Modern Physics Formulas

Complete collection of modern physics formulas with detailed explanations, notation meanings, units, and real-world applications. Master quantum mechanics and relativity.

⚛️

Photoelectric Effect

Photon Energy

E = hf = \frac{hc}{\lambda}

Photon energy equals Planck's constant times frequency, or Planck's constant times speed of light divided by wavelength.

Notation:

E

:Photon energy

h

:Planck's constant

f

:Frequency

c

:Speed of light

\lambda

:Wavelength

Units:

E

:Joule (J) or eV

h

:6.626 × 10⁻³⁴ J·s

f

:Hz

c

:3 × 10⁸ m/s

\lambda

Applications:

•Solar panels
•Photodetectors
•Quantum optics

Limitations:

Monochromatic light

Photoelectric Equation

hf = \phi + KE_{\text{max}}

Photon energy equals work function plus maximum kinetic energy of ejected electron.

Notation:

hf

:Photon energy

\phi

:Work function

KE_{\text{max}}

:Maximum kinetic energy

Units:

hf

:J or eV

\phi

:J or eV

KE_{\text{max}}

:J or eV

Applications:

•Photoelectric cells
•Material characterization
•Quantum efficiency

Limitations:

Clean metal surfaces

Stopping Potential

V_0 = \frac{KE_{\text{max}}}{e}

Stopping potential equals maximum kinetic energy divided by electron charge.

Notation:

V_0

:Stopping potential

KE_{\text{max}}

:Maximum kinetic energy

e

:Electron charge

Units:

V_0

KE_{\text{max}}

e

:1.602 × 10⁻¹⁹ C

Applications:

•Photoelectric experiments
•Electron energy measurement
•Quantum yield

Limitations:

Single electron processes

🌀

Wave-Particle Duality

De Broglie Wavelength

\lambda = \frac{h}{p} = \frac{h}{mv}

De Broglie wavelength equals Planck's constant divided by momentum, or Planck's constant divided by mass times velocity.

Notation:

\lambda

:De Broglie wavelength

h

:Planck's constant

p

:Momentum

m

:Mass

v

:Velocity

Units:

\lambda

h

:6.626 × 10⁻³⁴ J·s

p

:kg·m/s

m

:kg

v

:m/s

Applications:

•Electron microscopy
•Neutron diffraction
•Quantum tunneling

Limitations:

Non-relativistic speeds

Matter Wave Frequency

f = \frac{E}{h}

Matter wave frequency equals energy divided by Planck's constant.

Notation:

f

:Matter wave frequency

E

:Energy

h

:Planck's constant

Units:

f

:Hz

E

h

:6.626 × 10⁻³⁴ J·s

Applications:

•Quantum interference
•Wave function analysis
•Quantum coherence

Limitations:

Single particle states

Phase Velocity

v_{\text{phase}} = \lambda f = \frac{E}{p}

Phase velocity equals wavelength times frequency, or energy divided by momentum.

Notation:

v_{\text{phase}}

:Phase velocity

\lambda

:Wavelength

f

:Frequency

E

:Energy

p

:Momentum

Units:

v_{\text{phase}}

:m/s

\lambda

f

:Hz

E

p

:kg·m/s

Applications:

•Wave packet analysis
•Quantum mechanics
•Particle dynamics

Limitations:

Free particles

⚡

Special Relativity

Time Dilation

\Delta t = \frac{\Delta t_0}{\sqrt{1 - \frac{v^2}{c^2}}}

Time interval in moving frame equals proper time divided by Lorentz factor.

Notation:

\Delta t

:Time interval in moving frame

\Delta t_0

:Proper time

v

:Relative velocity

c

:Speed of light

Units:

\Delta t, \Delta t_0

v, c

:m/s

Applications:

•GPS satellite timing
•Particle accelerators
•Astronomical observations

Limitations:

Inertial reference frames

Length Contraction

L = L_0\sqrt{1 - \frac{v^2}{c^2}}

Length in moving frame equals proper length times Lorentz factor.

Notation:

L

:Length in moving frame

L_0

:Proper length

v

:Relative velocity

c

:Speed of light

Units:

L, L_0

v, c

:m/s

Applications:

•Particle physics
•Relativistic dynamics
•Space-time geometry

Limitations:

Inertial reference frames

Relativistic Energy

E = \frac{mc^2}{\sqrt{1 - \frac{v^2}{c^2}}}

Total relativistic energy equals rest mass times speed of light squared divided by Lorentz factor.

Notation:

E

:Total relativistic energy

m

:Rest mass

c

:Speed of light

v

:Velocity

Units:

E

m

:kg

c

:3 × 10⁸ m/s

v

:m/s

Applications:

•Nuclear reactions
•Particle physics
•Mass-energy equivalence

Limitations:

Inertial reference frames

🔮

Quantum Mechanics

Heisenberg Uncertainty Principle

\Delta x\Delta p \geq \frac{\hbar}{2}

Product of position and momentum uncertainties is greater than or equal to reduced Planck's constant divided by 2.

Notation:

\Delta x

:Position uncertainty

\Delta p

:Momentum uncertainty

\hbar

:Reduced Planck's constant

Units:

\Delta x

\Delta p

:kg·m/s

\hbar

:1.055 × 10⁻³⁴ J·s

Applications:

•Quantum measurements
•Wave function collapse
•Quantum limits

Limitations:

Simultaneous measurements

Schrödinger Equation

i\hbar\frac{\partial\psi}{\partial t} = \hat{H}\psi

Time-dependent Schrödinger equation relates wave function time derivative to Hamiltonian operator.

Notation:

i

:Imaginary unit

\hbar

:Reduced Planck's constant

\frac{\partial\psi}{\partial t}

:Wave function time derivative

\hat{H}

:Hamiltonian operator

\psi

:Wave function

Units:

\hbar

:1.055 × 10⁻³⁴ J·s

\psi

:m⁻³/²

t

Applications:

•Atomic structure
•Molecular dynamics
•Quantum systems

Limitations:

Non-relativistic quantum mechanics

Wave Function Normalization

\int|\psi|^2 dV = 1

Integral of wave function magnitude squared over all space equals one.

Notation:

\int

:Integral over all space

|\psi|^2

:Wave function magnitude squared

dV

:Volume element

Units:

|\psi|^2

:m⁻³

dV

:m³

Applications:

•Probability interpretation
•Quantum normalization
•Wave function analysis

Limitations:

Square integrable functions

✨

Quantum Phenomena

Quantum Tunneling

T \approx e^{-2a\sqrt{\frac{2m(V-E)}{\hbar}}}

Transmission probability approximately equals exponential of negative barrier width times square root of energy difference.

Notation:

T

:Transmission probability

a

:Barrier width

m

:Particle mass

V

:Barrier height

E

:Particle energy

\hbar

:Reduced Planck's constant

Units:

T

:dimensionless

a

m

:kg

V, E

\hbar

:1.055 × 10⁻³⁴ J·s

Applications:

•Scanning tunneling microscopy
•Nuclear fusion
•Quantum computing

Limitations:

Thin barriers, low transmission

Quantum Spin

S = \sqrt{s(s+1)}\hbar

Spin magnitude equals square root of spin quantum number times spin quantum number plus one times reduced Planck's constant.

Notation:

S

:Spin magnitude

s

:Spin quantum number

\hbar

:Reduced Planck's constant

Units:

S

:J·s

s

:dimensionless

\hbar

:1.055 × 10⁻³⁴ J·s

Applications:

•Magnetic resonance
•Particle physics
•Quantum information

Limitations:

Intrinsic angular momentum

Quantum Energy Levels

E_n = (n + \frac{1}{2})\hbar\omega

Energy of nth quantum level equals quantum number plus half times reduced Planck's constant times angular frequency.

Notation:

E_n

:Energy of nth level

n

:Quantum number

\hbar

:Reduced Planck's constant

\omega

:Angular frequency

Units:

E_n

n

:dimensionless

\hbar

:1.055 × 10⁻³⁴ J·s

\omega

:rad/s

Applications:

•Harmonic oscillator
•Molecular vibrations
•Quantum systems

Limitations:

Simple harmonic oscillator

Practice Problems

📝Calculate photon energy for 500nm light
📝Find de Broglie wavelength of 1keV electron
📝Calculate time dilation for 0.8c velocity

Try Interactive Calculators

Photoelectric Effect De Broglie Wavelength Relativistic Energy Relativistic Kinematics Quantum Tunneling Compton Scattering

Study Tips

💡Remember Planck's constant values
💡Use relativistic formulas for high speeds
💡Understand wave-particle duality