Quantum Fundamentals
The building blocks of quantum mechanics
Planck-Einstein Relation
E = hν = ħω
The fundamental relationship between energy and frequency in quantum mechanics.
It demonstrates that energy is quantized in discrete packets called quanta.
E = Energy
Energy of the photon in Joules (J)
h = Planck's constant
6.626 × 10⁻³⁴ J⋅s
ν = Frequency
Frequency in Hertz (Hz)
ħ = Reduced Planck constant
h/2π = 1.055 × 10⁻³⁴ J⋅s
ω = Angular frequency
2πν in rad/s
Formula Importance
100%
De Broglie Wavelength
λ = h/p = h/(mv)
All matter exhibits wave-like behavior, with wavelength inversely proportional to momentum.
This fundamental concept bridges particle and wave nature of matter.
λ = Wavelength
h = Planck's constant
p = Momentum
m = Mass
v = Velocity
Heisenberg Uncertainty Principle
Δx ⋅ Δp ≥ ħ/2
The fundamental limit to the precision with which certain pairs of physical properties
can be known simultaneously.
Δx = Position uncertainty
Δp = Momentum uncertainty
ħ = Reduced Planck constant
Photoelectric Effect
Kmax = hν - φ
The kinetic energy of emitted electrons depends on the frequency of incident light,
not its intensity. This provided crucial evidence for the quantum nature of light.
Kmax = Maximum kinetic energy
h = Planck's constant
ν = Light frequency
φ = Work function
Wave Mechanics
Mathematical framework of quantum waves
Schrödinger Equation (Time-dependent)
iħ ∂Ψ/∂t = ĤΨ
The fundamental equation of quantum mechanics describing how the quantum state
of a physical system changes with time.
i = Imaginary unit
ħ = Reduced Planck constant
Ψ = Wave function
Ĥ = Hamiltonian operator
Schrödinger Equation (Time-independent)
ĤΨ = EΨ
Describes the stationary states of a quantum system, where energy eigenvalues
are constant in time.
Ĥ = Hamiltonian operator
Ψ = Energy eigenfunction
E = Energy eigenvalue
Hamiltonian Operator
Ĥ = -ħ²/2m ∇² + V(x,t)
Represents the total energy of the system, including kinetic and potential energy.
-ħ²/2m ∇² = Kinetic energy operator
V(x,t) = Potential energy
∇² = Laplacian operator
Probability Density
ρ = |Ψ(x,t)|² = Ψ*Ψ
The probability of finding a particle at position x at time t.
ρ = Probability density
Ψ* = Complex conjugate of Ψ
Ψ = Wave function
Normalization Condition
∫ |Ψ(x,t)|² dx = 1
The wave function must be normalized to ensure total probability equals 1.
∫ = Integral over all space
|Ψ|² = Probability density
Momentum Operator
p̂ = -iħ∇
The quantum mechanical operator corresponding to classical momentum.
p̂ = Momentum operator
∇ = Gradient operator
i = Imaginary unit
Quantum Operators
Mathematical tools for quantum observables
Interactive Operator Visualization
Visualization of a quantum wave function and its probability density
Position Operator
x̂ψ(x) = xψ(x)
In position space, the position operator acts by multiplication with the position coordinate.
Angular Momentum Operator
L̂ = r̂ × p̂ = -iħ(r × ∇)
The cross product of position and momentum operators.
Energy Operator
Ê = iħ ∂/∂t
The operator corresponding to the total energy of the system.
Laplacian Operator
∇² = ∂²/∂x² + ∂²/∂y² + ∂²/∂z²
The differential operator representing the sum of second partial derivatives.
Quantum Applications
Real-world quantum phenomena and applications
Hydrogen Atom Energy Levels
En = -13.6 eV / n²
The energy levels of electron in hydrogen atom, derived from Bohr model.
n = Principal quantum number
E = Energy in electron volts
Particle in a Box
En = n²h²/(8mL²)
Energy levels for a particle confined in a 1D box of length L.
n = Quantum number (1,2,3,...)
m = Mass of particle
L = Length of box
Quantum Tunneling
T ≈ e^(-2κL)
Transmission probability for quantum tunneling through a barrier.
T = Transmission probability
κ = Decay constant
L = Barrier width
Bell's Inequality
|S| ≤ 2 (Classical) |S| = 2√2 (Quantum)
Distinguishes between quantum and classical predictions for entangled particles.
Fermi-Dirac Distribution
f(E) = 1/[e^(E-EF)/kT + 1]
Probability distribution for fermions at thermal equilibrium.
EF = Fermi energy
k = Boltzmann constant
T = Temperature
Bose-Einstein Distribution
f(E) = 1/[e^(E-μ)/kT - 1]
Probability distribution for bosons at thermal equilibrium.
μ = Chemical potential
k = Boltzmann constant
T = Temperature
Interactive Calculators
Apply quantum formulas with real-time calculations
De Broglie Wavelength Calculator
Result will appear here
Energy-Frequency Calculator
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Uncertainty Principle Calculator
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Quantum Physics Timeline
The evolution of quantum mechanics
1900
Max Planck
Introduces quantized energy levels and Planck's constant
1905
Albert Einstein
Explains photoelectric effect, introduces photon concept
1924
Louis de Broglie
Proposes wave-particle duality for matter
1926
Erwin Schrödinger
Develops wave equation for quantum mechanics
1927
Werner Heisenberg
Formulates uncertainty principle