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# 物理代写|核物理代考Nuclear Physics代写|Origin of the Yukawa potential

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## 物理代写|核物理代考Nuclear Physics代写|Origin of the Yukawa potential

The form (1.56), which is derived from quantum field theory, can be understood quite simply. Consider a de Broglie wave
$$\psi=\exp (-i(E t-\boldsymbol{p} \cdot \boldsymbol{r}) / \hbar)$$
The Schrödinger equation in vacuum is obtained by using $E=p^2 / 2 m$ and then taking the Laplacian and the time derivative. Assume now that we use the relativistic relation between energy and momentum :
$$E^2=p^2 c^2+m^2 c^4$$
(this is how Louis de Broglie proceeded initially). By taking second-order derivatives of (1.62) both in time and in space variables we obtain the KleinGordon equation :
$$\frac{1}{c^2} \frac{\partial^2 \psi}{\partial t^2}-\nabla^2 \psi+\mu^2 \psi=0$$
where we have set $\mu=m c / \hbar$. Originally, this equation was found by Schrödinger. He abandoned it because it did not lead to the correct relativistic corrections for the levels of the hydrogen atom. ${ }^5$ It was rediscovered later on by Klein and Gordon.
Forgetting about the exact meaning of $\psi$ in this context, (1.63) is the propagation equation for a relativistic free particle of mass $m$. In the case $m=0$, i.e., the photon, we recover the propagation equation for the electromagnetic potentials :
$$\frac{1}{c^2} \frac{\partial^2 \psi}{\partial t^2}-\nabla^2 \psi=0$$

## 物理代写|核物理代考Nuclear Physics代写|From forces to interactions

We have emphasized that, in quantum field theory, forces between particles are described by the exchange of virtual particles. The interactions can be described by (Feynman) “diagrams” like those shown in Fig. 1.13. Each diagram corresponds to a scattering amplitude that can be calculated according to the rules of quantum field theory. As we will see in Chap. 3, the effective potential is Fourier transform of the amplitude.
In quantum electrodynamics, the exchange of massless photons leads to the Coulomb potential. The exchange of massive particles leads to Yukawalike potentials. One example is the exchange of pions as shown in the first two diagrams in Fig. 1.13. These diagrams contribute to the nucleon-nucleon potential of Fig. 1.12 and lead to the binding of nucleons and to nucleonnucleon scattering.
Other massive particles can be exchanged between particles, for example the $Z^0$ boson
$$m_{\mathrm{Z}} c^2=91.188 \mathrm{GeV} \quad \frac{\hbar c}{m_{\mathrm{Z}} c^2}=2.11 \times 10^{-3} \mathrm{fm} .$$
This particle is the mediator of the neutral current sector of the weak interactions as illustrated in the fourth diagram of Fig. 1.13. It thus leads to a Yukawa-like potential between particles of the form (1.56). Compared to pion exchange, the Yukawa potential has a range about $10^{-3}$ times smaller. The effective coupling is also much smaller being of order
$$g_Z \sim \alpha$$
To compare with the strong interactions, we estimate an effective value of $V_0 R^2$ by taking $R \sim 2 \times 10^{-3} \mathrm{fm}$ and $V_0 \sim \alpha \hbar c / R$ :
$$V_0 R^2 \sim 3 \times 10^{-3} \mathrm{MeV} \mathrm{fm}^2$$
i.e. about 5 orders of magnitude smaller than that of the strong interactions (1.52) and (1.53). We conclude that $\mathrm{Z}^0$ can play no role in nucleon-nucleon binding or scattering.
On the other hand, $\mathrm{Z}^0$ exchange plays the essential role in neutrinonucleon elastic scattering. This is because the neutrino has only weak interactions.

# 核物理代考

## 物理代写|核物理代考Nuclear Physics代写|Origin of the Yukawa potential

$$\psi=\exp (-i(E t-\boldsymbol{p} \cdot \boldsymbol{r}) / \hbar)$$

$$E^2=p^2 c^2+m^2 c^4$$
(这就是路易·德布罗意最初的处理方式)。通过对(1.62)在时间和空间变量上的二阶导数，我们得到kleinggordon方程:
$$\frac{1}{c^2} \frac{\partial^2 \psi}{\partial t^2}-\nabla^2 \psi+\mu^2 \psi=0$$

$$\frac{1}{c^2} \frac{\partial^2 \psi}{\partial t^2}-\nabla^2 \psi=0$$

## 物理代写|核物理代考Nuclear Physics代写|From forces to interactions

$$m_{\mathrm{Z}} c^2=91.188 \mathrm{GeV} \quad \frac{\hbar c}{m_{\mathrm{Z}} c^2}=2.11 \times 10^{-3} \mathrm{fm} .$$

$$g_Z \sim \alpha$$

$$V_0 R^2 \sim 3 \times 10^{-3} \mathrm{MeV} \mathrm{fm}^2$$

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