# How to understand the transition amplitude in the Copenhagen interpretation

In Chapter 8 of Townsend's A Modern Approach to Quantum Mechanics, he states that the expression $$\langle x', t' | x_0, t_0 \rangle$$ gives the amplitude for a particle that is at position $$x_0$$ to at time $$t_0$$ to be at position $$x'$$ at time $$t'$$.

If we approach this statement from the Copenhagen interpretation, would we understand $$\langle x', t' | x_0, t_0 \rangle$$ as the amplitude that we will measure a particle at time $$t'$$ to have position $$x'$$ given that we measured its position at time $$t_0$$ and obtained a value of $$x_0$$ from that measurement?

• When we say amplitude in QM we simply mean the coefficient of a state with respect to a basis. Here the 'basis' is the position basis. Physically you expand a function into an integral of coefficients times Delta functions. Such coefficients of course are simply $\psi(x)$. The appearance of time into the game doesn't change anything. I hope it's clear – lcv Apr 11 at 20:33

Quantum theory uses point particles, but their description in terms of psi function cannot ever appear to be localized to a single point of space (or a system of particles cannot be localized to a single configuration). The reason is, the way the Born probability interpretation works is that $$|\psi|^2$$ gives probability density that when integrated over all space, gives 1. But in mathematics there is no normalizable psi function that could describe particle localized to a single point in a consistent way; there is no "square root" of delta distribution.
More correct way to understand things such as $$\langle x,t|x_0,t_0\rangle$$ is the way they were introduced or defined in the first place: as Green's function $$G(x,t;x_0,t_0)$$ of the time-dependent Schroedinger's equation. It is a function that solves the time-dependent Schroedinger equation for $$\psi(x',t')$$ for $$t'>t_0$$ for initial condition $$\psi(x',t_0) = \delta(x'-x_0)$$.
This Green's function allows us to express future psi function that comes out as result of evolution of any given initial condition psi function $$\psi_0(x)$$, as an integral of this function:
$$\psi(x,t) = \int G(x,t;x',t_0) \psi_0(x') dx'$$