Two particle scattering

Quantum Mechanics A

Schrödinger Equation The most fundamental equation of quantum mechanics; given a Hamiltonian ${\mathcal {H}}$, it describes how a state $|\Psi \rangle$ evolves in time.

Physical Basis of Quantum Mechanics

Basic Concepts and Theory of Motion UV Catastrophe (Black-Body Radiation) Photoelectric Effect Stability of Matter Double Slit Experiment Stern-Gerlach Experiment The Principle of Complementarity The Correspondence Principle The Philosophy of Quantum Theory

Schrödinger Equation

Brief Derivation of Schrödinger Equation Relation Between the Wave Function and Probability Density Stationary States Heisenberg Uncertainty Principle Some Consequences of the Uncertainty Principle

Operators, Eigenfunctions, and Symmetry

Linear Vector Spaces and Operators Commutation Relations and Simultaneous Eigenvalues The Schrödinger Equation in Dirac Notation Transformations of Operators and Symmetry Time Evolution of Expectation Values and Ehrenfest's Theorem

Motion in One Dimension

One-Dimensional Bound States Oscillation Theorem The Dirac Delta Function Potential Scattering States, Transmission and Reflection Motion in a Periodic Potential Summary of One-Dimensional Systems

Discrete Eigenvalues and Bound States

Harmonic Oscillator Spectrum and Eigenstates Analytical Method for Solving the Simple Harmonic Oscillator Coherent States Charged Particles in an Electromagnetic Field WKB Approximation

Time Evolution and the Pictures of Quantum Mechanics

The Heisenberg Picture: Equations of Motion for Operators The Interaction Picture The Virial Theorem

Angular Momentum

Commutation Relations Angular Momentum as a Generator of Rotations in 3D Spherical Coordinates Eigenvalue Quantization Orbital Angular Momentum Eigenfunctions

Central Forces

General Formalism Free Particle in Spherical Coordinates Spherical Well Isotropic Harmonic Oscillator Hydrogen Atom WKB in Spherical Coordinates

The Path Integral Formulation of Quantum Mechanics

Feynman Path Integrals The Free-Particle Propagator Propagator for the Harmonic Oscillator

Continuous Eigenvalues and Collision Theory

Differential Cross Section and the Green's Function Formulation of Scattering Central Potential Scattering and Phase Shifts Coulomb Potential Scattering

Classically, if we wish to consider a collection of identical particles, say billiard balls, it is always possible to label all the balls such that we can follow a single ball throughout interacting with others. It is not, however, possible to attach such labels to a quantum mechanical systems of, say, electrons. Quantum mechanical particles are far to small to attach such physical labels and there are not enough degrees of freedom to label each particle differently. Again considering the classical case, one could imagine simply recording the position of a given particle throughout its trajectory to distinguish it from any other particle. Quantum mechanically, however, we again fail in following a single particles trajectory since each time we make a measurement of position we disturb the system of particles in some uncontrollable fashion. If the wave functions of the particles overlap at all, then the hope of following a single particles trajectory is lost. We now attempt to study the consequences of such indistinguishably between identical quantum particles.

Scattering of Identical Particles

Let's look at the case of two identical bosons (spin 0) from their center of mass frame. To describe the system, we must use a symmetrized wave function. Under the exchange $\mathbf {r} _{1}\leftrightarrow \mathbf {r} _{2}\!$, $\mathbf {r} _{cm}=(\mathbf {r} _{1}+\mathbf {r} _{2})/2\!$ is invariant while $\mathbf {r} =\mathbf {r} _{1}-\mathbf {r} _{2}\!$ changes sign. So the center of mass wave function is already symmetric. Furthermore, the wave function has even parity. This implies that the only possible eigenstates of angular momentum of the two particles are those with even angular momentum quantum numbers. This is evident from the property of the associated Legendre polynomials.

But we have to symmetrize $\psi (\mathbf {r} )\!$ by hand:

$Y_{lm}(-\mathbf {\hat {r}} )=(-1)^{l}Y_{lm}(\mathbf {\hat {r}} )\!$

Under the transformation:

$\mathbf {\hat {r}} \rightarrow -\mathbf {\hat {r}} ,\theta \rightarrow \pi -\theta ,\phi \rightarrow \phi +\pi$

The first two terms of the symmetrized wave function represent the incident waves corresponding to the center of mass frame. Note that because we are considering identical particles we cannot distinguish the target particle from the incident one. Thus, each particle has equal amplitude of being either one.

The scattering amplitude is:

$f_{sym}(\theta ,\phi )=f(\theta ,\phi )+f(\pi -\theta ,\phi +\pi )\!$

$\theta \!$and $\pi -\theta \!$ can then be associated with the angle through which each particle is scattered. The total amplitude for particles to emerge at each angle is then exactly the sum of amplitudes for emerging at each angle, which is given above. The scattering amplitude remains consistent with the fact that we have two identical particles, and this gives us the differential cross section:

${\frac {d\sigma }{d\Omega }}=|f(\theta ,\phi )+f(\pi -\theta ,\phi +\pi )|^{2}=|f(\theta ,\phi )|^{2}+|f(\pi -\theta ,\phi +\pi )|^{2}+2\Re e[f(\theta ,\phi )f^{*}(\pi -\theta ,\phi +\pi )]$

Note:

The first two terms in the differential cross section is what we would get if we had two distinguishable particles, while the third term give the quantum mechanical interference that goes along with identical particles.

As an example, consider scattering through a 90 degree angle. We then have:

$f(\theta )=f(\pi -\theta )=f\left({\frac {\pi }{2}}\right)$

Now if the particles are distinguishable, the cross section for observing a scattered particle at 90 degrees is then:

$\left({\frac {d\sigma }{d\Omega }}\right)_{dis}=2\left|f\left({\frac {\pi }{2}}\right)\right|^{2}$

Where if the particles are indistinguishable, we see above that we will have:

$\left({\frac {d\sigma }{d\Omega }}\right)_{ind}=4\left|f\left({\frac {\pi }{2}}\right)\right|^{2}$

Thus the differential cross-section is exactly twice the distinguishable case when the particles are indistinguishable.