Scattering amplitude, link between quantum mechanics and QFT

• Kurret
In summary, the scattering amplitude f_k(\theta) in quantum mechanics can be defined as the coefficients of an outgoing spherical wave for two particles. This is obtained from the asymptotic behavior of the wave function of two scattering particles, interacting with a short range potential. However, in the low energy limit, this can also be computed in effective field theory using the effective Lagrangian L. The four point greens function can then be defined as \langle0|T\psi\psi\psi^\dagger\psi^\dagger|0\rangle and the scattering amplitude A can be obtained by subtracting the disconnected terms and factoring out an overall energy-momentum conserving delta function and propagators. This amplitude only depends on the
Kurret
In quantum mechanics, we can define the scattering amplitude $f_k(\theta)$ for two particles as the coefficients of an outgoing spherical wave. More precisely, the asymptotic behaviour (when $r\rightarrow\infty$) of a wave function of two scattering particles, interacting with some short range potential, is given by

$$\psi(r)=e^{ikz} + \frac{f_k(\theta)}{r}e^{ikr}$$

where the ingoing wave is the plane wave $e^{ikz}$. The full Hamiltonian is given by
$$H=\frac{1}{2m}p_1^2+\frac{1}{2m}p_2^2+V(r_{12})$$

The low energy limit can be obtained by expanding the scattering amplitude in partial waves and only include the lowest partial wave.

However, we can also compute this in effective field theory. In the low energy limit, the effective lagrangian is
$$L=\psi^\dagger\left(i\frac{\partial}{\partial t}+\frac{1}{2}\nabla^2\right)\psi-\frac{g_2}{4}(\psi^\dagger \psi)^2$$
We can then define the four point greens function as $\langle0|T\psi\psi\psi^\dagger\psi^\dagger|0\rangle$. We can then define the scattering amplitude A as, and I quote "It is obtained by subtracting the disconnected terms
that have the factored form $\langle0|T\psi\psi^\dagger|0\rangle\langle0|\psi\psi^\dagger|0\rangle$, Fourier transforming in all coordinates, factoring out an overall energy-momentum conserving delta function, and also factoring out propagators associated with each of the four external legs". For two particles, the amplitude A only depends on the total energy E. The claim is then that we have
$$f_k(\theta)=\frac{1}{8\pi}A(E=k^2)$$

My question is, although I understand that it is reasonable that there is a relation between these two quantities, I have no idea how to prove this and how to get the numerical factors right etc. So basically, what is the exact link between doing scattering computations in QM vs QFT? How can one show that the observables we are looking at is the same quantity?

The paper I am following is http://arxiv.org/abs/cond-mat/0410417 . The Effective field theory part I am referring starts at page 135 , especially the relation (295). The above quote of the definition of A is given on page 139.

1. What is scattering amplitude?

Scattering amplitude is a mathematical quantity that describes the probability of a particle interacting with other particles. It is used in quantum mechanics and quantum field theory to calculate the likelihood of particles colliding and scattering in a specific way.

2. How is quantum mechanics related to scattering amplitude?

Quantum mechanics is the theory that describes the behavior of particles at the microscopic level, while scattering amplitude is a key concept in this theory. It helps us understand and predict the behavior of particles during collisions and interactions, which is a fundamental aspect of quantum mechanics.

3. What is the link between scattering amplitude and quantum field theory (QFT)?

Quantum field theory is an extension of quantum mechanics that describes particles as excitations of underlying fields. Scattering amplitude is a crucial concept in QFT, as it allows us to calculate the probabilities of particle interactions and the creation and annihilation of particles in these fields.

4. How is scattering amplitude calculated in quantum mechanics and QFT?

In quantum mechanics, scattering amplitude is calculated using Feynman diagrams, which represent the different possible ways particles can interact. In QFT, it can be calculated using perturbation theory, where the amplitude is expressed as a series of terms that become more accurate as the number of terms increases.

5. What is the significance of scattering amplitude in physics?

Scattering amplitude is a fundamental concept in physics as it allows us to understand and predict the behavior of particles at the microscopic level. It is used in a variety of fields, from particle physics to condensed matter physics, and has played a crucial role in the development of our understanding of the fundamental laws of nature.

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