Wavefunction Normalization Calculator

Calculate Quantum Mechanical Properties

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Gaussian Wavefunction Normalizer

Calculate normalization constant for Gaussian wavefunctions.

Normalization Constant: -

Probability Density Calculator

Calculate probability density and cumulative probability.

Probability Density: -

Expectation Value Calculator

Calculate quantum mechanical expectation values.

Expectation Values: -

Understanding Wavefunction Normalization

What is Normalization?

Key concepts of quantum mechanics:

  • Probability interpretation
  • Normalization condition
  • Orthogonality
  • Completeness

Probability Density

Understanding probability:

  • Born interpretation
  • Square modulus
  • Integration rules
  • Boundary conditions

Expectation Values

Quantum observables:

  • Position
  • Momentum
  • Energy
  • Uncertainty relations

Applications

Normalization is important for:

  • Quantum mechanics
  • Molecular orbitals
  • Spectroscopy
  • Quantum chemistry
  • Particle physics

Essential Normalization Formulas

Normalization Condition

∫|ψ|²dx = 1

Over all space

Gaussian Normalization

N = (2α/π)^(n/4)

For n dimensions

Expectation Value

⟨A⟩ = ∫ψ*Âψdx

For operator Â