Categories: Physics, Quantum mechanics.


In quantum mechanics, the propagator K(xf,tf;xi,ti)K(x_f, t_f; x_i, t_i) gives the probability amplitude that a particle starting at xix_i at tit_i ends up at position xfx_f at tft_f. It is defined as follows:

K(xf,tf;xi,ti)xfU^(tf,ti)xi\begin{aligned} \boxed{ K(x_f, t_f; x_i, t_i) \equiv \matrixel{x_f}{\hat{U}(t_f, t_i)}{x_i} } \end{aligned}

Where U^exp(itH^/)\hat{U} \equiv \exp(- i t \hat{H} / \hbar) is the time-evolution operator. The probability that a particle travels from (xi,ti)(x_i, t_i) to (xf,tf)(x_f, t_f) is then given by:

P=K(xf,tf;xi,ti)2\begin{aligned} P &= \big| K(x_f, t_f; x_i, t_i) \big|^2 \end{aligned}

Given a general (i.e. non-collapsed) initial state ψi(x)ψ(x,ti)\psi_i(x) \equiv \psi(x, t_i), we must integrate over xix_i:

P=K(xf,tf;xi,ti)ψi(xi)dxi2\begin{aligned} P &= \bigg| \int_{-\infty}^\infty K(x_f, t_f; x_i, t_i) \: \psi_i(x_i) \dd{x_i} \bigg|^2 \end{aligned}

And if the final state ψf(x)ψ(x,tf)\psi_f(x) \equiv \psi(x, t_f) is not a basis vector either, then we integrate twice:

P=ψf(xf)K(xf,tf;xi,ti)ψi(xi)dxidxf2\begin{aligned} P &= \bigg| \iint_{-\infty}^\infty \psi_f^*(x_f) \: K(x_f, t_f; x_i, t_i) \: \psi_i(x_i) \dd{x_i} \dd{x_f} \bigg|^2 \end{aligned}

Given a ψi(x)\psi_i(x), the propagator can also be used to find the full final wave function:

ψ(xf,tf)=ψi(xi)K(xf,tf;xi,ti)dxi\begin{aligned} \boxed{ \psi(x_f, t_f) = \int_{-\infty}^\infty \psi_i(x_i) K(x_f, t_f; x_i, t_i) \:dx_i } \end{aligned}

Sometimes the name “propagator” is also used to refer to the fundamental solution GG of the time-dependent Schrödinger equation, which is related to KK by:

G(xf,tf;xi,ti)=iΘ(tfti)K(xf,tf;xi,ti)\begin{aligned} \boxed{ G(x_f, t_f; x_i, t_i) = - \frac{i}{\hbar} \: \Theta(t_f - t_i) \: K(x_f, t_f; x_i, t_i) } \end{aligned}

Where Θ(t)\Theta(t) is the Heaviside step function.