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

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.

2.4 Classical Limit: Stationary Phase Approximation

As ℏ β†’ 0, the phase S/ℏ oscillates rapidly. Only paths where S is stationary(Ξ΄S = 0) contribute significantly - these are the classical paths!

$$\delta S = 0 \quad \Rightarrow \quad \frac{\delta S}{\delta x(t)} = 0 \quad \Rightarrow \quad \text{Euler-Lagrange equation}$$

The stationary phase approximation gives:

$$K(x_b,t_b;x_a,t_a) \approx \sqrt{\frac{1}{2\pi i\hbar}\det\left(-\frac{\delta^2 S}{\delta x^2}\right)} e^{iS_{\text{cl}}/\hbar}$$

where Scl is the action evaluated on the classical path. This explains why we see classical behavior!

2.5 Connection to Operator Formalism

The propagator K can be written using the time evolution operator:

$$K(x_b,t_b;x_a,t_a) = \langle x_b|e^{-i\hat{H}(t_b-t_a)/\hbar}|x_a\rangle$$

For short times Ξ΅ = (tb-ta)/N:

$$e^{-i\hat{H}\epsilon/\hbar} \approx 1 - \frac{i\epsilon}{\hbar}\hat{H} = 1 - \frac{i\epsilon}{\hbar}\left(\frac{\hat{p}^2}{2m} + V(\hat{x})\right)$$

Insert complete sets of position eigenstates N times, and use:

$$\langle x|\hat{p}|p\rangle = p\langle x|p\rangle = p\frac{e^{ipx/\hbar}}{\sqrt{2\pi\hbar}}$$

After N iterations and taking N β†’ ∞, this reproduces the path integral formula!

2.6 Why Use Path Integrals?

Advantages

  • No operators! Just classical-looking functions
  • Manifestly Lorentz covariant in field theory
  • Direct connection to classical physics
  • Natural for perturbation theory
  • Easier for gauge theories
  • Generalizes to curved spacetime

Disadvantages

  • Mathematically not rigorous (measure theory issues)
  • Can't easily compute bound states
  • Time-dependent problems harder
  • Requires functional calculus

🎯 Key Takeaways

  • Path integral: K = βˆ«π’Ÿx(t) eiS/ℏ (sum over all paths)
  • Each path weighted by phase factor eiS/ℏ
  • Classical paths dominate when ℏ β†’ 0 (stationary phase)
  • Equivalent to SchrΓΆdinger equation formalism
  • Free particle: Exact Gaussian integral result
  • Provides deep insight into quantum-classical correspondence
  • Next: Extend this to quantum field theory!