1
$\begingroup$

Consider the time evolution operator $U(t_f, t_i)$ that controls the evolution of a wave function according to $|\psi(t_f \rangle = U(t_f, t_i) | \psi(t_i) \rangle$.

As I understand it, the Born rule says that we interpret $ \langle x | \psi(t) \rangle$ as the complex probability amplitude of measuring the system $\psi$ to have position $x$ and time $t$. Using the concept of the wave function, we write $\psi(x, t) = \langle x | \psi(t) \rangle$, and is the square norm of this wave function gives us a pdf.

Thus, it seems reasonable to interpret $ \langle x_f | U(t_f, t_i) | \psi(t_i) \rangle$ as the probability amplitude of measuring the postion of $\psi$ to be $x_f$ at time $t_f$. Let us say that $| \psi(t_i) \rangle = | x_i \rangle$. Then it seems reasonable to interpret $ K(x_f, t_f; x_i; t_i) := \langle x_f | U(t_f, t_i) | x_i \rangle$ as the "transition amplitude" of measuring a system to have position $x_f$ at time $t_f$ when we know the system had position $x_i$ at time $t_i$.

However, I have only seen the term "transition amplitude" used (in Townsend's "A Modern Approach to Quantum Mechanics, Chapter 8, and these notes, Section 1, http://www.blau.itp.unibe.ch/lecturesPI.pdf) used to refer to the notation $\langle x_f, t_f | x_i, t_i \rangle$, where the state $ |x_i, t_i \rangle$ is expressly NOT the time evolution of $ |x_i \rangle$ but rather the eigenfunction of the time evolved position operator in the Heisenberg picture (thus, $ |x_i, t_i \rangle = \exp\{i \hat{H} t_i\} | x_i \rangle$).

So, my question (finally) is: why does it seem that the term "transition amplitude" is applied to $\langle x_f, t_f | x_i, t_i \rangle$ and NOT (the equivalent) $K(x_f, t_f; x_i, t_i)$, when it seems that $K(x_f, t_f; x_i, t_i)$ has a very clear understanding as the transition amplitude of a particle starting at position $x_i$ at time $t_i$ to be measured at position $x_f$ at time $t_f$, whereas the interpretation of $\langle x_f, t_f | x_i, t_i \rangle$ seems much less straightforward?

$\endgroup$
0
$\begingroup$
  1. The overlap ${}_H\langle x_f, t_f | x_i, t_i \rangle{}_H$ is indeed the transition amplitude between the two instantaneous position eigenstates $| x_i, t_i \rangle{}_H$ and $| x_f, t_f \rangle{}_H$ in the Heisenberg picture.

  2. It is also equal to ${}_S\langle x_f | U(t_f, t_i) | x_i \rangle{}_S$ in the Schrödinger picture.

  3. In both pictures, it is often called a transition amplitude, see e.g. Section 2.2 & Section 2.5 in Sakurai.

  4. For a connection between the kernel and the Greens function, see e.g. this Phys.SE post.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.