (ν)

is Wien's law."

Error: the area (total energy density, obtained by integrating

u (ν)

over all

ν

) is the Stefan–Boltzmann result (

\propto T^{4}

); Wien's law

λ_{ma x} T = b

is about where the peak sits, not the area.

"Since

⟨ E ⟩ = h ν / (e^{h ν / k T} - 1)

has

h ν

on top, bigger

ν

means bigger average energy."

Error: the denominator grows exponentially with

ν

, overwhelming the linear numerator, so

⟨ E ⟩ \to 0

at high

ν

"Planck derived his law from photons of light travelling through space."

Error: Planck quantized the oscillators in the cavity walls, not light itself. Treating light as photons came later, from Einstein.

"The classical spectral energy density

u (ν) \propto ν^{2} k T

blows up because

k T

grows with

ν

Error:

k T

is fixed at a given temperature. The blow-up comes from the mode count

\propto ν^{2}

growing without bound while each still gets a full

k T

"The wavelength curve

u (λ)

is just the frequency curve

u (ν)

redrawn with

λ = c / ν

on the axis."

Error: you must also multiply by the Jacobian

c / λ^{2}

(

u_{λ} = u_{ν} c / λ^{2}

); skipping it gives the wrong shape and wrong peak.

Why questions

Why is the average energy per mode, not just the energy per photon, the key quantity?

Because the spectrum is built from many modes; you need each mode's thermal average

⟨ E ⟩

(weighted over how often it is excited), which is what the Boltzmann Distribution supplies.

Why does the Boltzmann factor

e^{- h ν / k T}

appear in the derivation?

It weights each allowed energy level by how likely thermal jiggling is to reach it; higher levels are exponentially less probable, which is what starves high-

ν

modes.

Why does the number of modes grow as

ν^{2}

Allowed standing waves sit on a 3-D integer grid; counting those near frequency

ν

counts points on a spherical shell whose area

\propto ν^{2}

, so the density of modes rises with

ν^{2}

Why does truncating

e^{h ν / k T} \approx 1 + h ν / k T

at first order recover the classical

k T

When

h ν ≪ k T

the step

h ν

is tiny compared to the thermal budget, so the discreteness is invisible; the leading linear term is the smooth-ramp approximation, and higher terms (the "staircase corrections") are negligible — exactly the regime where quantization doesn't matter, i.e. equipartition.

Why does making

h

smaller move Planck's world toward the classical one?

Smaller

h

shrinks each step

h ν

, so the staircase looks like a smooth ramp and quantization becomes invisible — recovering continuous classical energy.

Why does a small hole in a heated cavity behave as a near-perfect black body?

Light entering the hole bounces inside and is almost surely absorbed before escaping, so the hole absorbs (and thus emits) like an ideal black surface.

Why is Planck's constant relevant to chemistry, not just physics?

A mole of quanta near

λ \approx 400

nm carries ~300 kJ/mol, the scale of chemical bond energies, so quantized light can break or form bonds.

Why did equipartition succeed for heat capacities of gases yet fail for radiation?

Gases have a finite, small number of active modes at ordinary temperatures; radiation has an unbounded ladder of high-

ν

modes, and only there does the infinite sum diverge classically.

Edge cases

What happens to

⟨ E ⟩

ν \to 0

It approaches

k T

: for tiny

ν

e^{h ν / k T} - 1 \approx h ν / k T

, so

⟨ E ⟩ \to h ν / (h ν / k T) = k T

, the classical value.

What happens to

⟨ E ⟩

ν \to \infty

It goes to

0

: the exponential denominator explodes, so the mode is effectively frozen and contributes nothing.

What does Planck's law predict at

T = 0

for all

ν

Every mode's

⟨ E ⟩ \to 0

(denominator

\to \infty

), so no thermal radiation at any frequency — consistent with a cold body not glowing.

n = 0

(the empty oscillator) an allowed state, and does it matter?

Yes,

n = 0

is allowed (

E = 0

); it is the ground rung of the staircase and the first term (

= 1

) of the partition sum

Z

in the denominator of

⟨ E ⟩

If two black bodies have the same temperature but different sizes, do their spectral shapes differ?

No — the shape (

u (ν)

ν

) depends only on

T

; size changes total output but not where the peak sits or the curve's form.

When does the

u (ν)

-peak and

u (λ)

-peak difference matter most?

Whenever a problem hands you the peak of one and asks for the other — you must re-derive using

u_{λ} = u_{ν} c / λ^{2}

, not just substitute

λ = c / ν_{ma x}

Connections

Photoelectric Effect — where $E = h ν$ was applied to light quanta directly.
Bohr Model of the Atom & de Broglie Wavelength — later quantization built on Planck's step-idea.
Wien's Displacement Law — the peak law $λ_{ma x} T = b$ marked in the figures.
Stefan-Boltzmann Law — total emission $\propto T^{4}$ , the area under the curve.
Boltzmann Distribution — source of the $e^{- E / k T}$ weighting behind every $⟨ E ⟩$ .