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Quantum Mechanics2024-10-15

Introduction to Quantum States & Wavefunctions

Core Principle of Quantum Theory

In quantum mechanics, physical systems are described by a complex-valued mathematical entity called a wavefunction, typically denoted by the Greek letter Ψ(x,t)\Psi(x, t).

🎯 Quick Navigation & Key Topics

Schrödinger Equation

Time-dependent wave equation governing non-relativistic state evolution.

Born Probability

Statistical interpretation of wavefunction norm squared |Ψ|².

Hamiltonian Operator

Energy operator derived from kinetic and potential operators.

Normalization Condition

Total probability across all space sums to 1.

The Schrödinger Equation

The time-development of a non-relativistic quantum state is governed by the time-dependent Schrödinger Equation:

iℏ∂∂tΨ(x,t)=[−ℏ22m∂2∂x2+V(x,t)]Ψ(x,t)i\hbar \frac{\partial}{\partial t}\Psi(x, t) = \left[ -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} + V(x, t) \right]\Psi(x, t)

Where:

  • ℏ\hbar is the reduced Planck constant (ℏ≈1.054×10−34 J⋅s\hbar \approx 1.054 \times 10^{-34} \text{ J}\cdot\text{s}).
  • mm is the mass of the particle.
  • V(x,t)V(x, t) is the potential energy function.
  • H^=−ℏ22m∂2∂x2+V(x,t)\hat{H} = -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} + V(x, t) is the Hamiltonian operator.

Step-by-Step State Evolution Workflow

1
Formulate the Potential V(x)

Define the boundary conditions and potential energy barrier V(x)V(x) (e.g., Infinite Square Well or Harmonic Oscillator).

2
Solve the Time-Independent Equation

Separate spatial variables to find stationary energy eigenfunctions ψn(x)\psi_n(x) and eigenvalues EnE_n.

3
Construct the General Wavefunction

Superpose energy eigenstates with phase factors e−iEnt/ℏe^{-i E_n t / \hbar}.

Interactive Code Implementation

import numpy as np
 
def wave_packet(x, x0, sigma, k0):
    return (1.0 / (np.pi**0.25 * np.sqrt(sigma))) * np.exp(-0.5 * ((x - x0)/sigma)**2 + 1j * k0 * x)

Probability Density & Normalization

According to Max Born's statistical interpretation, the square of the magnitude of the wavefunction represents a probability density:

P(x,t)=∣Ψ(x,t)∣2=Ψ∗(x,t)Ψ(x,t)P(x, t) = |\Psi(x, t)|^2 = \Psi^*(x, t)\Psi(x, t)

This leads to the normalization condition, asserting that the particle must exist somewhere in space:

∫−∞∞∣Ψ(x,t)∣2 dx=1\int_{-\infty}^{\infty} |\Psi(x, t)|^2 \, dx = 1

Conservation of Probability

Due to the hermiticity of the Hamiltonian operator H^=H^†\hat{H} = \hat{H}^\dagger, the norm ∫∣Ψ∣2dx\int |\Psi|^2 dx remains constant in time.

Deep Dive: Hermitian Operators