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Quantum Mechanics Formulas

Complete collection of quantum mechanics formulas with detailed explanations.

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Quantum Basics

Schrödinger Equation

iℏ∂ψ∂t=H^ψi\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
ℏ:Reduced Planck's constant
∂ψ/∂t:Wave function time derivative
Ĥ:Hamiltonian operator
ψ:Wave function

Units:

ℏ:1.055 × 10⁻³⁴ J·s
ψ:m⁻³/²
t:s

Applications:

  • •Atomic structure
  • •Molecular dynamics
  • •Quantum systems

Limitations:

Non-relativistic quantum mechanics

Heisenberg Uncertainty

ΔxΔp≥ℏ2\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:

Δx:Position uncertainty
Δp:Momentum uncertainty
ℏ:Reduced Planck's constant

Units:

Δx:m
Δp:kg·m/s
ℏ:1.055 × 10⁻³⁴ J·s

Applications:

  • •Quantum measurements
  • •Wave function collapse
  • •Quantum limits

Limitations:

Simultaneous measurements

Energy-Time Uncertainty

ΔEΔt≥ℏ2\Delta E\Delta t \geq \frac{\hbar}{2}

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

Notation:

ΔE:Energy uncertainty
Δt:Time uncertainty
ℏ:Reduced Planck's constant

Units:

ΔE:J
Δt:s
ℏ:1.055 × 10⁻³⁴ J·s

Applications:

  • •Virtual particles
  • •Quantum tunneling
  • •Energy conservation

Limitations:

Short time intervals

de Broglie Wavelength

λ=hp\lambda = \frac{h}{p}

de Broglie wavelength equals Planck's constant divided by momentum.

Notation:

λ:de Broglie wavelength
h:Planck's constant
p:Momentum

Units:

λ:m
h:6.626 × 10⁻³⁴ J·s
p:kg·m/s

Applications:

  • •Wave-particle duality
  • •Electron microscopy
  • •Quantum interference

Limitations:

Non-relativistic particles

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Quantum Systems

Particle in a Box

En=n2h28mL2E_n = \frac{n^2h^2}{8mL^2}

Energy levels of particle in infinite potential well equals quantum number squared times Planck's constant squared divided by 8 times mass times length squared.

Notation:

E_n:Energy of level n
n:Quantum number
h:Planck's constant
m:Mass
L:Box length

Units:

E_n:J
n:dimensionless
h:6.626 × 10⁻³⁴ J·s
m:kg
L:m

Applications:

  • •Quantum confinement
  • •Nanostructures
  • •Electronic states

Limitations:

Infinite potential well

Harmonic Oscillator

En=(n+12)ℏωE_n = \left(n + \frac{1}{2}\right)\hbar\omega

Energy levels of quantum harmonic oscillator equals quantum number plus half times reduced Planck's constant times angular frequency.

Notation:

E_n:Energy of level n
n:Quantum number
ℏ:Reduced Planck's constant
ω:Angular frequency

Units:

E_n:J
n:dimensionless
ℏ:1.055 × 10⁻³⁴ J·s
ω:rad/s

Applications:

  • •Molecular vibrations
  • •Phonons
  • •Quantum optics

Limitations:

Simple harmonic motion

Hydrogen Atom Energy

En=−13.6 eVn2E_n = -\frac{13.6\text{ eV}}{n^2}

Energy levels of hydrogen atom equals negative 13.6 electron volts divided by principal quantum number squared.

Notation:

E_n:Energy of level n
n:Principal quantum number

Units:

E_n:eV
n:dimensionless

Applications:

  • •Atomic spectra
  • •Hydrogen-like atoms
  • •Quantum chemistry

Limitations:

Hydrogen atom only

Quantum Tunneling

T≈e−2a2m(V0−E)ℏT \approx e^{-2a\sqrt{\frac{2m(V_0-E)}{\hbar}}}

Transmission probability approximately equals exponential of negative twice barrier width times square root of twice mass times potential energy minus energy, all divided by reduced Planck's constant.

Notation:

T:Transmission probability
a:Barrier width
m:Mass
V₀:Barrier height
E:Particle energy
ℏ:Reduced Planck's constant

Units:

T:dimensionless
a:m
m:kg
V₀, E:J
ℏ:1.055 × 10⁻³⁴ J·s

Applications:

  • •Scanning tunneling microscopy
  • •Nuclear fusion
  • •Quantum computing

Limitations:

Thin barrier approximation

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Quantum Operators

Position Operator

x^ψ(x)=xψ(x)\hat{x}\psi(x) = x\psi(x)

Position operator acting on wave function equals position times wave function.

Notation:

x̂:Position operator
ψ(x):Wave function
x:Position

Units:

x̂:m
ψ(x):m⁻¹/²
x:m

Applications:

  • •Position measurements
  • •Wave function analysis
  • •Quantum mechanics

Limitations:

Position representation

Momentum Operator

p^=−iℏ∂∂x\hat{p} = -i\hbar\frac{\partial}{\partial x}

Momentum operator equals negative imaginary unit times reduced Planck's constant times partial derivative with respect to position.

Notation:

p̂:Momentum operator
i:Imaginary unit
ℏ:Reduced Planck's constant
∂/∂x:Partial derivative

Units:

p̂:kg·m/s
ℏ:1.055 × 10⁻³⁴ J·s

Applications:

  • •Momentum measurements
  • •Wave function analysis
  • •Quantum mechanics

Limitations:

Position representation

Angular Momentum

L^=r^×p^\hat{L} = \hat{r} \times \hat{p}

Angular momentum operator equals position operator cross product with momentum operator.

Notation:

L̂:Angular momentum operator
r̂:Position operator
p̂:Momentum operator
×:Cross product

Units:

L̂:kg·m²/s
r̂:m
p̂:kg·m/s

Applications:

  • •Atomic orbitals
  • •Rotational motion
  • •Quantum mechanics

Limitations:

Three dimensions

Spin Operator

S^=ℏ2σ\hat{S} = \frac{\hbar}{2}\sigma

Spin operator equals reduced Planck's constant divided by 2 times Pauli matrices.

Notation:

Ŝ:Spin operator
ℏ:Reduced Planck's constant
σ:Pauli matrices

Units:

Ŝ:kg·m²/s
ℏ:1.055 × 10⁻³⁴ J·s
σ:dimensionless

Applications:

  • •Electron spin
  • •Magnetic moments
  • •Quantum mechanics

Limitations:

Spin-1/2 particles

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Quantum Measurement

Expectation Value

⟨A⟩=∫ψ∗(x)A^ψ(x)dx\langle A \rangle = \int \psi^*(x)\hat{A}\psi(x)dx

Expectation value of operator A equals integral of complex conjugate of wave function times operator times wave function.

Notation:

⟨A⟩:Expectation value
ψ*(x):Complex conjugate of wave function
Â:Operator
ψ(x):Wave function

Units:

⟨A⟩:Depends on operator
ψ*(x), ψ(x):m⁻¹/²
Â:Depends on operator

Applications:

  • •Quantum measurements
  • •Observable quantities
  • •Quantum mechanics

Limitations:

Normalized wave functions

Probability Density

P(x)=∣ψ(x)∣2P(x) = |\psi(x)|^2

Probability density equals absolute square of wave function.

Notation:

P(x):Probability density
ψ(x):Wave function

Units:

P(x):m⁻¹
ψ(x):m⁻¹/²

Applications:

  • •Position measurements
  • •Wave function interpretation
  • •Quantum mechanics

Limitations:

Normalized wave functions

Commutator

[A^,B^]=A^B^−B^A^[\hat{A}, \hat{B}] = \hat{A}\hat{B} - \hat{B}\hat{A}

Commutator of operators A and B equals operator A times operator B minus operator B times operator A.

Notation:

[Â, B̂]:Commutator
Â, B̂:Operators

Units:

[Â, B̂]:Depends on operators
Â, B̂:Depends on operators

Applications:

  • •Operator algebra
  • •Uncertainty relations
  • •Quantum mechanics

Limitations:

Linear operators