Quantum Mechanics: Wave Function and Operators

Quantum Mechanics

• In quantum mechanics, a particle is distributed through space like a wave, described by a wave function Ψ(x) that contains all the dynamical information (position, momentum, energy) about the system.
• The wave function can be a complex number.

Operators

• Operators are mathematical formulations that act on a function to obtain a new function.
• Examples of operators include:
  • Derivation operator: Ĥ = d/dx
  • Second derivative operator: Ĥ = k d²/dx²
• When an operator acts on a function, a new function is obtained: (operator) (function) = (new function)
• Eigenfunctions are functions that, when acted upon by an operator, result in the same function multiplied by a constant (eigenvalue).

Eigenvalue Equations

• Eigenvalue equations: (operator) (function) = (eigenvalue) (function)
• These equations have solutions called eigenfunctions, where the eigenvalue corresponds to a specific physical property.

Operator Physics

• The state of an object is fully characterized by a wavefunction, Ψ(x).
• For each physical property, there is an operator, and the eigenvalue of the operator on the eigenfunction Ψ(x) gives the value of the property.

Schrödinger Equation

• The Schrödinger equation is a fundamental equation in quantum mechanics that relates the wavefunction and energy of a wave-particle to the potential energy: d²+/dx²+V(x) = E.
• Solution of the Schrödinger equation gives the wavefunction and energy of the system.
• The solutions satisfy the experimentally obtained De Broglie wavelengths.

Physical Meaning of Wavefunction

• The wavefunction contains all the dynamical information (position, momentum, energy) about the system.
• The wavefunction can be used to calculate the average position and momentum of a particle.
• The absolute value of the wavefunction, |Ψ|², represents the probability of finding a particle at a given position.

Complex Numbers

• Complex numbers have real and imaginary parts: z = a + bi, where i is the imaginary unit (i² = -1).
• Complex conjugate: replace i -> -i.
• Euler notation: ebix = cos(bx) + i sin(bx).

Restrictions on Wavefunctions

• Born interpretation puts severe restrictions on the acceptability of wavefunctions:
  • Ψ cannot be infinite anywhere.
  • Ψ must be single-valued.
  • Ψ must be continuous.
  • Ψ must have continuous slope.
  • Ψ cannot be zero everywhere.
  • |Ψ|² can be integrated.

Average Position and Momentum

• The average position of a particle is obtained by: = ∫x|Ψ|²dx.
• The momentum of a particle is obtained by: p̂ = -iℏ(d/dx).