Quantum Field Theory Course

Total: 440+ pages covering classical field theory through the Standard Model

Part III, Chapter 2

Path Integrals in Quantum Mechanics

Feynman's sum-over-paths: a revolutionary approach to quantum mechanics

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πŸ”—Course Connections

πŸ“šPrerequisites

Quantum Mechanics
Wave Functions & Postulates
Basic QM formalism
β†’
Mathematics
Variational Calculus
Action principle and Euler-Lagrange equations
β†’
▢️

Video Lecture

Lecture 8: Path Integral Formalism for Non-Relativistic QM - MIT 8.323

Feynman path integral formulation of quantum mechanics (MIT QFT Course)

πŸ’‘ Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.

2.1 The Path Integral Idea

In standard quantum mechanics, a particle evolves according to the SchrΓΆdinger equation. Feynman discovered an alternative formulation: the particle takes all possible paths, each weighted by eiS/ℏ.

πŸ’‘Classical vs Quantum Paths

Classical mechanics: Particle follows the path that minimizes (extremizes) the action S.

Quantum mechanics (Feynman): Particle "explores" all paths! But paths with S β‰ˆ Sclassicalinterfere constructively, while others cancel out.

This is why we see classical behavior for macroscopic objects - quantum interference averages out!

2.2 Transition Amplitude

The transition amplitude (propagator) from position xa at time tato position xb at time tb is:

$$\boxed{K(x_b,t_b;x_a,t_a) = \int \mathcal{D}x(t) \, e^{iS[x(t)]/\hbar}}$$

where the path integral βˆ«π’Ÿx(t) means "sum over all paths" from (xa,ta) to (xb,tb).

The action S for a path x(t) is:

$$S[x(t)] = \int_{t_a}^{t_b} dt \, L(x,\dot{x},t) = \int_{t_a}^{t_b} dt \left[\frac{1}{2}m\dot{x}^2 - V(x)\right]$$

What Does "Sum Over All Paths" Mean?

Mathematically, we discretize time into N steps:

$$K(x_b,t_b;x_a,t_a) = \lim_{N\to\infty} \int dx_1 \cdots dx_{N-1} \prod_{j=0}^{N-1} \sqrt{\frac{m}{2\pi i\hbar \epsilon}} \exp\left[\frac{i}{\hbar}\epsilon L(x_j, \frac{x_{j+1}-x_j}{\epsilon})\right]$$

where Ξ΅ = (tb-ta)/N β†’ 0. Each integral over xj sums over all possible positions at that time!

2.3 Example: Free Particle

For a free particle (V = 0), the action is:

$$S[x(t)] = \int_{t_a}^{t_b} dt \, \frac{1}{2}m\dot{x}^2$$

The path integral can be computed exactly (it's a Gaussian integral!):

$$\boxed{K(x_b,t_b;x_a,t_a) = \sqrt{\frac{m}{2\pi i\hbar(t_b-t_a)}} \exp\left[\frac{im(x_b-x_a)^2}{2\hbar(t_b-t_a)}\right]}$$

This matches the result from solving the SchrΓΆdinger equation! The path integral is equivalentto standard QM but provides new insights.

Key Concepts (This Page)

  • Path integral: $K = \int \mathcal{D}x(t)\, e^{iS/\hbar}$ sums over all paths
  • Each path is weighted by the phase factor $e^{iS[x]/\hbar}$
  • The measure is defined via time-slicing with N intermediate integrations
  • Free particle propagator is an exact Gaussian integral
  • Result matches the operator (Schrodinger) formalism exactly
Next: Harmonic Oscillator & Euclidean Path Integral