Path Integrals

$Ψ (x^{″}, t) = \int d x^{'} K (x^{″}, t, x^{'}, t_{0}) Ψ (x^{'}, t_{0})$ , $K (x^{″}, t, x^{'}, t_{0}) = ⟨ x_{N} | U (t, t_{0}) | x_{0} ⟩$ .

$K (x_{n}, t_{n}, x_{0}, t_{0}) = lim_{N \to \infty} {(\frac{m}{2 π i ℏ ε})}^{N / 2} \int_{\infty}^{\infty} d x_{1} \dots \int_{\infty}^{\infty} d x_{N - 1} \exp \frac{i}{ℏ} \sum_{n = 1}^{N} ε \sum \frac{m (x_{n} - x_{n - 1})^{2}}{2 ε^{2}} = \int_{x_{0}}^{x_{N}} D [x (t)] \exp \frac{i}{ℏ} \int_{t_{0}}^{t} L_{c l a s s i c a l} d t$

Free Particle

$t_{n} - t_{n - 1} = ε$ $L = \frac{1}{2} m {\dot{x}}^{2}, S = \int_{t_{0}}^{t_{N}} L d t = \sum_{n = 1}^{N} \frac{1}{2} m {(\frac{x_{n} - x_{n - 1}}{t_{n} - t_{n - 1}})}^{2} (t_{n} - t_{n - 1})$ Thus, $K (x_{N}, t_{N}, x_{0}, t_{0}) = \int_{x_{0}}^{x_{N}} D [x (t)] \exp \frac{i}{ℏ} S$ Hence, $D [x (t)] = {(\frac{m}{2 π i ℏ ε})}^{N / 2} \int_{\infty}^{\infty} d x_{1} \dots \int_{\infty}^{\infty} d x_{N - 1}$ .

Single Step

$Ψ (x^{″}, t) = \int_{R} K Ψ (x^{'}, t_{0})$ $K (x_{0} + η, t_{0} + ε; x_{0}, t_{0}) = {(\frac{m}{2 π i ℏ ε})}^{1 / 2} \exp \frac{i}{ℏ} \frac{m η^{2}}{2 ε}$ Let $x = x_{0} + η, t = t_{0} + ε$ . Then, $Ψ (x, t) = {(\frac{m}{2 π i ℏ ε})}^{1 / 2} \int_{R} d η Ψ (x_{0}, t_{0}) \exp \frac{i}{ℏ} \frac{m η^{2}}{2 ε} = {(\frac{m}{2 π i ℏ ε})}^{1 / 2} \int_{R} d η Ψ (x - η, t_{0}) \exp \frac{i}{ℏ} \frac{m η^{2}}{2 ε} = {(\frac{m}{2 π i ℏ ε})}^{1 / 2} \int_{R} d η [Ψ (x, t_{0}) - η \frac{\partial P s i}{\partial x} + \frac{1}{2} η^{2} \frac{\partial^{2} Ψ}{\partial x^{2}}] \exp \frac{i}{ℏ} \frac{m η^{2}}{2 ε} = {(\frac{m}{2 π i ℏ ε})}^{1 / 2} [Ψ (x, t_{0}) {(\frac{2 π i ℏ ε}{m})}^{1 / 2} + 0 + \frac{ℏ ε}{2 i m} {(\frac{2 π i ℏ ε}{m})}^{1 / 2} \frac{\partial^{2} Ψ}{h \partial x^{2}}] = Ψ (x, t_{0}) + \frac{ℏ i}{2 m} \frac{\partial^{2} Ψ}{\partial x^{2}} ε$

So, $Ψ (x, t_{0} + ε) - Ψ (x, t_{0}) = \frac{\partial Ψ}{\partial t} ε = \frac{i ℏ ε}{2 m} Ψ^{″}$ hence $i ℏ \dot{Ψ} = \frac{p^{2}}{2 m} Ψ$

Usefulness

Used in Field Theory but not in typical particle QM.