Categories: Physics, Quantum mechanics.

Propagator

In quantum mechanics, the propagator \(K(x_f, t_f; x_i, t_i)\) gives the probability amplitude that a particle starting at \(x_i\) at \(t_i\) ends up at position \(x_f\) at \(t_f\). It is defined as follows:

\[\begin{aligned} \boxed{ K(x_f, t_f; x_i, t_i) \equiv \matrixel{x_f}{\hat{U}(t_f, t_i)}{x_i} } \end{aligned}\]

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

\[\begin{aligned} P &= \big| K(x_f, t_f; x_i, t_i) \big|^2 \end{aligned}\]

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

\[\begin{aligned} P &= \bigg| \int_{-\infty}^\infty K(x_f, t_f; x_i, t_i) \: \psi_i(x_i) \dd{x_i} \bigg|^2 \end{aligned}\]

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

\[\begin{aligned} P &= \bigg| \iint_{-\infty}^\infty \psi_f^*(x_f) \: K(x_f, t_f; x_i, t_i) \: \psi_i(x_i) \dd{x_i} \dd{x_f} \bigg|^2 \end{aligned}\]

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

\[\begin{aligned} \boxed{ \psi(x_f, t_f) = \int_{-\infty}^\infty \psi_i(x_i) K(x_f, t_f; x_i, t_i) \:dx_i } \end{aligned}\]

Sometimes the name “propagator” is also used to refer to the so-called fundamental solution or Green’s function \(G\) of the time-dependent Schrödinger equation, which is related to \(K\) by:

\[\begin{aligned} \boxed{ G(x_f, t_f; x_i, t_i) = - \frac{i}{\hbar} \: \Theta(t_f - t_i) \: K(x_f, t_f; x_i, t_i) } \end{aligned}\]

Where \(\Theta(t)\) is the Heaviside step function. The definition of \(G\) is that it satisfies the following equation, where \(\delta\) is the Dirac delta function:

\[\begin{aligned} \Big( i \hbar \pdv{t_f} - \hat{H} \Big) G = \delta(x_f - x_i) \: \delta(t_f - t_i) \end{aligned}\]


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