Modern Physics Formula Sheet: Photons, Atoms, Nuclei

This modern physics formula sheet collects the results that describe light as particles, matter as waves, the structure of atoms and nuclei, and motion near the speed of light, all starting from the photon relation $E = h\nu = \frac{hc}{\lambda}$ . It covers the dual nature, atoms and nuclei chapters of Class 12 physics, the modern physics part of JEE Main and NEET, and the quantum and nuclear unit of AP Physics 2.

Please treat it as a study summary, not an exam document. Official formula sheets, where an exam supplies one, come from the exam board, and many Indian exams give no sheet at all, so you must learn these by heart. Each table row links to a page with the symbols, a derivation where boards ask for one, worked examples and a calculator.

Modern physics formula sheet: E = hν = hc / λ

Quick facts

Photon energy
$E = h\nu = \frac{hc}{\lambda}$
Photoelectric equation
$K_{max} = h\nu - \phi_0 = eV_0$
Matter wave
$\lambda = \frac{h}{p}$
Hydrogen levels
$E_n = -\frac{13.6\,Z^2}{n^2}$ eV
Decay law
$N = N_0 e^{-\lambda t}$
Mass energy
$E = mc^2$ , 1 u = 931.5 MeV
Handy constant
hc = 1240 eV nm, so E (eV) = 1240 / λ (nm)

How the modern physics formula sheet fits together

Almost every formula here is either a quantum rule or an energy balance. Follow the chain:

Light comes in packets. A photon of frequency ν carries $E = h\nu$ and momentum $p = \frac{h}{\lambda}$ .
Photons free electrons. The photoelectric equation is energy conservation for one photon and one electron.
Particles behave as waves. The same relation p = h/λ, read the other way, gives a wavelength to any moving particle.
Atoms have fixed levels. Quantised angular momentum fixes the orbits and energies in hydrogen, and jumps between levels make spectral lines.
Nuclei store mass as energy. Binding energy and decay follow from $E = mc^2$ and a simple probability law.

Two energy units run through every chapter. Use joules in SI work and electronvolts for atoms, 1 eV being 1.602 × 10⁻¹⁹ J. Many errors come from switching between the two halfway through a question.

Photons and the photoelectric effect

Photon and photoelectric formulas
Formula	What it gives	Unit
$E = h\nu = \frac{hc}{\lambda}$	Energy of one photon, see the energy formula	J or eV
$p = \frac{h}{\lambda} = \frac{E}{c}$	Momentum of a photon, linked to the momentum formula	kg m/s
$\phi_0 = h\nu_0 = \frac{hc}{\lambda_0}$	Work function from the threshold	J or eV
$K_{max} = h\nu - \phi_0$	Fastest photoelectron energy	J or eV
$eV_0 = K_{max}$	Stopping potential link, see the voltage formula	V for V₀
$\lambda_{min} = \frac{hc}{eV}$	Shortest X ray wavelength from a tube at voltage V	m

Frequency and wavelength are tied by $c = \nu\lambda$ ; see the wavelength formula and frequency formula pages if you need a refresher. The value of c is on the speed of light formula page. The full chapter, with intensity and photon flux, is on the dual nature chapter sheet.

Matter waves

de Broglie wavelength in its common forms
Formula	What it gives	Unit
$\lambda = \frac{h}{p} = \frac{h}{mv}$	Wavelength of any moving particle	m
$\lambda = \frac{h}{\sqrt{2mK}}$	Same, from kinetic energy K	m
$\lambda = \frac{h}{\sqrt{2meV}}$	Particle of charge e after a potential difference V	m
$\lambda = \frac{1.227}{\sqrt{V}}$ nm	Electron accelerated through V volts	nm

The last row is the second one with the electron's mass and charge already put in, so it only works for electrons and gives nanometres. For a proton or an alpha particle, go back to $\frac{h}{\sqrt{2mqV}}$ with the right mass and charge. A heavier particle at the same kinetic energy has a shorter wavelength. The unit check for all of these runs through the dimensional formula of Planck's constant, [M¹ L² T⁻¹].

Atoms: scattering, Bohr's model and spectra

Atomic structure formulas for one electron systems
Formula	What it gives	Unit
$r_0 = \frac{2kZe^2}{K}$	How near a head on alpha particle gets, the closest approach	m
$mvr = \frac{nh}{2\pi}$	Bohr's quantisation of angular momentum	J s
$r_n = 0.529\,\frac{n^2}{Z}$ Å	Radius of the nth orbit	Å (10⁻¹⁰ m)
$v_n = 2.18 \times 10^6\,\frac{Z}{n}$	Electron speed in the nth orbit	m/s
$E_n = -\frac{13.6\,Z^2}{n^2}$	Energy of level n	eV
$\frac{1}{\lambda} = RZ^2\left(n_1^{-2} - n_2^{-2}\right)$	Wavelength of a line, R = 1.097 × 10⁷ m⁻¹	m⁻¹
$N = \frac{n(n-1)}{2}$	Number of spectral lines from level n	no unit
$L = \sqrt{l(l+1)}\,\hbar$	Orbital angular momentum of an orbital	J s

The closest approach result is plain Coulomb energy; the electrostatic force formula and potential energy formula pages supply the background. In Bohr's model the kinetic energy of the electron equals −E_n and the potential energy equals 2E_n, a ratio that turns up in many one mark questions. Electrons also have spin angular momentum, which the orbital formula does not include.

Nuclei and radioactivity

Nuclear formulas
Formula	What it gives	Unit
$R = R_0 A^{1/3}$	Nuclear radius, R₀ ≈ 1.2 fm	m
$\Delta m = [Zm_p + (A - Z)m_n] - M$	Mass defect of a nucleus	u or kg
$E_b = \Delta m\,c^2$	Binding energy; 1 u gives 931.5 MeV	J or MeV
$N = N_0 e^{-\lambda t}$	Undecayed nuclei after time t	no unit
$T_{1/2} = \frac{\ln 2}{\lambda} = \frac{0.693}{\lambda}$	Half life	s
$\tau = \frac{1}{\lambda}$	Mean life	s
$A = \lambda N$	Activity, the decay rate	Bq (decays per second)

Because $R \propto A^{1/3}$ , the volume of a nucleus is proportional to A and nuclear density comes out roughly the same for every element. After n half lives the fraction left is $(1/2)^n$ : three half lives leave one eighth. The decay constant λ here is not a wavelength, so write it clearly in your working.

Special relativity

Relativity formulas
Formula	What it gives	Unit
$E = mc^2$	Rest energy of a mass, linked to the mass formula	J
$\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}$	Lorentz factor	no unit
$\Delta t = \gamma\,\Delta t_0$	Time dilation	s
$L = \frac{L_0}{\gamma}$	Length contraction	m
$E^2 = (pc)^2 + (mc^2)^2$	Energy and momentum together	J

Most school syllabi need only the mass energy relation for nuclear binding. College courses and some advanced school courses also use γ. For any speed below about a tenth of c, γ stays within half a percent of 1, so Newtonian answers are fine.

Study tips and exam traps

Learn hc = 1240 eV nm. It turns most photon questions into one division. Remember it needs λ in nm.
Keep units in one system. Either all SI or all eV with nm. Mixing gives answers off by 10⁻¹⁹ or 10⁹.
Intensity changes the number, not the energy. Brighter light gives more photoelectrons per second, but the same K_max.
Check the sign of E_n. Bound levels are negative. Ionisation energy from level n is +13.6Z²/n² eV.
Single atom or sample. One atom from level n gives at most n − 1 lines; a sample gives n(n − 1)/2.
Use 931.5 MeV per u. It saves converting masses to kilograms.
Half life is not half the mean life. T½ = 0.693τ.

Beyond this chapter, the whole syllabus is collected on separate sheets: Class 12 physics, JEE Main physics, NEET physics and AP Physics 2.

If an answer for an atomic energy comes out in thousands of eV, a unit conversion has almost certainly gone wrong.

Frequently asked questions

What does a modern physics formula sheet include?

It usually covers photon energy and momentum, the photoelectric equation, de Broglie wavelength, Bohr's model of hydrogen with its energy levels and spectral series, nuclear radius, binding energy, radioactive decay and the mass energy relation E = mc².

Can I use this page in place of the exam board sheet?

No, this page is a revision aid. Any official sheet comes from the exam board, and most Indian board and entrance exams expect you to remember these formulas.

Which single formula matters most in modern physics?

E = hν is the base of most of the chapter. Together with p = h/λ and E = mc² it explains the photoelectric effect, matter waves, atomic spectra and nuclear energy.

What is the value of hc in eV nm?

hc is about 1240 eV nm (1239.8 to four figures). Dividing 1240 by a wavelength in nanometres gives the photon energy in electronvolts.

Which modern physics formulas come up most in JEE and NEET?

Students meet the photoelectric equation, de Broglie wavelength, Bohr's energy levels, the number of spectral lines, half life and binding energy again and again in practice papers. Learn the conditions for each, not just the formula.