I Schrodinger equation for N particles in a box

Kashmir
Messages
466
Reaction score
74
[Pathria, statistical mechanics][1], pg2 ,when discussing ##N## particles in a volume ##V##

"...there will be a large number of different
ways in which the total energy E of the system can be distributed among the N particles
constituting it. Each of these (different) ways specifies a microstate, or complexion, of the
given system"
"Each of these (different) ways specifies a microstate, or complexion, of the
given system. In general, the various microstates, or complexions, of a given system can
be identified with the independent solutions ##ψ(r_1,..., r_N )## of the Schrodinger equation of
the system, corresponding to the eigenvalue E of the relevant Hamiltonian" .
If we have N particles in a box, we solve the energy eigenvalue for the total system ##H|E>=E|E>## and find the eigenvectors.

Using linear combination of these and boundary conditions we find the general solution ##ψ(r_1,..., r_N )##.So how does "the various microstates, or complexions, of a given system can
be identified with the independent solutions ##ψ(r1,..., rN )## of the Schrodinger equation of
the system, corresponding to the eigenvalue E of the relevant Hamiltonian" enter the discussion? [1]: https://www.google.co.uk/books/edition/Statistical_Mechanics/PIk9sF9j2oUC?hl=en
 
Physics news on Phys.org
We should investigate whether N particle of same kind are bosons or fermions for their placement in energy states.
 
anuttarasammyak said:
We should investigate whether N particle of same kind are bosons or fermions for their placement in energy states.
The author didn't mention that.
 
Here is my attempt:
Suppose we have two non interacting particles in identical environment.

The individual Hamiltonian for both is ##H_1## and the total Hamiltonian is ##H=H_1 \otimes 1+ 1\otimes H_2##
Also if ##|\phi_1>## and ##|\phi_2>## are the eigenstates of ##H_1## with eigenvalues ##E_1##,## E_2## then the eigenstates of the system is ##|\phi_1> \otimes|\phi_2>## with eigenvalue ##E_1 + E_2##

One of the state of the system is then
##|\psi >= e^(-iE_1t/\hbar)|\phi_1>
\otimes e^(-iE_2t/\hbar) |\phi_2>##

This state corresponds to particle 1 in state ##|\phi_1>## having energy ##E_1## & particle 2 in ##|\phi_2>## with energy ##E_2## defining a microstate.

Is this correct?
 
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. Towards the end of the first lecture for the Qiskit Global Summer School 2025, Foundations of Quantum Mechanics, Olivia Lanes (Global Lead, Content and Education IBM) stated... Source: https://www.physicsforums.com/insights/quantum-entanglement-is-a-kinematic-fact-not-a-dynamical-effect/ by @RUTA
If we release an electron around a positively charged sphere, the initial state of electron is a linear combination of Hydrogen-like states. According to quantum mechanics, evolution of time would not change this initial state because the potential is time independent. However, classically we expect the electron to collide with the sphere. So, it seems that the quantum and classics predict different behaviours!
Back
Top