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Since (\(S\) matrix) formulates the transition amplitude from initial state \(\left|i\right>\) to final state \(\left|f\right>\) through \(\left<i\right|S\left|f\right>\), it is a unitary operator—probability is conserved, meaning \(SS^* = I\). Now, having defined the transition operator through \(S = I + iT\), we can introduce another operator: \(K^{-1} = T^{-1} + iI\) [1].
Explain why this new matrix interesting.
Definition in terms of \(T\)-matrix.
Derive from \(K\)-matrix instead/as well.
A quantum mechanical state at rest with energy \(E_0\) can be described in terms of the wave function:
\[ \psi(t) = \psi_0 e^{-iE_0t} \]
Now, if we assume that the state has a decay width of \(\Gamma\), the probability density \(\psi^*\psi\) of this state can be described as:
\[ \psi^*\psi = \psi_0^*\psi_0 e^{-\Gamma t}. \]
The wave function itself then becomes:
\[ \psi(t) = \psi_0 e^{-iE_0t} e^{-\frac{\Gamma}{2} t} = \psi_0 e^{-i \left(E_0-\tfrac{i}{2}\Gamma\right) t}. \]
A particle with a finite decay width can therefore be described as a particle with complex energy:
\[ E' = E_0 - \frac{i}{2}\Gamma \]
Now, as an experimental physicist, one is interested in predicting the probability of observing the particle at energy \(E\) (we want to describe the observed invariant mass distributions). This can be achieved by applying a Fourier transform, so that \(\psi\) is described in terms of energy \(E\) (or frequency \(\omega\)) instead of time \(t\):
\[ \psi(E) \propto \psi_0 \int_0^\infty e^{i\left(E-E_0+\tfrac{i}{2}\Gamma\right)t}\,\mathrm{d}t \propto \frac{1}{\left(E-E_0\right) - \tfrac{i}{2}\Gamma} \]
The probability to observe the particle at energy \(E\) is therefore:
\[ \psi^*(E)\psi(E) \propto \frac{\frac{\Gamma^2}{4}}{\left(E-E_0\right)^2 + \frac{\Gamma^2}{4}} \]
Describe relation between \(\psi(E)\) and transition amplitude \(M\).
From this, one can see that the transition amplitude \(M\) is described by:
\[ M(E) \propto \frac{\frac{\Gamma}{2}}{\left(E-E_0\right) - i\frac{\Gamma}{2}} \]
because \(|\psi|^2\propto|M|\). This is called non-relativistic Breit–Wigner parametrization.
Describe how the relativistic Breit–Wigner formula:
\[ M(E) \propto \frac{m_0\Gamma}{m_0^2 - m_{ab}^2 - im_0\Gamma} \]
is derived and why this is important in case of \(\Gamma \gg m_0\).
from sympy import symbols
from sympy.plotting import plot
x, Gamma, E0 = symbols(R"x \Gamma E_0")
gamma2_4 = Gamma**2 / 4
breit_wigner = gamma2_4 / ((x - E0) ** 2 + gamma2_4)
plot(
breit_wigner.subs({R"\Gamma": 0.25, "E_0": 0.75}),
xlim=(0, 1.5),
title="Non-relativistic Breit–Wigner",
xlabel="$E$ (A.U.)",
);