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

💡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 ≈ S_classicalinterfere 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 x_a at time t_ato position x_b at time t_b 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 ∫𝒟x(t) means "sum over all paths" from (x_a,t_a) to (x_b,t_b).

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 ε = (t_b-t_a)/N → 0. Each integral over x_j 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 Schrödinger equation! The path integral is equivalentto standard QM but provides new insights.

2.4 Classical Limit: Stationary Phase Approximation

As ℏ → 0, the phase S/ℏ oscillates rapidly. Only paths where S is stationary(δS = 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 S_cl 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 ε = (t_b-t_a)/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 → ∞, 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 = ∫𝒟x(t) e^iS/ℏ (sum over all paths)
Each path weighted by phase factor e^iS/ℏ
Classical paths dominate when ℏ → 0 (stationary phase)
Equivalent to Schrödinger equation formalism
Free particle: Exact Gaussian integral result
Provides deep insight into quantum-classical correspondence
Next: Extend this to quantum field theory!

Quantum Field Theory Course

Path Integrals in Quantum Mechanics

🔗Course Connections

📚Prerequisites

Video Lecture

2.1 The Path Integral Idea