head on collision of two particles in schrodinger equation
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I hate to disappoint you, but it's really the same as Schrödinger’s equation for anything else.
The differences are just details, and can be generalised away. The fundamental Hamiltonian form just “works”.
How is that possible? Because Schrödinger's equation isn't at bottom about “particles”. All the details about “particle” behaviour are plugged into it, like information about ambient electrostatic fields and what-have-you, by way of the Hamiltonian it references.
Schrödinger's equation is a general statement about the time-evolution of a wide class of physical systems in terms of their space distribution, with a rather mind-stretching use of non-spatial phase information modelled as the complex arguments of the distribution. Nothing at all about the equation says what particles are in the system, in the same kind of way that Newton's theory of gravitation doesn't have anything to say about planets, except by being applied to them.
On the other hand, there isn't anything specially Schrödinger-ish about a Hamiltonian, or even particle-ish. Actually describing a system of particles is at least two removes away from Schrödinger's equation, and at least one remove from Hamiltonians.
Having said that, now look at
Two-Particle Systems
Here everything about the two particles is embedded in a nice simple Hamiltonian. This goes into the equation just as the Hamiltonian always does.
The differences are just details, and can be generalised away. The fundamental Hamiltonian form just “works”.
How is that possible? Because Schrödinger's equation isn't at bottom about “particles”. All the details about “particle” behaviour are plugged into it, like information about ambient electrostatic fields and what-have-you, by way of the Hamiltonian it references.
Schrödinger's equation is a general statement about the time-evolution of a wide class of physical systems in terms of their space distribution, with a rather mind-stretching use of non-spatial phase information modelled as the complex arguments of the distribution. Nothing at all about the equation says what particles are in the system, in the same kind of way that Newton's theory of gravitation doesn't have anything to say about planets, except by being applied to them.
On the other hand, there isn't anything specially Schrödinger-ish about a Hamiltonian, or even particle-ish. Actually describing a system of particles is at least two removes away from Schrödinger's equation, and at least one remove from Hamiltonians.
Having said that, now look at
Two-Particle Systems
Here everything about the two particles is embedded in a nice simple Hamiltonian. This goes into the equation just as the Hamiltonian always does.
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